"Science, Faculty of"@en .
"Physics and Astronomy, Department of"@en .
"DSpace"@en .
"UBCV"@en .
"Prat, Alain"@en .
"2015-03-23T21:37:43Z"@en .
"2015"@en .
"Doctor of Philosophy - PhD"@en .
"University of British Columbia"@en .
"This thesis investigates the merging of horizons which occurs when a black hole crosses a cosmological horizon. We study the simplest spacetime which has both a black hole and cosmological horizon, namely Schwarzschild-deSitter (SdS) spacetime. First we develop a new coordinate system for SdS spacetime, which allows us to properly illustrate and analyze the merging of horizons. We then use a combination of numerical and analytical methods to study the structure of the merging horizons, including the null generators which make up the horizon, as well as the presence of caustic points on the horizon. We find an analytical formula for the location in spacetime where the black hole and cosmological horizon first touch. Next we study the area of the horizons. Using numerical methods, we find several intriguing results regarding the behavior of horizon area on time, and in the limit of small black hole mass. The first result is that the time at which the black hole first touches the cosmological horizon is also the time at which the rate of horizon area increase is maximal. The second and third results concern the horizon area in the limit of small black hole mass. The second result is that in this limit, all of the increase in horizon area occurs prior to horizon merger. The third and final result is that in the limit of small black hole mass, the increase in horizon area can be thought of as being due in equal parts to two effects: to the joining of new generators not previously on the horizon, and the expansion of generators on the horizon for all times. The first and third results just mentioned are both corroborated using analytical techniques. Finally, we conclude by discussing how the study of merging horizons in this thesis is a valuable first step to undertaking a similar study of the horizons which occur in merging black hole binaries."@en .
"https://circle.library.ubc.ca/rest/handle/2429/52470?expand=metadata"@en .
"Merging Black Hole and CosmologicalHorizonsbyAlain PratB.Sc., The University of British Columbia, 2002M.Sc., The University of British Columbia, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)March 2015\u00C2\u00A9 Alain Prat 2015AbstractThis thesis investigates the merging of horizons which occurs when a black hole crosses a cos-mological horizon. We study the simplest spacetime which has both a black hole and cosmologicalhorizon, namely Schwarzschild-deSitter (SdS) spacetime. First we develop a new coordinate systemfor SdS spacetime, which allows us to properly illustrate and analyze the merging of horizons. Wethen use a combination of numerical and analytical methods to study the structure of the merginghorizons, including the null generators which make up the horizon, as well as the presence of causticpoints on the horizon. We find an analytical formula for the location in spacetime where the blackhole and cosmological horizon first touch. Next we study the area of the horizons. Using numericalmethods, we find several intriguing results regarding the behavior of horizon area on time, and in thelimit of small black hole mass. The first result is that the time at which the black hole first touchesthe cosmological horizon is also the time at which the rate of horizon area increase is maximal. Thesecond and third results concern the horizon area in the limit of small black hole mass. The secondresult is that in this limit, all of the increase in horizon area occurs prior to horizon merger. Thethird and final result is that in the limit of small black hole mass, the increase in horizon area can bethought of as being due in equal parts to two effects: to the joining of new generators not previouslyon the horizon, and the expansion of generators on the horizon for all times. The first and thirdresults just mentioned are both corroborated using analytical techniques. Finally, we conclude bydiscussing how the study of merging horizons in this thesis is a valuable first step to undertaking asimilar study of the horizons which occur in merging black hole binaries.iiPrefaceThis dissertation is original and independent work by the author, A. Prat. At the time of this writing,the results of this thesis have not been published.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1ivTable of Contents1.2 Merging black hole and cosmological horizons . . . . . . . . . . . . . . . . . . . 22 Preliminaries: Schwarzschild deSitter Spacetime . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Schwarzschild coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Radial timelike geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Horizons of stationary observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 r! \u00E2\u0080\u00A2 limit of SdS spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7 The dimensionless parameter e = HM . . . . . . . . . . . . . . . . . . . . . . . . 273 Lemaitre-Planar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 Construction of LP coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Metric components in LP coordinates . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Additional features of LP coordinates . . . . . . . . . . . . . . . . . . . . . . . . 43vTable of Contents4 Structure of Merging Black Hole and Cosmological Horizons . . . . . . . . . . . . . 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Null geodesic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Null geodesic calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4 Horizon shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.5 Merger point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105 Area of Merging Black Hole and Cosmological Horizons . . . . . . . . . . . . . . . 1325.1 Horizon area calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.3 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.1 Summary and discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.2 Applications and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . 176viTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189viiList of Figures2.1 Penrose diagram of Schwarzschild spacetime. . . . . . . . . . . . . . . . . . . . . 102.2 The Penrose diagram of the full spacetime manifold for SdS spacetime. . . . . . . 112.3 Penrose diagram of SdS spacetime with topological identification. . . . . . . . . . 122.4 Penrose diagram for deSitter spacetime. . . . . . . . . . . . . . . . . . . . . . . . 132.5 The hyperboloid for the 1+1 deSitter spacetime manifold. . . . . . . . . . . . . . . 142.6 The function f (r) in the SdS line element. . . . . . . . . . . . . . . . . . . . . . . 152.7 a) Penrose diagram of SdS spacetime, with the regions r < ra, ra < r < rb andr > rc indicated. b) Penrose diagram of SdS spacetime, with region covered bySchwarzschild coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.8 Effective potential diagram for radial timelike geodesic trajectories in SdS spacetime. 192.9 Penrose diagram with two trajectories shown: one stationary observer and one drift-ing observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23viiiList of Figures2.10 a) Penrose diagram with causal horizon (yellow lines) of an observer stationarynext to the black hole. b) Penrose diagram with causal horizon (yellow lines) of anobserver drifting away from the black hole. . . . . . . . . . . . . . . . . . . . . . 242.11 Penrose diagram showing the drifting observer, the causal horizon of this observerand the spacelike hypersurfaces of constant time. . . . . . . . . . . . . . . . . . . 253.1 The timelike (a) and spacelike (b) coordinate curves of LP coordinates, shown onthe Penrose diagram of SdS spacetime. . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Penrose diagram showing the region of SdS spacetime covered by Lemaitre coordi-nates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 a) Effective potential diagram for the timelike coordinate curves of the LP coordi-nate system. b) Effective potential diagram for the timelike coordinate curves of theLemaitre coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Curves illustrating the construction of LP coordinates. . . . . . . . . . . . . . . . . 364.1 The \u00E2\u0080\u009Ctrouser\u00E2\u0080\u009D shaped horizon of two merging black holes. . . . . . . . . . . . . . 584.2 a) The family of radial geodesic observers drifting away from r = re. b) Illustrationof the causal horizons (yellow) for three radial geodesic observers. . . . . . . . . . 604.3 The effective potential diagram for null geodesics in SdS spacetime. . . . . . . . . 644.4 Six frames showing the progression from black hole and cosmological horizons atearly times to a single cosmological horizon at late times. . . . . . . . . . . . . . . 85ixList of Figures4.5 3d plot of the merging of black hole and cosmological horizons. . . . . . . . . . . 894.6 Effective potential diagram with schematic illustration of the possible behaviors ofnull generators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.7 Three frames showing the merging of horizons, with a color scheme to indicate theorigin of each null generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.8 Comparison of the series and numerical solutions to the null geodesic equation. . . 1044.9 Plot showing the convergence of the series solution of the null geodesic equation. . 1085.1 Sample curve of normalized horizon area vs LP coordinate time. . . . . . . . . . . 1395.2 Examples plots of first and second derivative of area vs time for merging black holeand cosmological horizons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.3 a) Example plot of horizon area vs time for a small black hole. b) Plot of nondi-mensional horizon area increase at merger time for small values of the ratio ofSchwarzschild to Hubble radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.4 Plot of the maximum nondimensionalized rate of area increase for various values ofthe ratio of Schwarzschild to Hubble radius. . . . . . . . . . . . . . . . . . . . . . 1485.5 Plot of the nondimensional time interval HDT =H(Tmerger\u0000Thal f ) for several smallvalues of the paramter e\u00CB\u0086 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150xList of Figures5.6 Ratio of area increase due to existing generators and area increase due to new gen-erators, for several values of the parameter e\u00CB\u0086 . . . . . . . . . . . . . . . . . . . . . 153xiAcknowledgementsI want to thank my supervisor, family, colleagues and friends for creating the kind of environmentthat made the research in this thesis possible and enjoyable. First and foremost, my supervisor\u00E2\u0080\u0099spositive attitude and affable nature helped me persevere through the more difficult moments ofmy doctoral work. My parents\u00E2\u0080\u0099 support and encouragement was also of great help. Finally, mycolleagues and friends in the department helped create a positive research environment that made theresearch all the more enjoyable. This work was partly supported by an NSERC PGS D scholarship.xiiDedicationThis thesis is dedicated to my late uncle Fr\u00C3\u00A9d\u00C3\u00A9ric Prat, whose passion for science was inspirational.xiiiChapter 1Introduction1.1 OverviewThis thesis investigates the merger of horizons that occurs when a black hole crosses an observer\u00E2\u0080\u0099scosmological horizon. Our main motivation is to use a simple spacetime with merging horizonsas a mathematical laboratory where we can investigate various questions about the mathematicsof the merging horizons. This thesis is broadly organized as follows: in section 1.2 we introducethe concept of a cosmological horizon, and discuss the motivation for considering the merger ofa black hole with a cosmological horizon. In chapter 2 we review some mathematical propertiesof the spacetime (Schwarzschild deSitter spacetime; SdS spacetime from now on) which will beconsidered in this thesis and give a more precise fomulation of the problem statement consideredhere. In chapter 3 we introduce a new coordinate system for SdS spacetime, developed specificallyfor the purposes of this thesis. In chapters 4 and 5, we present the main results of this thesis: ananalysis of the structure and area of the horizons that result from a merger of a black hole with acosmological horizon. These results are obtained using both numerical and analytical methods. Inchapter 6, we use the main results from the previous chapters and discuss how they could be relatedto the merging of horizons that occurs in binary black holes. As well, we summarize the results anddiscuss avenues for future research.11.2. Merging black hole and cosmological horizons1.2 Merging black hole and cosmological horizonsThe cosmological horizon The measured acceleration of the universe, as inferred from supernovadata [24, 26], indicates that either our theory of gravity on large scales is in serious need of repair,or that the universe contains a mysterious fluid, often called dark energy, which is driving thisaccelerated expansion. The revision of cosmological models to include dark energy, typically in theform of vacuum energy or a cosmological constant, has led to striking revisions of our understandingof the universe. For example, constraints on cosmological parameters strongly suggest continuedexpansion of the universe [27], even if the energy density of the universe is greater than the criticaldensity required to make it flat. Another striking feature of a universe with accelerated expansionis the presence of a cosmological horizon1, which in a perfectly homogeneous universe is a sphere2surrounding any observer in spacetime such that beyond this sphere, no information can ever reachthe observer.The cosmological horizon is not to be confused with the particle horizon. The particle horizonseparates objects that are sufficiently close (as measured using comoving distance) that the lightthey emitted in the past has had time to reach us, so that we may currently observe them. Theparticle horizon limits what we can currently observe; beyond the current particle horizon, distantobjects have not yet come into view, so to speak. The cosmological horizon, on the other hand, limitswhich part of the universe will ever be accessible to a hypothetical eternal observer. Currently thereare distant objects (galaxies and so on) so far that the light they emit will never reach us, no matterhow long we would be willing to wait (see pages 128-129 of [15] for a careful discussion of particleand cosmological horizons for deSitter spacetime).This inability to receive light from objects beyond the cosmological horizon is often described interms of distant objects travelling faster than light. However, this \u00E2\u0080\u009Cfaster than light\u00E2\u0080\u009D description is1For FRW universes accelerated expansion implies a cosmological horizon; for more general cosmological modelsone can find exceptions ( See [21].)2In inhomogeneous cosmological models the cosmological horizon can of course be nonspherical. However thetopology of the cosmological horizon in these cases is nonetheless spherical.21.2. Merging black hole and cosmological horizonsincorrect for a number of reasons. First, the proper distance to distant objects is an inherently am-biguous concept since it depends on the choice of the time coordinate. Second, even in cases whereone can define an unambiguous proper distance, such as in homogeneous FRW cosmological mod-els, only in pure deSitter spacetime does the faster than light limit coincide with the cosmologicalhorizon (see [5] for a discussion of this and other misconceptions).The cosmological horizon is better understood in terms of stretching of space between an observerand distant objects. In cosmological models where the expansion of the universe is accelerating,the distance between an observer and a distant object may grow so quickly that even light will nottraverse this great distance, even after an eternity. Another equally valid way of describing thecosmological horizon is in terms of infinite redshift. Currently there are objects whose light, whenit finally reaches us in the far future, will have been stretched by the expansion of the universe intolight of unfathomably long wavelengths. As one moves further away from these objects, one findsa point where objects would suffer an infinite redshift, and the light emitted by them would take aninfinite amount of time to reach us. These objects lie at the boundary of our current cosmologicalhorizon (see page 129 of [15] for a discussion of this infinite redshift in the context of deSitterspacetime).Although one can only speak of objects being inside or outside the cosmological horizon at any onetime, it is possible for an object to cross the cosmological horizon. For example, in a universe witha cosmological constant and a single object of negligible mass, we would essentially have deSitterspacetime. Thus the cosmological horizon would be at the same proper distance from the observerat all times, but the proper distance between the object and the observer would be increasing dueto the expansion of the universe. Consequently, there would be a critical time where the objectwould cross the observer\u00E2\u0080\u0099s cosmological horizon (page 129, [15]). On can similarly deduce thatin a universe with a cosmological constant, an observer and a small black hole would move awayfrom each other such that the black hole would effectively cross the cosmological horizon, and itshorizon would merge with the cosmological horizon. The study of this merging of horizons and itsconsequences is the subject of this thesis.31.2. Merging black hole and cosmological horizonsThe inability to access information beyond the cosmological horizon is reminiscent of the inabilityto access information inside a black hole (see page 300 of [32] for the definition of a black hole). Ina universe with a cosmological horizon, the observable universe is separated from a possibly muchlarger universe by two types of horizons: the black hole horizons and the cosmological horizon.An important distinction between these two types of horizons is that the cosmological horizon isobserver dependent, whereas the black hole horizons are thought of as observer independent. Allobservers outside a black hole should agree on the location of the black hole horizon, regardlessof their location and trajectory in spacetime. On the other hand, each observer in a universe withaccelerated expansion has a cosmological horizon centered on their location. We will come back tothis distinction in section 2.4, where we will explain how to incorporate both types of horizons intoa single definition of horizon. This will be crucial for exploring the merger of the two horizons.Motivation Here we seek to address to address a number of questions about the horizon whichresults when a black hole crosses and merges with a cosmological horizon. This could be, forexample, a supermassive black hole in the center of a distant galaxy, merging with what is currentlyour cosmological horizon. There are several motivations for considering such a merger. Our primarymotivation for considering this merger is the similarity between it and the merger of horizons thatoccurs during a head on collision in a binary black hole merger (see [23] for a thorough numericalexploration of a head on binary black hole merger).By studying the merger of black hole and cosmological horizons, we create a mathematical labora-tory where we can investigate the structure and area of the horizons during the merger. The resultsobtained in this study then lead to natural questions about the merger of binary black hole horizons.An alternative approach to studying the structure and area of merging horizons is to use perturbationtheory to construct the spacetime of an extreme mass-ratio binary black hole system (as in [14]), orto use a Rindler horizon approximation for the larger black hole horizon, as in [13].The analysis of the structure and the area of the cosmological and black hole horizons are in chapters4 and 5, respectively. Our analysis of the structure of the horizons will focus on the location and41.2. Merging black hole and cosmological horizonsstructure of the caustic, as well as the structure of the horizon at late times. Our analysis of thehorizon area can be seperated into coordinate dependent and independent results. The coordinateindependent results include an analysis of the relative importance of different horizon generatorsto the final horizon area, as well as a characterization of the time of maximal area increase. Thecoordinate dependent results will include a quantitative analysis of the horizon area in the extrememass ratio limit. Although these latter results are coordinate dependent, in the sense that the choiceof time coordinate can affect the area, we will argue that we can make general statements about thequalitative behavior of the horizon area in the extreme mass ratio limit which should hold regardlessof the coordinate system.Although our main motivation is the application of our results to the horizons of extreme mass ratiobinary black holes, there are other applications of our results as well. In the context of cosmology,examining the combined effect of both black hole and cosmological horizons allows us to give amore precise answer to the question: which part of the universe is in principle observable? Althoughboth black holes and the cosmological horizon independently shroud parts of the universe from viewand influence, a full understanding requires combining these two types of horizons.A full understanding of the observable part of the universe is also related to the paradoxes (orperceived paradoxes) created by the presence of horizons. For example, many of the questionsrelated to information loss in the context of black hole horizons have recently been extended tocosmological horizons as well (see for example, [10]). By considering a case where black hole andcosmological horizons meet, we are examining the horizons for a spacetime where these two sets ofquestions overlap.Still in the context of cosmology, calculations of the total entropy of the observable universe rely onestimating the area of the cosmological horizon, as well as the area of the black hole horizons insidethe cosmological horizon (see [7] for an example of such a calculation). In addition, calculations ofthe rate of change of entropy of the observable universe requires knowledge of the rate of changeof horizon area as black holes merge with the cosmological horizon (see [31] for an example ofcalculating the rate of change of entropy for FRW cosmologies). Both of these types of calculations51.2. Merging black hole and cosmological horizonshave so far ignored the corrections due to distortions in the shapes of black hole and cosmologi-cal horizons as they approach one another. Here we provide the first step towards including suchcorrections.Another motivation for the work in this thesis is the possibility of using a simple analytical spacetimewith merging horizons as a useful analytical testbed for the numerical event horizon solvers usedin binary black hole simulations. To our knowledge, the only other analytically known spacetimewhich has been shown to contain event horizon mergers is the exact solution known as the Kastor-Traschen solution [19]. This solution has been used as a test bed for a numerical event horizonsolver in [4].Our work is also related to the broader question of the topological transition in event horizons. Eversince the discovery of black string instability and pinch off in five dimensional spacetime [12], therehas been interest in topological transitions of event horizons in higher dimensional spacetimes. Forexample, Emparan and Hassad [9] have studied the self-similar geometry at the intersection of ablack hole and cosmological horizon merger for spacetimes with dimension greater than six. Itshould be noted that this merger is fundamentally different from the one studied in this thesis, sinceit occurs due to a changing parameter in the spacetime, as opposed to being due to the movement ofan observer.We address our basic questions regarding the merger of horizons by considering the simplest cosmo-logical spacetime which has both a black hole and a cosmological horizon: Schwarzschild deSitter(SdS) spacetime. This choice of spacetime is motivated not only by the fact that the metric is knownanalytically, but also by the fact that it is a good approximation to the late time spacetime structureof our own universe, according to the L-CDM model.6Chapter 2Preliminaries: Schwarzschild deSitterSpacetime2.1 IntroductionIn 1916, Karl Schwarzschild published the first analytical solution [29] to Einstein\u00E2\u0080\u0099s newly devel-oped field equations of gravitation [8]. This solution for the spacetime metric is the now famousSchwarzschild metric, which represents the vacuum spacetime geometry outside any sphericallysymmetric body. If the entire spacetime is considered to be devoid of matter, so that the energy-momentum tensor vanishes, and one uses so-called Schwarzschild coordinates to represent the met-ric, one obtains the well known line elementds2 = \u00E2\u009C\u00931\u0000 2Mr \u00E2\u0097\u0086dt2\u0000\u00E2\u009C\u00931\u0000 2Mr \u00E2\u0097\u0086\u00001 dr2\u0000 r2(sin2 qdf 2 +dq 2) , (2.1)where 0 < r < \u00E2\u0080\u00A2. In the above line element, there is a coordinate singularity at r = 2M and acurvature singularity at r = 0. In the decades that followed, it was gradually understood that thereis an event horizon at r = 2M, and that Schwarzschild spacetime corresponds to the simplest blackhole geometry one can imagine: a static non-rotating black hole in a vacuum spacetime. It was alsounderstood that the curvature singularity at r = 0 represent a breakdown of the classical descriptionof spacetime geometry.72.1. IntroductionAnother historically important exact solution to Einstein\u00E2\u0080\u0099s equation is the so-called deSitter space-time, first discovered by Willem deSitter [6]. Mathematically, the deSitter spacetime is the maxi-mally symmetric vacuum solution of Einstein\u00E2\u0080\u0099s equation with a cosmological constant. One com-monly used coordinate system is the so-called static coordinates of deSitter spacetime, in which theline element takes the formds2 = (1\u0000Hr)dt2\u0000 (1\u0000Hr)\u00001 dr2\u0000 r2(sin2 qdf 2 +dq 2) , (2.2)where H =pL/3 is the Hubble constant and 0 < r < \u00E2\u0080\u00A2. Provided that one ignores the coordinatesingularity at r = 1/H in the above, the static coordinates can be said to cover the whole of thedeSitter manifold. The physical interpretation of deSitter spacetime is obtained by consideringhomogeneous spacelike hypersurface slicings of the deSitter manifold. This results in an FRWcosmological model for a universe devoid of matter, but with a cosmological constant driving theexpansion of space. One can obtain different cosmologies depending on the choice of spatial slices.For example, if one uses the so-called \u00E2\u0080\u009Cclosed\u00E2\u0080\u009D slicing of deSitter spacetime, one can interpretthe full deSitter manifold as a so-called \u00E2\u0080\u009Cbig bounce\u00E2\u0080\u009D universe. This is an FRW cosmology withspatially homogeneous constant time slices of spherical geometry, and with a scale factor that firstshrinks to a minimum value and then grows indefinitely. It is from this behavior of the scale factorthat the big bounce universe gets its curious name (the name is in keeping with the names \u00E2\u0080\u009Cbig bang\u00E2\u0080\u009Dand \u00E2\u0080\u009Cbig crunch\u00E2\u0080\u009D). Another cosmological model is the so-called deSitter universe or steady-stateuniverse. This is also an FRW cosmology, where the spatial geometry of the constant time slicesis flat and Euclidean and the scale factor increases exponentially in time. In the so-called planarcoordinates of the deSitter universe, the line element takes the formds2 = dt2\u0000 e2Ht \u0000dx2 +dy2 +dz2\u0000 , (2.3)where t is the cosmic time and (x,y,z) are the Euclidean coordinates on the flat slices. It is this lattercosmological model that will be of interest to us in this thesis. As discussed below, we will considera part of Schwarzschild deSitter spacetime which can be interpreted as a black hole embedded in adeSitter universe.82.1. IntroductionSchwarzschild deSitter (SdS) spacetime is a generalization of both Schwarzschild and deSitterspacetime, and encompasses both as special cases. It can be thought of as the generalization ofSchwarzschild spacetime obtained when one allows for a positive cosmological constant in Ein-stein\u00E2\u0080\u0099s equations. Thus it can be interpreted as the spacetime geometry of the simplest black hole inthe presence of the cosmological constant. Like Schwarzschild spacetime, it is a static and spher-ically symmetric spacetime. A well-known but nevertheless remarkable fact about Schwarzschildspacetime that it is the unique vacuum spherically symmetric spacetime, as encapsulated by Birkhoff\u00E2\u0080\u0099stheorem [3]. It is possible to generalize this theorem to SdS spacetime and include the cosmologicalconstant (see [28] for a proof of the theorem). The static and spherically symmetric nature of SdSspacetime is captured by its four Killing vectors, one of which is associated with time and three ofwhich are associated with spherical symmetry. The two parameters characterizing SdS spacetimeare the mass M of the black hole and the value L of the cosmological constant, with SdS space-time reducing to Schwarzschild spacetime for M = 0 and deSitter spacetime for L = 0. As will beshown in section 2.7, these two parameters can be combined into a single dimensionless parametere = MpL/3 in such a way that, up to constant conformal rescaling, the spacetime only depends onthe single parameter e . Thus the essential character of the spacetime geometry only depends on asingle parameter, and without loss of generality we only need to consider the effect of changing theparameter e on the spacetime structure.SdS spacetime is rarely used in an astrophysical context due to the fact that the corrections to theSchwarzschild or Kerr metric due to the cosmological constant are negligible for most observation-ally relevant astrophysical phenomena (however, see [30] for an example of an application). HereSdS spacetime is used as the simplest example of a spacetime with both a black hole and cosmo-logical horizon. Although SdS spacetime is rarely used in an astrophysical context, it is interestingto point out that according to the L-CDM model of cosmology, SdS spacetime is relevant to thelate time spacetime structure of our universe. The cosmological constant is becoming the dominantcomponent of energy density in the universe, and localized matter structures are slowly approachingincreasingly isolated supermassive black holes. In the far future, an observer in outer space wouldpresumably find spacetime structure to be approximately that of a single isolated black hole in auniverse with a cosmological constant; that is, Schwarzschild deSitter spacetime (or more precisely,92.1. IntroductionFigure 2.1: Penrose diagram of Scharzschild spacetime. The spacetime can be thought of as con-sisting of two regions: a black hole region and a white hole region. The line r = 2M with negativeslope separates these two regions.Kerr deSitter spacetime if one takes into account rotation of the black hole).The causal structure of the full SdS spacetime manifold has many subtleties, some of which can bereadily understood by considering the Penrose diagram of the spacetime (see [20] for a more de-tailed discussion of the Penrose diagram of SdS spacetime). In everything that follows, we restrictourselves to values of L and M such that 9LM2 < 1. In this case the Penrose diagram is as shownin figure 2.2. Some of the features of this Penrose diagram can be understood by first recalling thePenrose diagram of Schwarzschild spacetime, shown in figure 2.1. Like Schwarzschild spacetime,SdS spacetime has both black hole and white hole regions, which are time reverses of each other.In contrast to Schwarzschild spacetime, however, SdS spacetime can be viewed as having an infi-nite series of alternating black hole and white hole regions, as shown in figure 2.2. These can bethought of as distinct black and white holes, each existing in causally separated parallel universes.More commonly however, one constructs SdS spacetime using the topological identification processshown in figure 2.3 (the resulting spacetime is what is normally called SdS spacetime; the case withan infinite number of black holes is rarely discussed). When the identification process in figure 2.3is used, the spacelike hypersurfaces of constant time can be taken to have spherical topology, as inthe big bounce universe of deSitter spacetime discussed previously. The topological identificationalso means that there is only one black hole and one white hole region. On the other hand, when SdS102.1. IntroductionFigure 2.2: The Penrose diagram of the full spacetime manifold for SdS spacetime. There are aninfinite number of black hole and white hole regions continuing indefinitely in both directions. Theblack hole regions can be thought of as black holes embedded in a steady state universe, and thewhite hole regions are simply the time reverse of the black hole regions.spacetime is viewed as having an infinite number of black holes, one usually thinks of the spacelikeslices of constant time as non-compact and with infinite spatial extent. These considerations of theglobal topology of SdS will not be important for our purposes since we will be limiting ourselves toa part of the spacetime containing a single black hole (such as one of the black hole regions shownin 2.2).We can further understand the causal structure of SdS spacetime by recalling the Penrose diagramof deSitter spacetime, shown in figure 2.4. As with SdS spacetime, there is an identification proce-dure which results in a spacetime where the spacelike hypersurfaces can be chosen to have sphericaltopology. Furthermore, with an appropriate choice of slicing the spacelike hypersurfaces are perfectthree spheres, and we obtain the \u00E2\u0080\u009Cbig bounce\u00E2\u0080\u009D cosmological model discussed previously. Alterna-tively, one can consider spacelike slices covering only the upper half of deSitter spacetime, alsoshown in figure 2.4. In this case, we have an FRW cosmology with flat spacelike hypersurfacesand a scale factor growing exponentially in time. The resulting cosmological model is sometimescalled the deSitter universe, as discussed above, although it now often also goes under the nameof the steady state universe. The reason for this latter name is that the deSitter universe obeys theso-called \u00E2\u0080\u009Cperfect cosmological principle\u00E2\u0080\u009D, which extends the usual \u00E2\u0080\u009Ccosmological principle\u00E2\u0080\u009D byrequiring that not only all vantage points in space be equivalent, but all vantage points in spacetimebe equivalent.In addition to the use of a Penrose diagram, one can also represent deSitter spacetime as a hy-perboloid embedded in three dimensional Minkowski spacetime, as shown in figure 2.5. The hy-112.1. IntroductionFigure 2.3: Penrose diagram of SdS spacetime. The identification procedure involves gluing thetwo lines of crosses above. The resulting spacetime has spacelike slices of constant time whichhave spherical topology.perboloid results from suppressing two of the spatial dimensions of deSitter spacetime. It nicelyillustrates the slices used in both the big bounce and steady state cosmology of deSitter spacetime(see [18] for similar diagrams constructed for SdS spacetime).As with deSitter spacetime, one can consider only half of SdS spacetime (this is the region labelled\u00E2\u0080\u009CSchwarzschild coordinates region\u00E2\u0080\u009D in figure 2.7). In the limit M = 0 this part of SdS spacetimereduces to the steady state universe previously discussed, and in the limit L = 0 it reduces to theblack hole half of Schwarzschild spacetime. Putting these two limits together, we interpret theregion of SdS spacetime in figure 2.7 as a black hole embedded in a steady-state universe. It iswell known that in the steady-state universe there is a cosmological horizon surrounding each freelyfloating observer, and that the proper distance between such observers increases exponentially withtime. Based on this we expect the black hole embedded in the steady-state universe to eventuallymerge with the cosmological horizon of a freely falling observer drifting away from the black hole.122.1. IntroductionFigure 2.4: Penrose diagram for deSitter spacetime. The diagram on the left shows the full deSittermanifold, where it is understood that the left and right vertical edges are to be glued through anidentification process. The spacelike hypersurfaces of constant time (the horizontal lines) are threespheres. In the diagram on the right, only the upper half of the deSitter manifold is considered. Herethe spacelike hypersurfaces are three dimensional flat Euclidean space. In terms of the cosmology,the situtation on the left is the big bounce cosmology and on the right we have the steady statecosmology. Figure from \u00C2\u00A9 [15], page 127, by permission from publisher.The coordinates used to cover this part of the spacetime will be developed in chapter 3. The space-like hypersurfaces of these coordinates reduce to the flat spacelike hypersurfaces of deSitter space-time in the limit that M = 0 (as shown in figures 2.4 and 2.5). In some sense, they can be thought ofas a generalization of what has traditionally been called the planar coordinates of deSitter spacetime.In the limit M = 0, these coordinates are closely related to the well known Lemaitre coordinates [22](this is discussed in section 3.3.2). These coordinates will have several advantages over traditionalcoordinate systems such as the Schwarzschild coordinates (section 2.2). For example, they will befree of coordinate singularities and will have geodesic timelike coordinate curves.In section 2.2 we review the Schwarzschild coordinates for SdS spacetime and discuss some addi-tional features of the spacetime which will be relevant in later sections. In section 2.3 we find theequations for the Schwarzschild coordinates of radial timelike geodesics. These equations will beimportant for the coordinate system developed in chapter 3. In section 2.4 we discuss the horizonsof observers stationary next to the black hole. Traditionally, it is these stationary Killing horizons132.1. IntroductionFigure 2.5: The hyperboloid on the left is the full deSitter manifold. The slices of constant time arecircles. These shrink to a minimum size and then grow again, giving rise to the so-called big bouncecosmology. The diagram on the left shows a set of constant time slices which only cover half of thehyperboloid. These are the constant time slices of the steady state universe. They are spatially flatand grow exponentially. Figure from \u00C2\u00A9 [15], page 125, by permission from publisher.142.2. Schwarzschild coordinatesFigure 2.6: Function f (r) = 1\u00002M/r\u0000H2r2.which are called the horizons of SdS spacetime. Understanding these stationary horizons is an im-portant starting point before undertaking the study of merging horizons in SdS spacetime. In section2.5 we give a more precise formulation of the problem statement considered in this thesis, althoughthe full formulation of the problem will not come until chapter 4. In section 2.7 we define a di-mensionless parameter e which will play an important role in calculations throughout this thesis.Finally, in section 2.6 we discuss how the r ! \u00E2\u0080\u00A2 limit of SdS spacetime can be well approximatedby deSitter spacetime (in a sense that we will make more precise). This fact about SdS spacetimewill be important when we set up the equations for the null geodesic generators of the merginghorizons in chapter 4.152.2. Schwarzschild coordinates2.2 Schwarzschild coordinatesSdS spacetime is traditionally presented in Schwarzschild coordinates, where the line element isds2 = f (r)dt2\u0000 f (r)\u00001dr2\u0000 r2(sin2 qdf 2 +dq 2) , (2.4)f (r) = 1\u0000 2Mr \u0000 L3 r2 . (2.5)The Schwarzschild coordinates of SdS spacetime encompass as special cases both the Schwarzschildcoordinates of Schwarzschild spacetime (eq. 2.1) and the static coordinates of deSitter spacetime(eq. 2.2). Throughout this thesis, we will use units such that G = c = 1 and take the metric sig-nature to be (+,\u0000,\u0000,\u0000). In the above, the 2-surfaces (t,r) = constant are spheres of area 4pr2,with q and f the polar and azimuthal angles on these spheres, respectively. The spheres r = rb andr = rc can be interpreted as the location of black hole and cosmological horizons, respectively, fora set of static observers. This will be discussed in more detail in section 2.4. The Schwarzschildcoordinates can be used to cover either the black hole or white hole part of SdS spacetime (i.e. anyof the triangular regions in figure 2.2). Figure 2.7b illustrates how the Schwarzschild coordinateswould cover the black hole region of SdS spacetime. This region of spacetime can be thought of asconsisting of three parts: the interior of a black hole for r < rb, an external expanding universe forr > rc, and an intermediate region rb < r < rc. The regions r < rb, rb < r < rc and r > rc are shownon the Penrose diagram in figure 2.7a. The coordinate t is timelike in the region rb < r < rc, andspacelike in the regions r < rb and r > rc. In the intermediate region rb < r < rc observers can inprinciple remain stationary with (r(t),f(t),q(t)) = constant, neither caught up in the expansionof the universe nor irrevocably swallowed by the black hole. Such observers would in general havea proper acceleration, except at a critical equilibrium radius re 2 (rb,rc), where it is possible to havea stationary observer outside the black hole whose worldline is a timelike geodesic. This can beseen by looking at the effective potential for timelike radial geodesics, as shown in figure 2.8 (theeffective potential is found in section 2.3). This unstable equilibrium can be thought of as resultingfrom the balance of the expansion pulling the observer to larger r and the gravitational pull of theblack hole pulling the observer to smaller r. The timelike trajectory r(t) = re is shown in figure 2.9.The equilibrium radius re will play an important role in the coordinate system we will develop in162.3. Radial timelike geodesicschapter 3. Its value can be calculated as re = 1H (MH) 13 , (2.6)where H =pL/3. H can be interpreted as the Hubble constant in the case of pure deSitter space-time. Another value which we be needed in many calculations further on in this thesis is f (re).From (2.4) and (2.6), we have f (re) = 1\u00003(MH)2/3 . (2.7)2.3 Radial timelike geodesicsLet g(t)= (t(t),r(t),0,p/2) be the Schwarzschild coordinates of an arbitary timelike radial geodesic.SdS spacetime is static and has the Killing vectorx (t) = (1,0,0,0) .This leads to the following conservation law for g(t):hdgdt ,x (t)i= E = constant .That is: dtdt = Ef (r) , (2.8)where (2.4) has been used. Another equation for g(t) comes from the requirement that t be propertime:hdgdt , dgdt i= 1 .172.3. Radial timelike geodesics(a)(b)Figure 2.7: a) Penrose diagram of SdS spacetime with the regions r < ra, ra < r < rb and r > rcindicated. b) The black hole portion of SdS spacetime covered by Schwarzschild coordinates. Thisis the region of the spacetime that will be considered in this thesis.182.3. Radial timelike geodesicsFigure 2.8: Effective potential diagram for radial timelike geodesic trajectories in SdS spacetime.There is an unstable equilibrium at r= re. The black line represent the motion of a geodesic observermoving away from the equilibrium location.This gives f (r)\u00E2\u009C\u0093 dtdt\u00E2\u0097\u00862\u0000 f (r)\u00001\u00E2\u009C\u0093 drdt\u00E2\u0097\u00862 = 1 . (2.9)Substituting (2.8) into the above gives the equation for r(t):E2\u0000\u00E2\u009C\u0093 drdt\u00E2\u0097\u00862 = f (r) .The above equation can be recast in effective potential form as\u00E2\u009C\u0093 drdt\u00E2\u0097\u00862 +Veff(r) = E2 ,where Veff(r) = f (r) .192.4. Horizons of stationary observersA plot of the effective potential Veff(r) is shown in figure 2.8. We see that there is an unstableequilibrium point at r = re. If we impose the requirementr(t =\u0000\u00E2\u0080\u00A2) = re (2.10)on a trajectory, this leads to the following value for E:E = f (re)1/2 .Substituting the above into (2.8) and (2.9) leads to the following equations for t(t) and r(t):dtdt = f (re)1/2f (r) , (2.11)drdt = \u00C2\u00B1( f (re)\u0000 f (r))1/2 , (2.12)where the + sign is for trajectories drifting towards larger values of r and the \u0000 sign is for tra-jectories drifting towards r = 0. There is also a trajectory with r(t) = re for all t . The family oftrajectories satisfying the above equations will play an important role in the coordinate system devel-oped in chapter 3. As well, the horizons considered in this thesis will be those an observer driftingaway from the equilbrium (as shown in figures 2.8 and 2.9), whose coordinates are described by asolution of the above equation (these horizons will be discussed more in section 2.5).2.4 Horizons of stationary observersConsider the Penrose diagram of SdS spacetime, as shown in figure 2.7a. Notice first from thisdiagram that unlike Schwarzschild spacetime, there is no clear distinction between future null in-finity and future timelike infinity, and instead the two combine into what could be called futurenull/timelike infinity (see section 5.2 of [15] for a discussion of future null/timelike infinity fordeSitter spacetime). Because of this, it is not possible to use the usual definition of a black holeevent horizon as the boundary of the causal past of future null infinity (see section 12.1 of [32] for202.4. Horizons of stationary observersa detailed definition of a black hole event horizon). Instead, we need an alternative definition ofevent horizon. One possibility is to define the black hole horizon as the boundary of the causal pastof future null/timelike infinity. However, since we are ultimately interested in describing the partof the universe that is accessible to a specific observer, we will use the concept of a causalhorizon,which is based on the causal past of a timelike trajectory. The causalhorizon of an observer (i.e.timelike trajectory) is defined as the boundary of the causal past of the observer\u00E2\u0080\u0099s trajectory. For-mally, if g(t) is the observer\u00E2\u0080\u0099s trajectory, and g(R) is the image of the real line under the mappingthat is g(t), then the causal horizon of g(t) is Bd(J\u0000(g(R))). The terminology \u00E2\u0080\u009Ccausal horizon\u00E2\u0080\u009Dmay not be familiar to some readers, and has not yet gained widespread and popular usage. Forexample, some authors prefer to use the words \u00E2\u0080\u009Ccosmological horizon\u00E2\u0080\u009D in the cosmological context,and \u00E2\u0080\u009CRindler horizon\u00E2\u0080\u009D in the context of accelerated observers, even though these are both observerdependent horizons which can be subsumed under the broader concept of causal horizons. Here wefollow the terminology in [17] by using the words \u00E2\u0080\u009Ccausal horizon\u00E2\u0080\u009D.The causal horizon can naturally incorporate both the black hole and cosmological horizons. Forexample, for an observer stationary outside a Schwarzschild black hole, the causal horizon of thisobserver would coincide with what we normally think of as the black hole horizon; the sphericalsurface at the Schwarzschild radius. More generally, the formal definition of a black hole eventhorizon (as in section 12.1 of [32]) coincides with the causal horizon of any observer with a timelikewordline reaching future timelike infinity. In a cosmological context, it has long been known thatfor a geodesic observer in a deSitter universe, the causal horizon is a spherical surface centered onthis observer, and is at a cosmological proper distance equal to the Hubble radius. Historically, thiscosmological horizon of the deSitter universe was one of the first examples where it was necessaryto generalize the concept of the event horizon by considering observer dependent causal horizons.For example, the concept of the causual horizon is used in a seminal article on the thermodynamicsof cosmological event horizons [11], although it is simply called an \u00E2\u0080\u009Cevent horizon\u00E2\u0080\u009D in that article.For simplicity, from now on we will often use the word horizon, where it is implicitly understoodthat we are refering to the causal horizon, unless otherwise specified. We will also abuse ter-minology slightly and use the word horizon to mean either the full horizon, viewed as a 3d null212.4. Horizons of stationary observershypersurface, or the 2d surface which is formed by taking the intersection of this null hypersurfacewith a 3d spacelike hypersurface of constant time. For example, when we use the word horizonin the context of the deSitter universe, we could be referring to the full horizon, which is a nullhypersurface obtained by taking the boundary of the causal past of a geodesic observer\u00E2\u0080\u0099s trajectory.However, we could also be referring to the intersection of this null hypersurface with a spacelikehypersurface t = constant, where t is the time coordinates from the planar coordinates leading to theline element in (2.3). This intersection results in a spherical surface surrounding the observer, andis normally what one thinks of when referring to the horizon.To illustrate the presence of both cosmological and black hole horizons in SdS spacetime, considerfirst the trajectory of an observer with r(t) = constant and rb < r(t) < rc, where rb and rc are thetwo roots of f (r) = 0 (we assume that 0< 9LM2 < 1, so that f (r) has precisely two real roots). Theexample trajectory of r(t) = re is shown in figure 2.9, and other trajectories with r(t) = constantwould have this same shape. From the Penrose diagram in figure 2.9, we can deduce that the causalhorizon of this observer consists of the spheres r = rb and r = rc (also see figure 2.10a). The outersphere r = rc is the cosmological horizon and the inner sphere rb is the black hole horizon. Nowconsider any observer satisfying rb < r(t)< rc for all t 2R. From the Penrose diagram, we can onceagain conclude that r = rc and r = rb are the cosmological and black hole horizons, respectively.Thus the concentric spheres r = rb and r = rc are the horizons for a large set of observers who neitherfall into the black hole by crossing r = rb, nor get irrevocably caught up in the accelerated expansionof the space by crossing r = rc. In addition to being the horizons for a large family of observers,the horizons r = rb and r = rc are both Killing horizons associated with the Killing vector of thespacetime. For these reasons, it is natural to call r = rb and r = rc the horizons of SdS spacetime,and to our knowledge, all previous work on the horizons of SdS spacetime have been dealing withthese horizons exclusively.222.5. Problem statementFigure 2.9: Penrose diagram with two observer trajectories shown. The green trajectory correspondsto an observer with r(t) = re for all t . The red trajectory corresponds to an observer drifting awayfrom the unstable equilibrium r = re (see figure 2.8).2.5 Problem statementHere we will study the horizons for an observer caught up in the expansion of space and driftingaway from the black hole along a geodesic. This trajectory is shown on the Penrose diagram in figure2.9. This is a natural trajectory to consider since it resembles what we are currently experiencing inour universe. According to the L-CDM model of cosmology, we are currently in the L dominatedphase of the universe\u00E2\u0080\u0099s expansion, and black holes are constantly drifting away from us and crossingour cosmological horizon.The horizons for such an observer consist of a black hole merging with a cosmological horizon.Since this merging of horizons is fundamentally 3+1 dimensional, it cannot be deduced from the1+1 dimensional Penrose diagram alone (see figure 2.10 for an illustration of the past causal horizonof both stationary and drifting observers on the Penrose diagram). In order to deduce this mergingof horizons and study it, we must calculate the individual trajectories of light rays eminating fromthe observer\u00E2\u0080\u0099s trajectory at late times. Schwarzschild coordinates are inadequate for such a calcu-lation, and so we must develop another coordinate system, as will be discussed in chapter 3. Thecalculations of the lights rays which make up the horizon and the analysis of this horizon will be in232.5. Problem statement(a)(b)Figure 2.10: a) Penrose diagram with causal horizon (yellow lines) of an observer stationary next tothe black hole. b) Penrose diagram with causal horizon (yellow lines) of an observer drifting awayfrom the black hole.242.5. Problem statementFigure 2.11: Penrose diagram showing the drifting observer (green curve), the causal horizon ofthis observer (yellow lines) and the spacelike hypersurfaces of constant time (red curves). The redcurves are the spacelike hypersurfaces for the coordinate system developed in chapter 3. The upperand lower red curves correspond to very late times and very early times, respectively. Notice howthe intersection of the lower red curves and the yellow lines is arbitrarily close to r = rb (left side) orr = rc (right side) at early times. This is what allows us to conclude that at early times, the horizonsof the drifting observer consist of the two concentric spheres r = rb and r = rc.chapter 4.Although the merging of horizons discussed in the previous paragraph cannot be deduced explic-itly by looking at the Penrose diagram, its existence can be inferred by using our knowledge of theobserver\u00E2\u0080\u0099s horizon at both early times and late times. In chapter 3 we will introduce a coordinatesystem whose spacelike hypersurfaces are shown in the Penrose diagram in 2.11. From this dia-gram, we see that at early times, the spacelike hypersurfaces of constant time intersect the horizonarbitrarily close to the spheres rb and rc. Thus the early time horizons of the drifting observer are thesame as the horizons of the stationary observer; they are concentric spheres, with the inner spherebeing the black hole horizon and the outer sphere being the cosmological horizon.We can also deduce the shape of the horizon for our drifting observer at late time. At late times,the spacetime in the neighborhood of the drifting observer can be well approximated by the deSitteruniverse, and the trajectory of the drifting observer approaches the trajectory of an observer ex-panding with the Hubble flow in the deSitter universe. Both of these facts will be discussed in moredetail in section 2.6 below. Thus the horizon surrounding the drifting observer will approach that of252.6. r! \u00E2\u0080\u00A2 limit of SdS spacetimean observer caught up in the Hubble flow of deSitter spacetime. That is, the horizon will approacha closed surface surrounding the observer (the surface will be spherical for the appropriate choiceof time slicing).In conclusion, if at early times the horizons are two concentric spherical surfaces and at late timesthe horizon is a single closed surface, then we can deduce that at some point in time there is atransition between the two, and this transition must necessarily involve the merger of the black holeand cosmological horizons. It is this transition which we study in this thesis.2.6 r! \u00E2\u0080\u00A2 limit of SdS spacetimeIn section 2.5, we claimed that in the limit that r ! \u00E2\u0080\u00A2, SdS spacetime in some sense approachesdeSitter spacetime, and that this could be used to conclude that the late time behavior of a driftingobserver\u00E2\u0080\u0099s horizon must be the same as that of an observer caught up in the Hubble flow of thedeSitter universe (also known as the steady state universe). This conclusion about the late timebehavior of the observer\u00E2\u0080\u0099s horizon is not only important for the heuristic arguments used in section2.5, but also critical for the calculations to be performed in chapter 4, where we will solve theequations for the null geodesic generators which make up the horizon. These equations will requireinitial conditions, or more precisely, final conditions, which will be obtained using knowledge ofthe late time behavior of the horizon. We will use the fact that at late times the drifting observer isin a part of SdS spacetime well approximated by deSitter spacetime, so that the horizon at late timesshould resemble the horizon of an observer drifting in the deSitter universe. That is, at late times thehorizon should be a closed surface surrounding the observer. Notice that for r > rc, the coordinater in SdS spacetime is timelike, so that the limit r! \u00E2\u0080\u00A2 is the late time limit.The claim that SdS spacetime approaches deSitter spacetime in the limit r ! \u00E2\u0080\u00A2 is an inherentlyambiguous one, since it requires a way of comparing two spacetimes. The definition we will useis that SdS spacetime approaches deSitter spacetime as r ! \u00E2\u0080\u00A2, if there exists a coordinate system262.7. The dimensionless parameter e = HMof SdS spacetime where the lapse, shift and 3-metric of SdS spacetime all approach the lapse,shift and 3-metric in a coordinate system of deSitter spacetime. Usually comparing two spacetimesby comparing their metric components in specific coordinate systems is hopelessly difficult, sinceone cannot disentangle the difference in metric components due to coordinate changes from thosedue to genuine changes in spacetime geometry. Fortunately, the Schwarzschild coordinates of SdSspacetime are linked in a simple and natural way to the static coordinates of deSitter spacetime,so that according to our definition, we can claim that at late times SdS spacetime does indeedapproach deSitter spacetime (see (2.4) and (2.2) for the line elements of SdS and deSitter spacetimein Schwarzschild and static coordinates, respectively).Given the late time behavior of SdS spacetime in the sense defined above, it follows that the late timebehavior of the null geodesics and timelike geodesics in SdS spacetime must approach the late timebehavior of null and timelike geodesics in deSitter spacetime. Again, there is an inherent ambiguityinvolving the definition of closeness used when comparing null geodesics in one spacetime to nullgeodesics in another spacetime. This can be resolved using a specific coordinate system as wasdone when comparing spacetimes above.To summarize, we use the r ! \u00E2\u0080\u00A2 behavior of SdS spacetime to infer that the late time behaviorof the horizon must be a closed surface surrounding the drifting observer. This knowledge will inturn be used in chapter 4 to set up the \u00E2\u0080\u009Cinitial\u00E2\u0080\u009D conditions (when flowing backwards in time) for thegeodesic equations of the null generators which make up the horizon.2.7 The dimensionless parameter e = HMAlthough the metric components of SdS spacetime depend on both the black hole mass M and thecosmological constant L, the essential geometry of the spacetime can be characterized by a single272.7. The dimensionless parameter e = HMdimensionless parameter e , defined as follows:e = HM , (2.13)where H = pL/3 is the Hubble constant. Once e is specified, changing M or L simply amountsto a constant rescaling of the metric. This can be seen by introducing the dimensionless time t\u00C2\u00AF andradial coordinate r\u00C2\u00AF as follows: t\u00C2\u00AF = Ht,r\u00C2\u00AF = Hr .The line element (2.4) now becomesds2 = 1H2 \"\u00E2\u009C\u00931\u0000 2e\u00C2\u00AFr \u0000 r\u00C2\u00AF2\u00E2\u0097\u0086dt\u00C2\u00AF2\u0000\u00E2\u009C\u00931\u0000 2e\u00C2\u00AFr \u0000 r\u00C2\u00AF2\u00E2\u0097\u0086\u00001 dr\u00C2\u00AF2\u0000 r\u00C2\u00AF2(sin2 qdf 2 +dq 2)# (2.14)and we see that apart from a constant overall rescaling by the Hubble length 1/H, the line elementonly depends on the dimensionless parameter e . Thus all values of M and L which yield the samevalue for e = HM lead to conformally equivalent spacetimes. Due to the conformal equivalence ofthese spacetimes, the essential character of the geometry only depends on the parameter e . For thisreason, throughout this thesis we will often discuss the dependence of the metric on the parametersM and L by referring to the single parameter e .The parameter e is related to the relative size of the spherical black hole and cosmological horizonsof stationary observers (as measured using the circumference of a great circle, for example). Fore \u00E2\u008C\u00A7 1 the black hole is much smaller than the cosmological horizon and as e ! e\u0000c , where ec =1/3p3, the black hole size approaches that of the cosmological horizon size. For e > ec there isneither a black hole or cosmological horizon, and instead we have a naked singularity [20]. In thisthesis we will restrict ourselves to values ofM andL such that 0< e < ec, in which case the structureof the spacetime is as shown in the Penrose diagram in figure 2.9, and the function f (r) will haveprecisely two positive roots, rb and rc. The spheres r = rb and r = rc are the locations of the blackhole and cosmological horizons for an observer stationary outside the black hole (stationary in the282.7. The dimensionless parameter e = HMsense of the timelike Kiling vector of (2.4)). The roots rb and rc, as well as the ratio rb/rc, can becomputed for e \u00E2\u008C\u00A7 1 by using pertubation methods to find the solution to f (r) = 0. This givesrb = 2M(1+O(e2)) ,rc = 1H \u00001\u0000 e +O(e2)\u0000 , (2.15)rbrc = e +O(e2) .From the above we see that for e \u00E2\u008C\u00A7 1 the black hole radius approaches the Schwarzschild radiusrb = 2M and the cosmological horizon approaches the Hubble radius rc = 1H . As well, we see that therelative size of rb and rc is characterized by the parameter e , as mentioned previously. Throughoutthis thesis we will take the point of view that the case where e = 0 corresponds to pure deSitterspacetime (i.e. M = 0), with the introduction of e \u00E2\u008C\u00A7 1 equivalent to introducing a small black holeinto the spacetime. In this way considering e \u00E2\u008C\u00A7 1 amounts to considering a perturbation of deSitterspacetime caused by a small black hole. This perturbation of deSitter spacetime can be seen in theline element (2.14), where e = 0 gives the line element for deSitter spacetime and e \u00E2\u008C\u00A7 1 introducesa small perturbation to the gt\u00C2\u00AFt\u00C2\u00AF and gr\u00C2\u00AFr\u00C2\u00AF metric components. The limit e \u00E2\u008C\u00A7 1 will be exploited in someof the analytical calculations in this thesis. For example, it will be used to approximate the causticstructure of merging horizons in chapter 4 and the horizon area in chapter 5. It is a very natural limitto consider, given that for a typical supermassive black hole and the currently accepted value of L,we would have M \u00E2\u0087\u00A1 1012m and L\u00E2\u0087\u00A1 10\u000052m\u00002, giving e \u00E2\u0087\u00A1 10\u000014.29Chapter 3Lemaitre-Planar CoordinatesWe wish to develop a coordinate system that acomplishes several purposes. First, we would likethese new coordinates to cover the same region of spacetime as covered by Schwarzschild coordi-nates (see figure 2.7b), so that we may interpret the spacetime region under consideration as a blackhole in the L dominated phase of an expanding universe. Second, in contrast to Schwarzschild co-ordinates, we would like these coordinates to be free of coordinate singularities. Also in contrastwith Schwarzschild coordinates, if (T,R,q ,f) are the new coordinates, we would like the metricsignature to be (+,\u0000,\u0000,\u0000) on the whole region of spacetime covered by the coordinates, so that Tand R can always be interpreted as timelike and spacelike variables, respectively. Lastly, we wouldlike the T = constant hypersurfaces to be interpreted as the hypersurfaces of constant time of a ho-mogeneous and isotropic universe containing a black hole. This interpretation will come from thefact that the coordinate T will be the proper time for a family of geodesic observers, along with thefact that these hypersurfaces reduce to the usual hypersurfaces of an FRW cosmology in the limitM ! 0. The requirement that the spacelike hypersurfaces intersect the black hole can be verifiedby, for example, comfirming the existence of an apparent horizon on these hypersurfaces.The coordinates which will achieve all of these purposes will be developed in this chapter, and area special case of the general class of coordinate systems known as Gaussian normal (GN) coordi-nates. These coordinates are built using a family of freely falling observers (i.e. timelike geodesics)and hypersurfaces orthogonal to these observers. Given an arbitary spacetime, it is not possible ingeneral to set up global GN coordinates over the whole of that spacetime, and one must content30Chapter 3. Lemaitre-Planar Coordinates(a)(b)Figure 3.1: a) The timelike coordinate curves of LP coordinates, shown on the Penrose diagram ofSdS spacetime. b) The spacelike coordinate curves of LP coordinates.31Chapter 3. Lemaitre-Planar Coordinatesone\u00E2\u0080\u0099s self to a local use of GN coordinates. However, according to the Frobenius theorem (see sec-tion 2.3.3 of [25], or section B.3 of [32]), if we have a family of timelike trajectories (i.e. timelikecongruence) with vanishing vorticity tensor, then there exists a time coordinate such that the hyper-surfaces t = constant are everywhere orthogonal to this congruence. One set of spacetimes wherewe have such a congruence, and it is possible to set up global GN coordinates, are the familiarspacetimes of FRW cosmology. For example, GN coordinates for the steady state universe lead tothe familiar line element ds2 = dt2\u0000 e2Ht \u0000dr2 +r2(dq 2 + sin2 qdf 2)\u0000 (3.1)which has flat hypersurfaces and an exponential scale factor. The (t,r,q ,f) coordinates above areoften called planar coordinates. The GN coordinates which we will develop for SdS spacetime willreduce to the planar coordinates of the steady state universe in the limit M = 0, and can be thoughtof as a generalization of these coordinates to include the case where there is a single Schwarzschildblack hole. They are also closely related to the well known Lemaitre coordinates [22] in the casewhere the cosmological constant is zero (this will be discussed in section 3.3.2). For this reason,we will call them Lemaitre-planar (LP) coordinates. They will be based on a family of timelikegeodesics eminating from the equilibrium point r = re (see figure 3.3a for the effective potentialdiagram and figure 2.9 for the timelike trajectory r(t) = re on the Penrose diagram). The family ofsuch trajectories is shown on the Penrose diagram in figure 3.1a. The spacelike hypersurfaces of LPcoordinates are shown in figure 3.1b.As an interesting historical note, it would seem that the earliest construction of GN coordinatesfor Schwarzschild deSitter spacetime go all the way back to Lemaitre [22]. These coordinates(let us simply call them Lemaitre coordinates) reduce to the well known Lemailtre coordinates forSchwarzschild spacetime in the limit that L! 0. They also have the attractive feature of havingspatially flat hypersurfaces. However, the part of spacetime covered by these coordinates is not thepart of the spacetime we are interested in considering (see figure 3.2), and so these coordinates willnot be useful for our purposes. The difference between Lemaitre coordinates and LP coordinatescan be understood by comparing the family of radial timelike geodesics used as timelike coordinate32Chapter 3. Lemaitre-Planar CoordinatesFigure 3.2: Penrose diagram showing the region of SdS spacetime covered by Lemaitre coordinates.(a) (b)Figure 3.3: Effective potential diagrams for radial timelike geodesic trajectories. a) The blackline and green lines indicate two families of radial timelike geodesics trajectories: those goingtowards r = \u00E2\u0080\u00A2 and those going to r = 0. The \u00E2\u0080\u009Cenergy at infinity\u00E2\u0080\u009D is chosen so that these trajectoriesall eminate from the equilibrium r = re. These two families are the trajectories used as timelikecoordinate curves for the LP coordinate system. b) The black line indicates a family of outgoingradial timelike geodesics. The \u00E2\u0080\u009Cenergy at infinity\u00E2\u0080\u009D for these trajectories is E = 1; it turns out this isthe only value which produces flat spacelike hypersurfaces. This family of trajectories are used asthe timelike coordinate curves of Lemaitre\u00E2\u0080\u0099s coordinate system for SdS spacetime [22].333.1. Construction of LP coordinatescurves for these two coordinate systems, as shown on the effective potential diagram in figure 3.3.The geodesics in Lemaitre coordinates are all radially outgoing, whereas the geodesics in LP coor-dinates either fall into the singularity or drift out to r = \u00E2\u0080\u00A2, always starting from r = re. In Lemaitrecoordinates, the family of timelike coordinate curves are all essentially the same: the radial coor-dinate as a function of proper time, r(t), is the same for all of them, and they only differ in theirrelationship between coordinate time and proper time. In LP coordinates, on the other hand, wehave two families of trajectories whose radial coordinate as a function of proper time are all thesame; those moving radially inward and those moving radially outward.We construct LP coordinates in section 3.1. This is done by starting from Schwarzschild coordinatesand integrating along the coordinate curves of LP coordinates. This allows us to find the explicitmapping from Schwarzschild to LP coordinates. This mapping is used in section 3.2 to find themetric components of SdS spacetime in LP coordinates. In section 3.3 we discuss some additionalfeatures of LP coordinates which will be useful in later sections.3.1 Construction of LP coordinatesIn terms of observers, the timelike coordinate curves of LP coordinates can be thought of as a setof observers drifting radially away from the equilibrium r = re, as shown on the effective potentialdiagram in figure 3.3a and on the Penrose diagram in figure 3.1a. One set of observers drift intothe black hole while the other ones drift away from the black hole, caught up in the expansion ofspace. The time coordinate is defined as the time measured by clocks carried by each of theseobservers. We place a requirement that the clocks of two sufficiently nearby observers should, tolowest approximation in the distance between them, agree on the time of an event taking place onthe midpoint of a rod connecting them. Mathematically this amounts to the requirement of havingspacelike hypersurfaces T = constant orthogonal to the timelike coordinate curves. The spatialcoordinates consist of the angles (q ,f), which remain constant for any of our observers, and aradial variable R, which is also constant for any observer, and whose value is the Schwarzschild343.1. Construction of LP coordinatesradius of the observer\u00E2\u0080\u0099s position when that observer\u00E2\u0080\u0099s clock value reads some chosen value T = T \u00E2\u0087\u00A4.We now turn to the mathematical formulation of LP coordinates. Our construction of LP coordinatesis based on four requirements. The first two requirements specify the coordinate curves of the coor-dinate system, and the second two requirements specify the coordinates which are to be used alongthese curves. Suppressing the (q ,f) variables for the moment, let (T,R) be the LP coordinates. Werequire that:(i) The timelike coordinate curves R= constant are radial timelike geodesics satisfying r(t =\u0000\u00E2\u0080\u00A2)=re, where r(t) is the Schwarzschild radial coordinate of the trajectory, parametrized by proper time,and re is the equiibrium radius (see figure 3.3a for the effective potential diagram and figure 3.1afor a plot of the timelike coordinate curves on the Penrose diagram).(ii) The spacelike coordinate curves T = constant are orthogonal to the timelike coordinate curves(see figure 3.1b for a plot of the spacelike coordinate curves on the Penrose diagram).(iii) The timelike coordinate T is chosen such that any spacelike coordinate curve T = constantintersects a chosen timelike coordinate curve R = R\u00E2\u0087\u00A4 > re at a point with coordinates (T,R) =(t,R\u00E2\u0087\u00A4), where t is the proper time along the curve R = R\u00E2\u0087\u00A4. This intersection of curves is shown infigure 3.4a. As will be shown in section 3.2, R\u00E2\u0087\u00A4 can be thought of as essentially arbitrary, in thesense that its value does not affect the metric components in the LP coordinate system.(iv) The spacelike coordinate R is chosen such that any curve R = constant intersects a chosenspacelike coordinate curve T = T \u00E2\u0087\u00A4 with R = r, where r is the radial Schwarzschild coordinate of theintersection point. This intersection of curves is shown in figure 3.4b. As will be shown in section3.2, the effect of changing T \u00E2\u0087\u00A4 on the metric components in the LP coordinate system amounts tosimply shifting the time variable T , and thus T \u00E2\u0087\u00A4 is essentially arbitrary.Translated into precise mathematical language, conditions (i) and (ii) specify the directions of the353.1. Construction of LP coordinates(a)(b)Figure 3.4: a) The green curve is the arbitrarily chosen timelike coordinate curve R = R\u00E2\u0087\u00A4 and thered curves are the spacelike coordinate curves T = constant. The value of the time coordinate T onany one of these spacelike curves is assigned to be T = t , where t is the value of the proper timealong the curve R = R\u00E2\u0087\u00A4 at the point of intersection between the two curves. b) A similar procedureis used to assign a value to R for the timelike coordinate curves R = constant (shown in red). Thecoordinate R is defined such that R = r, where r is the value of the Schwarzschild coordinate at theintersection point of a timelike coordinate curve with the chosen spacelike curve T = T \u00E2\u0087\u00A4 (shown ingreen).363.1. Construction of LP coordinatestimelike and spacelike coordinate tangent vectors \u00E2\u0088\u0082\u00E2\u0088\u0082T and \u00E2\u0088\u0082\u00E2\u0088\u0082R , respectively. Expressed in the basisof coordinate tangent vectors in Schwarzschild coordinates, \u00E2\u0088\u0082\u00E2\u0088\u0082 t and \u00E2\u0088\u0082\u00E2\u0088\u0082 r , these directions are\u00E2\u0088\u0082\u00E2\u0088\u0082T \u00C2\u00B5 f (re) 12f (r) \u00E2\u0088\u0082\u00E2\u0088\u0082 t + sgn(r\u0000 re) [ f (re)\u0000 f (r)]1/2 \u00E2\u0088\u0082\u00E2\u0088\u0082 r , (3.2)\u00E2\u0088\u0082\u00E2\u0088\u0082R \u00C2\u00B5 sgn(r\u0000 re) [ f (re)\u0000 f (r)] 12 \u00E2\u0088\u0082\u00E2\u0088\u0082 t + f (re)1/2 f (r) \u00E2\u0088\u0082\u00E2\u0088\u0082 r . (3.3)The first relation above comes from the geodesic equations for timelike radial geodesics (equations(2.11)-(2.12)). The second relation is obtained by simply requiring that h \u00E2\u0088\u0082\u00E2\u0088\u0082T , \u00E2\u0088\u0082\u00E2\u0088\u0082Ri = 0. Conditions(iii) and (iv) can also be translated into precise mathematical language. Letting r(T,R) be thefunction mapping (T,R) coordinates to the Schwarzschild coordinate r, we have:(iii), r(T,R\u00E2\u0087\u00A4) = r0(t = T ;R\u00E2\u0087\u00A4), where r0(t;R\u00E2\u0087\u00A4) is a timelike geodesic satisfying r0(t =\u0000\u00E2\u0080\u00A2) = reand r0(t = T \u00E2\u0087\u00A4) = R\u00E2\u0087\u00A4,(iv), r(T \u00E2\u0087\u00A4,R) = R.Consider any of the timelike coordinate curves R = constant. Suppressing the coordinate R for amoment since it is held constant, let r(T ) and t(T ) be the Schwarzschild coordinates along thiscoordinate curve. From condition (i) above, we have that r(T ) is a monotonic function of T (exceptfor the one curve where r(T ) = re for all T ). Therefore we can define the inverse T (r) and thefunction t(r) = t(T (r)). By dividing the components of the tangent vector in equation (3.2) weobtain that t(r) satisfies the following:dtdr \u0000\u0000\u0000\u0000R=constant = f (re)1/2f (r)( f (re)\u0000 f (r))1/2 sgn(r\u0000 re) . (3.4)Next we apply the same procedure to a spacelike coordinate curve T = constant. We suppress the Tcoordinate and consider the functions r(R) and t(R) along the curve. For T = T \u00E2\u0087\u00A4, we have r(R) = Rand the function r(R) is monotonic. It then follows by continuity that for values of T sufficientlynear T \u00E2\u0087\u00A4, r(R) is once again monotonic. Therefore, for values of T near T \u00E2\u0087\u00A4, we can define the373.1. Construction of LP coordinatesfunction t(r) = t(R(r)). We divide the components of the tangent vector in equation (3.3) to obtaindtdr \u0000\u0000\u0000\u0000T=constant = ( f (re)\u0000 f (r))1/2f (re)1/2 f (r) sgn(r\u0000 re) . (3.5)We will obtain the explicit mapping from LP coordinates (T,R) to Schwarzschild coordinates (t,r)by integrating equations (3.4) and (3.5). Note that the right hand side of (3.4) is ill-defined wheref (r) = 0 and where f (r) = f (re) (i.e. at r = rb, r = rc and r = re) and the right hand side of(3.5) is ill-defined for f (r) = 0. This difficulty can be overcome by temporarily restricting thecoordinate transformation from LP to Schwarzschild coordinates to any region small enough thatit does not intersect r = rb, r = re or r = rc. We have thus placed two temporary restrictions onthe coordinate mapping from LP to Schwarzschild coordinates: the first is that T be sufficientlynear T \u00E2\u0087\u00A4 and the other is that the region considered not intersect r = rb, r = re or r = rc. Oncethe explicit mapping from LP coordinates to Schwarzschild coordinates in this restricted region isobtained, these temporary restriction will be lifted and the mapping will be extended to the fullSchwarzschild region (i.e. the region shown in figure 3.1). This will be done by simply defining theLP coordinates using the formula obtained for the functions t(T,R) and r(T,R). The justificationfor this procedure will ultimately come from the fact that the resulting metric components in LPcoordinates will be well defined over the region of spacetime of interest (the metric componentswill be derived in section 3.2).Let us now derive the functions r(T,R) and t(T,R). To do this, consider two paths joining anarbitrary point (T,R) to the point (T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4), where it is understood that (T,R) and R\u00E2\u0087\u00A4 are chosen sothat the restrictions mentioned in the previous paragraph are met. The first path connects the point(T,R) to (T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) by first going along R = constant until the point (T \u00E2\u0087\u00A4,R), and then going alongthe path T = T \u00E2\u0087\u00A4 until the point (T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) is reached. The second path connects the point (T,R) to(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) by first going along T = constant until the point (T,R\u00E2\u0087\u00A4), and then going along the pathR = R\u00E2\u0087\u00A4 until the point (T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) is reached. Notice that both of these paths consist entirely of LPcoordinate curves.We now integrate dt/dr along these two paths. Moving along the first path, we integrate (3.4) from383.1. Construction of LP coordinates(T,R) to (T \u00E2\u0087\u00A4,R) along the coordinate curve R = constant, and then integrate (3.5) from (T \u00E2\u0087\u00A4,R) to(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) along the coordinate curve T = T \u00E2\u0087\u00A4. This gives the following equation:r(T \u00E2\u0087\u00A4,R)\u00CB\u0086r(T,R) dtdr \u0000\u0000\u0000\u0000R=constant dr+ r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)\u00CB\u0086r(T \u00E2\u0087\u00A4,R) dtdr \u0000\u0000\u0000\u0000T=constant dr = t(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)\u0000 t(T,R) . (3.6)Moving along the second path, we integrate (3.5) from (T,R) to (T,R\u00E2\u0087\u00A4) along the coordinate curveT = constant, and then integrate (3.4) from (T,R\u00E2\u0087\u00A4) to (T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) along the coordinate curve R =constant. This gives the following equation:r(T,R\u00E2\u0087\u00A4)\u00CB\u0086r(T,R) dtdr \u0000\u0000\u0000\u0000T=constant dr+ r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)\u00CB\u0086r(T,R\u00E2\u0087\u00A4) dtdr \u0000\u0000\u0000\u0000R=constant dr = t(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)\u0000 t(T,R) . (3.7)We can find the function r(T,R) by first combining (3.6) and (3.7) into the following equation:r(T \u00E2\u0087\u00A4,R)\u00CB\u0086r(T,R) dtdr \u0000\u0000\u0000\u0000R=constant dr+ r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)\u00CB\u0086r(T \u00E2\u0087\u00A4,R) dtdr \u0000\u0000\u0000\u0000T=constant dr = r(T,R\u00E2\u0087\u00A4)\u00CB\u0086r(T,R) dtdr \u0000\u0000\u0000\u0000T=constant dr+ r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)\u00CB\u0086r(T,R\u00E2\u0087\u00A4) dtdr \u0000\u0000\u0000\u0000R=constant dr .Manipulating the limits of integration in the above and rearranging terms, we obtain the following:r(T,R)\u00CB\u0086r(T \u00E2\u0087\u00A4,R) \u00E2\u009C\u0093 dtdr \u0000\u0000\u0000\u0000R=constant\u0000 dtdr \u0000\u0000\u0000\u0000T=constant\u00E2\u0097\u0086dr = r(T,R\u00E2\u0087\u00A4)\u00CB\u0086r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) \u00E2\u009C\u0093 dtdr \u0000\u0000\u0000\u0000R=constant\u0000 dtdr \u0000\u0000\u0000\u0000T=constant\u00E2\u0097\u0086dr . (3.8)Using (3.4) and (3.5), we havedtdr \u0000\u0000\u0000\u0000R=constant\u0000 dtdr \u0000\u0000\u0000\u0000T=constant = 1f (re)1/2 sgn(r\u0000 re)( f (re)\u0000 f (r))1/2 .From condition (iii) above, and the geodesic equation (2.12), we have that r(T,R) satisfies thefollowing: \u00E2\u0088\u0082 r\u00E2\u0088\u0082T = sgn(r\u0000 re)( f (re)\u0000 f (r))1/2 (3.9)393.1. Construction of LP coordinatesCombining the previous two equations, we havedtdr \u0000\u0000\u0000\u0000R=constant\u0000 dtdr \u0000\u0000\u0000\u0000T=constant = 1f (re)1/2 \u00E2\u009C\u0093 \u00E2\u0088\u0082 r\u00E2\u0088\u0082T \u00E2\u0097\u0086\u00001 . (3.10)Substituting the above into (3.8), we obtainr(T,R)\u00CB\u0086r(T \u00E2\u0087\u00A4,R) \u00E2\u009C\u0093 \u00E2\u0088\u0082 r\u00E2\u0088\u0082T \u00E2\u0097\u0086\u00001 dr = r(T,R\u00E2\u0087\u00A4)\u00CB\u0086r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) \u00E2\u009C\u0093 \u00E2\u0088\u0082 r\u00E2\u0088\u0082T \u00E2\u0097\u0086\u00001 dr . (3.11)Interpreted physically, the above is the statement that the timelike curves R = constant correspondto a family of observers whose clocks are synchronized. It is for this reason that Gaussian normalcoordinates are sometimes also called synchronous coordinates. Let us evaluate the integral on theright hand side above explictly. We do so by making a change of variables. Fixing R = R\u00E2\u0087\u00A4, definer(T ) = r(T,R\u00E2\u0087\u00A4). r(T ) is a monotonic function and can be inverted to give the function T (r). UsingT (r) as the new variable of integration, the integral becomesr(T,R\u00E2\u0087\u00A4)\u00CB\u0086r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) \u00E2\u009C\u0093 \u00E2\u0088\u0082 r\u00E2\u0088\u0082T \u00E2\u0097\u0086\u00001 dr = T\u00CB\u0086T \u00E2\u0087\u00A4 dt = T \u0000T \u00E2\u0087\u00A4 . (3.12)Using the above and (3.9) in (3.11), we obtain the following implicit equation for the functionr(T,R): r(T,R)\u00CB\u0086R dr( f (re)\u0000 f (r))1/2 = sgn(R\u0000 re)(T \u0000T \u00E2\u0087\u00A4) , (3.13)where we have used r(T \u00E2\u0087\u00A4,R) = R (condition (iv) above). We have also removed sgn(r\u0000 re) fromthe integral and replaced it with sgn(R\u0000 re). It is possible to do this because our definition of thetimelike coordinate curves (condition (i) at the beginning of this section) ensures that none of themcross r = re, and furthermore our definition of the coordinate R (condition (iv) above) ensures thatsgn(r\u0000 re) = sgn(R\u0000 re). The integral on the left hand side above can be found analytically. Oncethis is done it is possible to write an explicit formula for r(T,R) in terms of elementary functionsand their inverses. This will only be useful for the purpose of extending the coordinate mapping toall values of the Schwarzschild coordinates, and so we relegate it to section 3.3.3.403.1. Construction of LP coordinatesTo obtain the function t(T,R), we manipulate the limits of integration and rearrange terms in (3.7)to gett(T,R) = t(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)+ r(T,R\u00E2\u0087\u00A4)\u00CB\u0086r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) \u00E2\u009C\u0093 dtdr \u0000\u0000\u0000\u0000R=constant\u0000 dtdr \u0000\u0000\u0000\u0000T=constant\u00E2\u0097\u0086dr+ r(T,R)\u00CB\u0086r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) dtdr \u0000\u0000\u0000\u0000T=constant dr . (3.14)The first integral on the right hand side above is the same as the right hand side of (3.8), and has infact already been found. Combining (3.10) and (3.12), we haver(T,R\u00E2\u0087\u00A4)\u00CB\u0086r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) \u00E2\u009C\u0093 dtdr \u0000\u0000\u0000\u0000R=constant\u0000 dtdr \u0000\u0000\u0000\u0000T=constant\u00E2\u0097\u0086dr = 1f (re)1/2 (T \u0000T \u00E2\u0087\u00A4) .Substituting the above and (3.5) into (3.14), we obtain the following formula for the function t(T,R):t(T,R) = t(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4)+ 1f (re)1/2 (T \u0000T \u00E2\u0087\u00A4)+ r(T,R)\u00CB\u0086r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) ( f (re)\u0000 f (r))1/2f (re)1/2 f (r) sgn(r\u0000 re)dr . (3.15)The integral on the right hand side can be found analytically. Again, this explicit formula is not nec-essary except for the purpose of extending the mapping, and so we relegate it to section 3.3.3. Notethat the function t(T,R) above is only known once the function r(T,R) has been found from (3.13).The equations (3.13) and (3.15), taken together, provide a change of coordinates from Schwarzschildcoordinates (t,r) to the LP coordinates (T,R). In so doing they allow us to find the metric compo-nents of SdS spacetime in LP coordinates, as will be done in the next section. Additional detailsregarding LP coordinates will be dealt with in section 3.3. For example, in section 3.3.3, we dealwith the issue of ensuring that (3.13) and (3.15) provide a mapping over the whole region of SdSspacetime covered by Schwarzschild coordinates. That is, we ensure that the mapping from LPcoordinates to Schwarzschild coordinates can indeed be extended beyond the restricted region forwhich it was originally derived.413.2. Metric components in LP coordinates3.2 Metric components in LP coordinatesWe are interested in obtaining the metric components of SdS spacetime in LP coordinates, and so wemust first obtain the four components of the Jacobian matrix of partial derivatives associated withthe coordinate transformation from Schwarzschild to LP coordinates. This is achieved by takingpartial derivatives of (3.13) and (3.15). Taking the partial derivative of (3.13) with respect to T orR, we get the first two Jacobian components:\u00E2\u0088\u0082 r\u00E2\u0088\u0082T = sgn(r\u0000 re)( f (re)\u0000 f (r)) 12 , (3.16)\u00E2\u0088\u0082 r\u00E2\u0088\u0082R = \u00EF\u00A3\u00BF f (re)\u0000 f (r)f (re)\u0000 f (R)\u00001/2 . (3.17)Next, taking the partial derivative of (3.15) and using the above where necessary, we get the othertwo components of the Jacobian:\u00E2\u0088\u0082 t\u00E2\u0088\u0082T = f (re) 12f (r) , (3.18)\u00E2\u0088\u0082 t\u00E2\u0088\u0082R = [ f (re)\u0000 f (r)]sgn(R\u0000 re)f (re)1/2 f (r) [ f (re)\u0000 f (R)]1/2 . (3.19)In the above, it is understood that r = r(T,R) is a function of T and R. Using these four Jacobiancomponents, we can apply the tensor transformation law to the Schwarzschild metric components(2.4) to obtain the line element in LP coordinates. This yieldsds2 = dT 2\u0000 f (re)\u0000 f (r)f (re) [ f (re)\u0000 f (R)]dR2\u0000 r2(dq 2 + sin2 qdf 2) , (3.20)where r = r(T,R) is given implicitly by (3.13). The constant R\u00E2\u0087\u00A4, which was used as part of thedefinition of the coordinates (condition (iii) in section 3.1), does not appear explicitly in the aboveline element, nor does it appear in the equation (3.13) defining r(T,R). Thus we see that the valueof R\u00E2\u0087\u00A4 has no impact on the metric and is essentially arbitrary. The constant T \u00E2\u0087\u00A4 (defined in condition(iv) of section 3.1) affects the above line element through its presence in (3.13). However, as willbe discussed in section 3.3.5, there is a freedom in shifting the variable T by a constant amount. For423.3. Additional features of LP coordinatesthis reason, T \u00E2\u0087\u00A4 can be thought of as arbitrary and we will take T \u00E2\u0087\u00A4 = 0 from now on. Notice that themetric component gRR above is undefined at R = re. However, we can define gRR at R = re in sucha way that it is continuous, as will be discussed in section 3.3.4.3.3 Additional features of LP coordinatesIn what follows, we describe some additional features of LP coordinates which will be useful inlater sections, or which shed further light on the coordinate system.3.3.1 The case M = 0: deSitter spacetime in planar coordinatesIn the case that M = 0, (3.20) reduces to the deSitter line element in planar coordinates, as in (3.1).This can be seen by first noting that when M = 0, we have:f (r) = 1\u0000H2r2,f (re) = 1 ,where re is given by (2.6) and f (r) is given by (2.5). Using the above, the equation (3.13) definingr(T,R) can be readily integrated to give r(T,R) = ReHT ,where recall that T \u00E2\u0087\u00A4 = 0. Substituting the above into (3.20) and setting M = 0, we getds2 = dT 2\u0000 e2HT \u0000dR2 +R2(dq 2 + sin2 qdf 2)\u0000 , (3.21)which is precisely the same as (3.1). The above is the line element of an FRW cosmology withflat spatial hypersurfaces and an exponentially growing scale factor. It has traditionally been asso-433.3. Additional features of LP coordinatesciated with the cosmology of a \u00E2\u0080\u009Csteady state\u00E2\u0080\u009D universe obeying the perfect cosmological principle.The above line element also approximates the geometry of a universe going through a period ofinflation. The coordinates (T,R,q ,f) are often called planar coordinates, though in the context ofinflation the name inflationary coordinates is used as well. Mathematically, the coordinates coverthe upper half of the hyperboloid corresponding to deSitter spacetime, as illustrated in figure 2.5.Physically, the coordinate curves R = constant correspond to a family of freely floating observers ina homogeneous and isotropic universe, and the coordinate T corresponds to proper time measuredby these observers. The coordinate R is sometimes called the comoving distance and the coordinateT is sometimes called the cosmic time.3.3.2 The case H = 0: Schwarzschild spacetime in Lemaitre coordinatesIn the case that H = 0, we have f (r) = 1\u0000 2Mr ,f (re) = 1 ,where re is given by (2.6) and f (r) is given by (2.5). The line element (3.20) reduces tods2 = dT 2\u0000\u00E2\u009C\u0093Rr \u00E2\u0097\u0086dR2\u0000 r2dW2 . (3.22)Let us introduce the new variable r as r = 23R3/2(2M)1/2 . (3.23)(3.22) now becomes ds2 = dT 2\u0000 2Mr dr2\u0000 r2dW2 .The above is the line element of Schwarzschild spacetime, with (T,r,q ,f) the well known Lemaitrecoordinates [22]. In the case that H = 0, the LP coordinates are therefore closely related to the443.3. Additional features of LP coordinatesLemaitre coordinates, the only difference being the redefinition of the radial variable through (3.23).The function r(T,r) in the line element above can be found by first obtaining r(T,R) from (3.13),and then replacing R by r using (3.23). This givesr(T,R) = \u00E2\u009C\u009332\u00E2\u0097\u00862/3(2M)1/3(r\u0000T )2/3 .The region of spacetime covered by these coordinates is the black hole half of Schwarzschild space-time (the upper half in figure 2.1). The spacelike hypersurfaces T = constant in this coordinatesystem are intrinscially flat. To see this, first set T = constant in (3.22) to getds2 =\u0000\u00E2\u009C\u0093Rr \u00E2\u0097\u0086dR2\u0000 r2dW2 . (3.24)Now consider the Jacobian component (3.17) with H = 0:\u00E2\u0088\u0082 r\u00E2\u0088\u0082R = \u00E2\u009C\u0093Rr \u00E2\u0097\u00861/2 .Using the above to transform the line element (3.24) to (r,q ,f) coordinates, we obtain the lineelement of three dimensional Euclidean space in spherical coordinates. Physically, the coordinatecurves R= constant correspond to a family of observers falling into the black hole from infinity, andall with the same \u00E2\u0080\u009Cenergy at infinity\u00E2\u0080\u009D. They only differ in the reading on their clocks when they passthrough some chosen value of the Schwarzschild radius r. For example, an observer with coordinateR = R0 would reach the singularity r = 0 at time T = 23q R302M . The coordinate T measures propertime as measured by any one of these observers.3.3.3 Explicit coordinate transformation from Schwarzschild to LP coordinatesIn deriving the coordinate transformation from LP to Schwarzschild coordinates, as given by equa-tions (3.13) and (3.15), we placed certain restrictions on the values of the coordinates (T,R). Thiswas to avoid certain technical difficulties, such as the presence of a divergent integrand in (3.15).The easiest way to overcome these restrictions is to perform the integrals in (3.13) and (3.15) ana-453.3. Additional features of LP coordinateslytically, and then use the resulting explicit form of the coordinate transformation as the definitionof LP coordinates. Thus the true definition of the LP coordinates is provided in this section, in thesense that the definition given by equations (3.13) and (3.15) does not apply to the entire spacetimeregion of interest. Since the explicit coordinate transformation given in this section is obtained byfinding the antiderivatives of the integrands in (3.13) and (3.15), it will lead to the same Jacobian,as given by (3.16)-(3.19), and thus to the same line element (3.20).We start by giving the mapping from LP to Schwarzschild coordinates. By performing the integralin (3.13), we obtain the following formula for the Schwarzschild coordinate r as a function of theLP coordinates (T,R): r(T,R) =8><>:reF\u000011 \u00E2\u0087\u00A3F1\u00E2\u0087\u00A3 Rre\u00E2\u008C\u0098eHT\u00E2\u008C\u0098 , R\u00EF\u00A3\u00BF rereF\u000012 \u00E2\u0087\u00A3F2\u00E2\u0087\u00A3 Rre\u00E2\u008C\u0098eHT\u00E2\u008C\u0098 , R\u0000 re (3.25)where F1(r) = \u00E2\u0087\u00A3r +1+pr(r +2)\u00E2\u008C\u0098\" 1\u0000rp3r(r +2)+2r +1# 1p3 ,F2(r) = \u00E2\u0087\u00A3r +1+pr(r +2)\u00E2\u008C\u0098\" r\u00001p3r(r +2)+2r +1# 1p3 . (3.26)Recall that we have taken T \u00E2\u0087\u00A4 = 0 in the above. The functions F1(r) and F2(r) are strictly monotonic,and therefore invertible, for r \u00EF\u00A3\u00BF 1 and r \u0000 1, respectively. The ranges of these functions are F1 \u0000 0and F2 \u0000 0, so that the domains of the inverses F\u000011 (y) and F\u000012 (y) are well defined for all y\u0000 0.The formula for the Schwarzschild time coordinate t(T,R) is a little more involved. First we rewrite(3.15) as t(T,R) = t\u00E2\u0087\u00A4+ 1f (re)1/2 0@T + r\u00CB\u0086r\u00E2\u0087\u00A4 sgn(r\u0000 re)( f (re)\u0000 f (r))1/2f (r) dr1A , (3.27)where t\u00E2\u0087\u00A4 = t(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) and r\u00E2\u0087\u00A4 = r(T \u00E2\u0087\u00A4,R\u00E2\u0087\u00A4) and we have once again taken T \u00E2\u0087\u00A4 = 0. Recall that R\u00E2\u0087\u00A4 doesnot appear anywhere in the line element (3.20), so that its choice is essentially arbitrary. In addition463.3. Additional features of LP coordinatesto the freedom in choosing R\u00E2\u0087\u00A4, there is a freedom in shifting the variable T by an arbitrary constantamount (this is discussed in section 3.3.5 below). Because of these two freedoms, the choice for thevalues of t\u00E2\u0087\u00A4 and r\u00E2\u0087\u00A4 are arbitrary, and we take t\u00E2\u0087\u00A4 = 0 and r\u00E2\u0087\u00A4 = (re + rc)/2 without loss of generality.By evaluating the integral in the equation above, we obtain the following formula for t(T,R):t(T,R) = 1f (re)1/2 (T +A(r(T,R))\u0000A(r\u00E2\u0087\u00A4)) , (3.28)where r(T,R) is given above, and A(r) is the antiderivative of the integrand in (3.27):A(r) = qr2b +2rbre (re\u0000 rb)H (ra\u0000 rb)(rb\u0000 rc) arctanh0@ rbre + rer + rbrqr2b +2rbrepr2 +2rre1A\u0000pr2c +2rcre (re\u0000 rc)H (ra\u0000 rc)(rb\u0000 rc) arctanh rcre + rer + rcrpr2c +2rcrepr2 +2rre!\u0000 ln\u00E2\u009C\u00931+rre + 1repr2 +2rer\u00E2\u0097\u0086 (3.29)\u0000pr2a +2rare (re\u0000 ra)H (ra\u0000 rb)(ra\u0000 rc) ln 2rare + rer + rar +pr2a +2rarepr2 +2rrere (r\u0000 ra) ! .In the above, ra, rb and rc are the roots of f (r) = 0, with ra < 0 < rb < re < rc. Notice thatthe mapping from LP to Schwarzschild coordinates, as given by (3.25)-(3.29), is asymptoticallydiscontinuous at r = rb and r = rc. That is, as one approaches values of (T,R) such that r(T,R) = rbor r(T,R) = rc, one finds that t ! \u00E2\u0080\u00A2 or t !\u0000\u00E2\u0080\u00A2. These discontinuities are a consequence of thefact that the Schwarzschild coordinates are not defined along r = rb and r = rc. Therefore, strictlyspeaking, the equations (3.25)-(3.29) provide three mappings from LP coordinates to Schwarzschildcoordinates: one for 0 < r < rb, one for rb < r < rc, and one for r > rc. For any value of theSchwarzschild variables (t,r) in any one of these three regions, we can solve (3.25) and (3.28) fora unique value of the LP coordinates (T,R), with R > 0. This leads to the inverse mapping, whichtakes Schwarzschild coordinates into LP coordinates, and is given byR(t,r) =8><>:reF\u000011 \u00E2\u0087\u00A3F1\u00E2\u0087\u00A3 rre\u00E2\u008C\u0098e\u0000HT (t,r)\u00E2\u008C\u0098 , r \u00EF\u00A3\u00BF rereF\u000012 \u00E2\u0087\u00A3F2\u00E2\u0087\u00A3 rre\u00E2\u008C\u0098e\u0000HT (t,r)\u00E2\u008C\u0098 , r \u0000 re (3.30)473.3. Additional features of LP coordinatesT (t,r) = f (re)1/2t +A(r\u00E2\u0087\u00A4)\u0000A(r) . (3.31)(3.30)-(3.31) provide the inverse mapping over any one of the three regions previously discussed.We have thus constructed a coordinate mapping from LP to Schwarzschild coordinates which coversthe region of SdS spacetime of interest, namely the region covered by Schwarzschild coordinates, asshown in figure 2.7b (where it is understood that points with r = rb and r = rc should not be includedin the region covered by Schwarzschild coordinates). The fact that the coordinate transformationfrom LP to Schwarzschild coordinates is not defined for r = rb and r = rc does not present any realdifficulty, since in the end what matters is that the metric components in LP coordinates, as givenby equation (3.20), are well defined for all values of r. The issue of the continuity of the metriccomponents is dealt with in section 3.3.4.3.3.4 Continuity of metric componentsConsider the gRR component of the line element (3.20):gRR =\u0000 f (re)\u0000 f (r)f (re) [ f (re)\u0000 f (R)] , (3.32)where r = r(T,R), so that the above is a function of both T and R. Notice that the above is undefinedat R = re. However, by the very definition of our coordinate system (see conditions (i) and (iv) insection 3.1), we have r(T,Re) = re, so that the numerator vanishes as well as R! re. We will nowshow that the limit of gRR as R! re is well-defined, so that by defining gRR at R = re as being equalto this limit, we will have a continuous metric. Using (3.17), we can rewrite the above asgRR =\u0000 1f (re)\u00E2\u009C\u0093 \u00E2\u0088\u0082 r\u00E2\u0088\u0082R\u00E2\u0097\u00862 . (3.33)From the above, we see that we can find the limit of gRR as R! re by calculating \u00E2\u0088\u0082 r/\u00E2\u0088\u0082R at R = re.To do this, we find the lowest order behavior of r(T,R) near R. First consider F1(r) and F2(r) in483.3. Additional features of LP coordinates(3.26). For r near 1, these have the following asymptotic forms:F1(r) \u00E2\u0087\u00A0 \u00E2\u0087\u00A32+p3\u00E2\u008C\u0098\u00E2\u009C\u00931\u0000r6 \u00E2\u0097\u0086 1p3 ,F2(r) \u00E2\u0087\u00A0 \u00E2\u0087\u00A32+p3\u00E2\u008C\u0098\u00E2\u009C\u0093r\u000016 \u00E2\u0097\u0086 1p3 .The inverses F\u000011 (y) and F\u000012 (y) therefore have the following asymptotic forms:F\u000011 (y) \u00E2\u0087\u00A0 \u00006\u00E2\u009C\u0093 y2+p3\u00E2\u0097\u0086p3 +1 ,F\u000012 (y) \u00E2\u0087\u00A0 6\u00E2\u009C\u0093 y2+p3\u00E2\u0097\u0086p3 +1 .Substituting the two preceeding equations into 3.25, we obtainr(T,R) = reF\u00001i \u00E2\u009C\u0093Fi\u00E2\u009C\u0093 Rre\u00E2\u0097\u0086eHT\u00E2\u0097\u0086\u00E2\u0087\u00A0 re +(R\u0000 re)ep3HT ,where i = 1,2. Notice that the asymptotic form of r(T,R) near R = re is the same for both R \u00EF\u00A3\u00BF reand R \u0000 re. Also, as one would expect, r(T,R) is linear in R near R = re. From the above we cancompute thatlimR!re \u00E2\u0088\u0082 r\u00E2\u0088\u0082R = ep3HT .Substituting the above in (3.33), we getlimR!re gRR =\u0000 1f (re)e2p3HT ,so that the limit of gRR as R ! re is well defined. Let us now rewrite the line element for LPcoordinates in (3.20) as ds2 = dT 2\u0000gRR dR2\u0000 r2(dq 2 + sin2 qdf 2) ,with gRR =8><>:\u0000 f (re)\u0000 f (r)f (re)[ f (re)\u0000 f (R)] for R 6= re,\u0000 1f (re)e2p3HT for R = re .493.3. Additional features of LP coordinatesUsing the above definition, we have a line element which is defined and continuous for all valuesof R > 0 and \u0000\u00E2\u0080\u00A2 < T < \u00E2\u0080\u00A2. As was shown in section 3.3.3, these values of LP coordinates coverthe whole region of SdS spacetime relevant to this thesis (i.e. the region covered by the coordinatecurves shown in figure 3.1). Thus we have a well-defined and continuous line element over the re-gion of spacetime considered in this thesis. This will be essential when computing the cosmologicaland black hole horizons in chapter 4.3.3.5 Killing symmetriesIt was claimed in section 3.3.3 that there is a fundamental ambiguity in the definition of the LPcoordinate T , and that one can add an arbitrary constant to this variable and still obtain essentiallythe same coordinate system. In this section, we show how this ambiguity in the definition of Tis related to one of the Killing symmetries of SdS spacetime, and how moving along the flowassociated with this Killing field maps the LP coordinate curves onto themselves, and thereforedoes not change the coordinate system in any fundamental way.The absence of the Schwarzschild coordinate t in the line element (2.4) allows us to deduce theexistence of the Killing vector x (t) = \u00E2\u0088\u0082t for SdS spacetime. The integral curves of this Killingfield are those for which the other three Schwarzschild coordinates (r,q ,f) are held constant. Tounderstand the role of this Killing symmetry in LP coordinates, we first introduce a new radialcoordinate r as r(R) = 1H ln\u00E2\u009C\u0093F2\u00E2\u009C\u0093 Rre\u00E2\u0097\u0086\u00E2\u0097\u0086 , (3.34)where F2 is defined in (3.26). In the above, and in everything that follows, we are restricting ourattention to R > re. Notice that this corresponds to \u0000\u00E2\u0080\u00A2< r < \u00E2\u0080\u00A2. The role of the Killing symmetryin SdS spacetime is the same for both R< re and R> re, so that considering only R> re is sufficient.Furthermore, the Killing symmetry does not map points with R < re to points with R > re or vice-versa, so that these two regions can be considered separately without any difficulty. From the above,503.3. Additional features of LP coordinateswe have dr = ( f (re)\u0000 f (R))\u00001/2 dR .Using the above, we can transform the line element (3.20) from LP coordinates into (T,r,q ,f)coordinates to get ds2 = dT 2\u0000 ( f (re)\u0000 f (r)) dr2\u0000 r2(dq 2 + sin2 qdf 2) , (3.35)where r = r(T,r). By combining (3.34) with (3.25), we find that r(T,r) is given byr(T,r) = reF\u000012 \u00E2\u0087\u00A3eH(r+T )\u00E2\u008C\u0098 .Since the line element (3.35) only depends on r and T through r(T,r), it follows that any transfor-mation of r and T that does not change r will be a symmetry of the spacetime. By examining theexpression for r(T,r) above, we have that for any constant T0, the transformationT ! T +T0,r ! r\u0000T0will leave r unchanged and will be a symmetry of the spacetime. The above symmetry transforma-tion is precisely the same as the one associated with the Killing field x (t) in Schwarzschild coordi-nates, as can be deduced by simply realizing that both have the same integral curves, namely thoseobtained by setting (r,q ,f) to constant values. Using (3.34), we can take the above transformationand find the equivalent transformation in LP coordinates. This givesT ! T +T0, (3.36)R ! reF\u000012 \u00E2\u009C\u0093F2\u00E2\u009C\u0093 Rre\u00E2\u0097\u0086e\u0000HT0\u00E2\u0097\u0086 . (3.37)Since the right hand sides of (3.36) and (3.37) depend only on T and R, respectively, the abovetransformation takes the LP coordinate curves T = constant and R = constant and maps them ontothemselves. Thus the LP coordinate system that we have created is in fact a family of coordinatesystems, all of which have the same line element as given by (3.20), and which are all related by the513.3. Additional features of LP coordinatestransformation given above. Which of these coordinate systems we choose is arbitary. This choiceof coordinates is the freedom in shifting the coordinate T to we alluded to in section 3.3.3. Notethat this freedom is still present even after one has chosen the values for the constants T \u00E2\u0087\u00A4 and R\u00E2\u0087\u00A4introduced in section 3.1.3.3.6 Spacelike hypersurfaces: intrisinsic geometryLet us consider the spacelike hypersurfaces T = constant. The line element on these hypersurfacesis found by setting dT = 0 in (3.20). This givesds2 =\u0000 f (re)\u0000 f (r)f (re) [ f (re)\u0000 f (R)]dR2\u0000 r2(dq 2 + sin2 qdf 2) . (3.38)Now consider using the coordinates (r,q ,f) on the spacelike hypersurface T = constant, where ris the Schwarzschild radial variable first introduced in (2.4). When used as a coordinate on thehypersurface T = constant, r is a spacelike variable (this would not be the case if one attempted tocombine the variable r with the time coordinate T from LP coordinates into a \u00E2\u0080\u009Chybrid\u00E2\u0080\u009D coordinatesystem (T,r,f ,q); in such a case the variable r would be timelike at some locations in spacetime,much as it is in Schwarzschild coordinates). From (3.17), we havedR = \u00EF\u00A3\u00BF f (re)\u0000 f (r)f (re)\u0000 f (R)\u0000\u00001/2 dr .Substituting the above into (3.20), we getds2 =\u0000 1f (re)dr2\u0000 r2(dq 2 + sin2 qdf 2) . (3.39)We recognize the above as the line element for a three dimensional cone with opening angle ofb = arcsin\u0000 f (re)1/2\u0000. Unlike the familiar two dimensional cone, this three dimensional cone is notlocally intrinsically flat in the sense of having a vanishing Riemann curvature tensor. In fact the523.3. Additional features of LP coordinatesRiemann tensor can be calculated to beRqfqf = 3r2(sin2 q)e2/3 ,where it is understood that there are other non-zero components of the Riemann tensor, related tothe above by an appropriate shuffling of indices. The associated Ricci scalar isR =\u00006r2 e2/3 ,where recall that e = HM. The above reveals a curvature singularity at r = 0. Recognizing that thespacelike hypersurfaces are not flat will be important in section 4.4.3, where we find that the latetime shape of the horizons is not perfectly spherical, in spite of the fact that the spacetime at latetimes is well approximated by deSitter spacetime. This non-sphericity of the horizon is an artifactof choosing spacelike hypersurfaces which are not intrinsically flat, as will be explained.3.3.7 Spacelike hypersurfaces: polar coordinatesIn this section we describe the coordinates which will be used on the spacelike hypersurfaces T =constant when analyzing and illustrating the horizon in chapter 4. The horizon we will find canbe thought of as a series of 2-surfaces, each living on one of the hypersurfaces T = constant. Wewill use (r,f ,q) as coordinates on these hypersurfaces, leading to the line element (3.39), andthe 2-surfaces will be visualized as living in 3d Euclidean space, with (r,f ,q) playing the role ofspherical coordinates. This way of representing the horizon introduces metrical distortions sincethe hypersurfaces T = constant have the geometry of a cone and are not truly Euclidean. However,visualizing the horizon as a series of 2-surfaces in 3d Euclidean space has the advantage that it easilyallows one to illustrate the basic change in horizon topology that occurs during merger. As well,it will be useful for identifying and illustrating key features of the horizons, such as the presenceof caustic points. In our graphical representation we will also suppress the q coordinate by settingq = p/2, so that the horizon will be illustrated as a series of curves in the Euclidean plane, with533.3. Additional features of LP coordinates(r,f) as polar coordinates. The spherical symmetry of SdS spacetime allows us to recover anynull geodesic from one constrained to q = p/2 by a simple rotation, and the full 2-surface can bevisualized by rotating the curves (such as those in figure 4.4) about the x-axis. Another advantage ofusing (r,f ,q) instead of (R,f ,q) coordinates is that the line element (3.39) is considerably simplerthan (3.38), and this will greatly simplify calculations of horizon area in chapter 5.54Chapter 4Structure of Merging Black Hole andCosmological Horizons4.1 IntroductionThe observer dependent causal horizonwe would like to find is defined as the boundary of the causalpast of the trajectory of the observer moving radially away from the black hole. The trajectory ofthe observer is shown on the effective potential diagram in figure 3.3a and the causal horizon isillustrated schematically in figure 2.10b. Our objective is to approximate this causal horizon using afamily of null geodesics called the null generators. Before performing such calculations, we outlinea justification for our procedure, as well as a strategy for computing the null generators.4.1.1 Null generators of the horizonWe first introduce some notation. Let g(t) be the timelike geodesic trajectory of our observer and letg(R) be the image of the real line under the mapping g(t). g(R) is the set of spacetime events thatmake up the world line of our observer. Let J\u0000(g(R)) and I\u0000(g(R)) be the causal and chronologicalpasts of g(R), respectively, and letH be the horizon, so thatH \u00E2\u008C\u0098 Bd(J\u0000(g(R))).554.1. IntroductionNext we give a definition of the null generators, and justify their use in describing the horizon.Let us define the null generators of H to be those null geodesics which lie entirely within H .To understand why null generators give a complete description of the horizon, we invoke theorem8.1.6 of [32], and apply it to the closed set g(R). Notice that g(R) is a closed set since Bd(g(R)) =g(R). According to the theorem, every point in S \u00E2\u008C\u0098 Bd(I\u0000(g(R)))\u0000 g(R) lies on at least onenull geodesic which is contained entirely in S. Since the set S is precisely the horizon H , thisguarantees that the null generators give a complete description ofH . The fact that S =H followsfrom Bd(J\u0000(g(R))) = Bd(I\u0000(g(R))) and g(R)\Bd(I\u0000(g(R))) = /0, where the former is discussedat the bottom of page 191 of [32], and the latter follows from the fact that g(R)\u00E2\u0087\u00A2 I\u0000(g(R)), alongwith the fact that I\u0000(g(R)) is an open set (page 190 of [32]).The third step is to develop a strategy for computing the null generators. This strategy will dependcrucially on the late time behavior of the generators. The basic idea will be to use knowledge of thelate time behavior of the generators as \u00E2\u0080\u009Cinitial conditions\u00E2\u0080\u009D for the null geodesic equations.Before developing our strategy, it is useful to get a better understanding of the future behavior ofthe generators. We once again use theorem 8.1.6 of [32], this time using the second part of thetheorem and applying it to the null geodesics associated with the closed set g(R). As discussed inthe previous paragraph, these null geodesics are the null generators of the horizon. The theoremstates that these generators are either future inextendible or have a future endpoint on g(R). Letus show that none of these generators have a future endpoint on g(R), and so by the theorem justcited, they are in fact all future inextendible. The proof is by contradiction. Let C (l ) be a generatorof the horizon, and suppose it has a future endpoint p 2 g(R). Since g(R) \u00E2\u0087\u00A2 I\u0000(g(R)), we havep2 I\u0000(g(R)), and therefore, p2 Int(J\u0000(g(R))) (where we have used I\u0000(S) = Int(J\u0000(S)); page 191of [32]). Since Int(J\u0000(g(R))) is an open set (page 190 of [32]) and p 2 Int(J\u0000(g(R))), there is aneighborhood O of the point p such that O \u00E2\u0087\u00A2 Int(J\u0000(g(R))). By the definition of future endpoint(page 193 of [32]), there must be a value l0 such that C (l ) 2 O for all l > l0. We have thusshown that C (l ) 2 Int(J\u0000(g(R))) for all l > l0, which contradicts the fact that we must alwayshave C (l ) 2H = Bd(J\u0000(g(R))). That is, it contradicts the fact that a generator, by definition, iscontained entirely on the horizonH , as discussed in the paragraph above. We have thus established564.1. Introductionthat the generators of the horizon are future inextendible. By definition, this means that they do nota have a future endpoint (note that this is a stronger statement than claiming that they do not have afuture endpoint in g(R); hence the power of the theorem). Formally, the absence of a future endpointfor a generator C means that for every point p 2 C , there exists a neighborhood O of p such thatgiven any l0 2 R, it is the case that g(l ) /2 O for some l > l0. Less formally, the absence of afuture endpoint means that the generators either run into a singularity or continue indefinitely. Bycontrast, a future endpoint would mean that a null geodesic was stopped abruptly, and could in somesense be extended by continuing where it left off (these issues are discussed in detail on page 193of [32])). The horizon generators which we are interested in here are inextendible due to the factthat they continue indefinitely towards future timelike/null infinity, as illustrated by the 45 degreeyellow lines in the Penrose diagram (figure 2.10b).In order to implement our strategy for computing the generators, we need a more precise undertand-ing of the future behavior of the generators. Having established that the family of null generatorsare future inextendible, we can use the Penrose diagram (figure 2.10b) to conclude that they willsatisfy r(l = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2, where l is an affine parameter and r(l ) is the Schwarzschild coordinateof such a curve. If we use the time coordinate T from the LP coordinates developed in chapter 3as the parameter along the null generators, the condition r(l = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2 becomes r(T = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2instead (this can be seen by looking at the spacelike coordinate curves of the LP coordinate system,shown in figure 3.1b). We can gain further insight into the generators which make up the horizonby using our knowledge of SdS spacetime as r ! \u00E2\u0080\u00A2. As discussed in section 2.6, SdS spacetimeapproaches deSitter spacetime in the limit that r ! \u00E2\u0080\u00A2. We know the shape of the causal horizonfor any geodesic observer in deSitter spacetime consists of a closed surface surrounding that ob-server. Note that by the shape of the causal horizon, here we mean the intersection of a spacelikehypersurface with the causal horizon H . Let us denote the spacelike hypersurfaces T = constantby ST and the intersection with H by ST = ST \H . Based on our knowledge of the shape of thecausal horizon in deSitter spacetime, we expect ST to approach a closed surface surrounding ourobserver as T ! \u00E2\u0080\u00A2. Such a surface is shown in the last frame of figure 4.4 (it is a curve since onedimension is suppressed). It is simply the cosmological horizon surrounding the observer once theblack hole and observer have drifted sufficiently far apart. The null generators which make up this574.1. IntroductionFigure 4.1: The \u00E2\u0080\u009Ctrouser\u00E2\u0080\u009D shaped horizon of two merging black holes. The green curves are horizongenerators which enter through caustic points on the \u00E2\u0080\u009Cinseam\u00E2\u0080\u009D of the trouser. Notice that thesegenerators cross at the caustic points.closed surface will satisfy r(T =\u00E2\u0080\u00A2) =\u00E2\u0080\u00A2 and f(T =\u00E2\u0080\u00A2) = 0. Furthermore, since the null generatorswhich make up the horizon are all future inextendible, they will all reach this final closed surface asT ! \u00E2\u0080\u00A2.We now have a strategy for finding the null generators which make up the horizon: set up thenull geodesic equations, and look for a family of solutions which form a closed surface as T !\u00E2\u0080\u00A2 and satisfy r(T = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2 and f(T = \u00E2\u0080\u00A2) = 0. Since the null geodesic equations cannot besolved analytically except in certain special circumstances, we use a combination of analytical andnumerical methods. In section 4.4.3, analytical methods are used to find an approximate seriessolution for the desired family of geodesics in the limit that T ! \u00E2\u0080\u00A2. This approximation is thenused to set initial conditions for the numerical solution of the equations. The numerical methodused is discussed in 4.3.3 and the numerical results are presented in 4.4.1.Let us conclude this section by making some additional remarks about null generators and thestructure of the causal horizon. As discussed, H is a null hypersurface generated by a family offuture inextendible null geodesics. It thus enjoys the same properties as a traditional black hole584.1. Introductionevent horizon. In particular, we have the following (page 203 of [25]): (i) new generators can enterthe horizon through special points on the horizon called caustic points, (ii) once a generator entersthe horizon, it can never leave, and (iii) through each point onH , there is either a unique generatorgoing through that point, or it is a caustic point where new generators enter. A well known exampleof caustics in the context of black holes are those on the \u00E2\u0080\u009Cinseam\u00E2\u0080\u009D of the \u00E2\u0080\u009Ctrouser\u00E2\u0080\u009D shaped horizonassociated with the head-on merger of two non-rotating black holes, as shown in figure 4.1. Noticehow generators cross at the caustic points. In the merger of black hole and cosmological horizonsstudied in this thesis, there is also a set of caustic points. These are illustrated in the diagram shownin figure 4.5. There is some resemblance with figure 4.1, the main difference being that at earlytimes we have one horizon (the black hole) inside a larger horizon (the comosmological horizon).One of the focal points of this chapter will be the analysis of the mathematical structure of thecaustic points associated with the merging of the black hole and cosmological horizons. Anotherfocal point in the analysis of the horizons will be a precise determination of the merger point wherethe black hole and cosmological horizon first touch. The analysis of the merger point will be donein section 4.5.4.1.2 The family of radial geodesic observersSo far we have been describing the causal horizonH for any observer drifting away from the blackhole along a radial geodesic satisfying r(t = \u0000\u00E2\u0080\u00A2) = re and r(t) > re. The observer\u00E2\u0080\u0099s trajectory isshown on the effective potential diagram in figure 3.3 and on the Penrose diagram in figure 2.10b.This trajectory is part of a family of observers all drifting away from the black hole along radialgoedesics (this family is plotted on the Penrose diagram in figure 4.2a). Let us prove that the familyof causal horizons associated this family of observers are all equivalent, in the sense that any onecan be obtained from any other by applying a symmetry transformation of the spacetime. Noticefirst that this family of geodesics precisely coincides with the timelike coordinate curves in the LPcoordinate system which satisfy r > re (these are the curves to the right of the green curve in figure3.1a). As was shown in section 3.3.5, the flow associated with the Killing vector field x (t) mapsthis family of curves onto itself. It follows that this family of geodesics, and therefore also their594.1. Introduction(a)(b)Figure 4.2: a) The family of radial geodesic observers drifting away from r = re. Notice that thisset of trajectories was used in the construction of the timelike coordinate curves of the LP coordi-nate system (figure 3.1a). b) Illustration of the causal horizons (yellow) for three radial geodesicobservers. The observer trajectories, as well as the causal horizons, are related by the symmetrytransformation associated with the Killing vector x (t).604.1. Introductioncausal horizons, are related to each other by the flow associated with x (t), and thus are related by asymmetry of the spacetime. This family of horizons is illustrated in figure 4.2b.4.1.3 Non-radial and non-geodesic observersMore generally, the observers moving along radial geodesic trajectories are part of an even largerfamily of observers moving along trajectories which satisfy the following two requirements:(i) rb < r(t =\u0000\u00E2\u0080\u00A2) < rc,(ii) the trajectory approaches one of the outward radial geodesics as t ! \u00E2\u0080\u00A2 (as measured using theproper distance along the hypersurfaces T = constant, for example).Physically, conditions (i)-(ii) require the observer to start close to the black hole, then move alonga possibly non-radial and/or accelerated trajectory, and finally end up caught up in the Hubble flowof spacetime. We will now sketch a proof that the causal horizons for the observers in the family oftrajectories satisfying condition (i)-(ii) are all equivalent, in the sense of being related by a Killingsymmetry. As was explained at the beginning of this section, the causal horizon H of any radialgeodesic observer is calculated by propagating null geodesics backwards in time, starting from a setof light rays forming a closed surface surrounding the observer. Observers statisfying condition (ii)above have the same late time behavior as radial geodesic observers, and so their causal horizons atlate times are also expected to have the same behavior: a closed surface surrounding the observer,corresponding to a deSitter horizon for a homogeneous expanding universe. This deSitter horizonis the cosmological horizon in the steady state universe. Since the late time behavior of these causalhorizons is the same, and this late time behavior uniquely determines the causal horizon for all priortimes, the causal horizon of the observers satisfying (i)-(ii) above are expected to match those ofthe radial geodesic observers. In the preceding, we have attempted to motivate the idea that it isthe late time behavior of the observers\u00E2\u0080\u0099 trajectories which determines the causal horizon, so that the614.1. Introductionhorizons calculated in this thesis apply to a broader class of observers than simply those which aremoving on radial geodesic trajectories. We have not given a rigourous proof however.An interesting further generalization of the above considerations would be to ask the followingquestion: are the casual horizons of all observers reaching r = \u00E2\u0080\u00A2 the same, in the sense of beingrelated by a Killing symmetry transformation? This would include trajectories which are non-geodesic and non-radial for all times. If the casual horizon for such observers are indeed all thesame, one could give a complete categorization of the possible casual horizons of observers in SdSspacetime (when considering the part of the spacetime covered by LP coordinates; see figure 3.1).Observers with rb < r(t)< rc for all t would have the familiar spherical Killing horizons r = rb andr = rc (as discussed in section 2.4), and observers with r(t =\u00E2\u0080\u00A2) =\u00E2\u0080\u00A2would have the causal horizonsdiscussed in this thesis. Note that all observers which do not fall into the black hole and reach thesingularity at r = 0 must in fact be in one the two categories just mentioned. This can be deducedby looking at the Penrose diagram. The question of the causal horizon of observers moving alongnon-radial and non-geodesic trajectories for all times will not be explored in this thesis, however.Instead it will be addressed in a forthcoming publication.4.1.4 Chapter organizationThis chapter is organized as follows. In section 4.2, we set up the null geodesic equations whichwill be used to calculate the causal horizon H described above. This is accomplished by firstexploiting the symmetries of SdS spacetime to find the null geodesic equations in Schwarzschildcoordinates (section 4.2.1). These equations contain conserved quantitites which can be used tonaturally incorporate the initial direction of propagation of a backwards light ray. The equations inSchwarzschild coordinates are then used to find the equations for the functions r(T,a) and f(T,a),where T is the time variable from LP coordinates, r is the radial Schwarzschild coordinate, f isthe azimuthal angle common to both of these coordinate systems (the angle q will be set to p/2and thus will be irrelevant), and a is an angle parametrizing the initial direction of propagation of a624.1. Introductionlight ray (section 4.2.2). As was discussed in section 3.3.7, when considering a single hypersurfaceT = constant, the illustration and analysis of the horizon structure is particularly simple if instead ofusing LP coordinates (R,f ,q), one instead uses (r,f ,q) as coordinates on these three dimensionalhypersurfaces.In section 4.3, we outline the method used for solving the null geodesic equations. First, the equa-tions for r(T ) and f(T ) are modified by compactifying both T and r (section 4.3.1). This allowsfor a more efficient numerical integration of the equations, and gives a method for dealing with thetricky issue of setting up initial conditions for light rays whose starting point is effectively futurenull/timelike infinity. The issue of initial conditions (or more precisely, asymptotic requirements asT ! \u00E2\u0080\u00A2) is dealt with in section 4.3.2. The numerical method used is discussed in section 4.3.3.In section 4.4, we analyze and illustrate the overall shape of the horizons. First, the results of thenumerical integration are used to display the qualitative structure of the horizons by plotting theirshape for various times (section 4.4.1). Analytical methods are then used to study in detail variousaspects of the global horizon structure. We analyze the motion of lightlike geodesics using effectivepotentials, and categorize their possible behavior based on their initial conditions (section 4.4.2).This analysis corroborates our numerical results, and identifies key quantities whose calculation isuseful for later analytical results. Next we consider the structure of the horizons at late times (section4.4.3). This allows us to set up an explicit formula which can be used to set the initial conditionswhen numerically integrating the null geodesic equations (note that these initial conditions will infact already have been used for numerical purposes in section 4.4.1, even though section 4.4.3 comesafter section 4.4.1).Finally, we focus on the analysis of the location of the merger point when the two horizons firstmerge (section 4.5). We give an explicit formula for the location of the merger point, in the limitthat e ! 0 (section 4.5).634.2. Null geodesic equationsFigure 4.3: The effective potential diagram for null geodesics. The red curve is a plot of V (r) \u00E2\u008C\u0098Ve f f (r)sin2a = 1r2 \u00001\u0000 2Mr \u0000. The green and blue curves are schematic illustrations of trajectories withand without turning points, respectively. The existence of turning points depends on the \u00E2\u0080\u009Cinitialconditions\u00E2\u0080\u009D, as characterized by the value of the parameter a . The precise conditions are sin2a \u00EF\u00A3\u00BF27H2M2 (no turning points) and sin2a > 27H2M2 (turning point). The parameter values used toproduce the plot are M = 1, 27H2M2 = 14 , with sin2a \u00E2\u0087\u00A1 3/4 and sin2a \u00E2\u0087\u00A1 2/9 for the green andblue trajectories, respectively.4.2 Null geodesic equationsIn the following section, we exploit the symmetries of SdS spacetime to find the equations for t(l ),r(l ) and f(l ), where (t,r,f) are the usual Schwarzschild coordinates and l is an affine parameteralong a backwards null geodesic. In section 4.2.2, we use the equations for r(l ) and t(l ) to find theequation for T (l ), where T is the time variable in LP coordinates. This equation is then combinedwith the equations for r(l ) and f(l ) to yield the equations for r(T ) and f(T ).644.2. Null geodesic equations4.2.1 Schwarzschild coordinatesAs discussed in section 2.1, SdS spacetime possesses four Killing vectors. The vectors x (f), x (1),x (2) are associated with spherical symmetry and the vector x (t) is associated with the stationarityof the spacetime. The conservation law associated with x (f) can be used to deduce that there is afamily of geodesics whose trajectories lie entirely on the hypersurface q = p/2. The symmetriesassociated with x (1) and x (2) are then used to conclude that all other geodesics can be obtained fromthis family by applying the transformations associated with the flow corresponding to x (1) and x (2).The upshot of this is that only the geodesics with q = p/2 need to be found, and without loss ofgenerality we can set q = p/2 immediately in the Schwarzschild line element. Next, we use theconservation laws associated with the two Killing vectorsx (t) = (1,0,0,0) = \u00E2\u0088\u0082\u00E2\u0088\u0082 t ,x (f) = (0,0,1,0) = \u00E2\u0088\u0082\u00E2\u0088\u0082f .Letting (t(l ),r(l ),f(l )) be the coordinates of our null geodesic trajectory, the conservation lawsassociated with these Killing vectors aredtdl f (r) = e = constant,dfdl r2 = l = constant .The requirement that the trajectory be null leads tof (r)\u00E2\u009C\u0093 dtdl \u00E2\u0097\u00862\u0000 f (r)\u00001\u00E2\u009C\u0093 drdl \u00E2\u0097\u00862\u0000 r2\u00E2\u009C\u0093dfdl \u00E2\u0097\u00862 = 0 .654.2. Null geodesic equationsCombining the above three equations, we have the equations for the Schwarzschild coordinates ofa null geodesic trajectory: dtdl = ef (r) ,dfdl = lr2 ,\u00E2\u009C\u0093 drdl \u00E2\u0097\u00862 +\u00E2\u009C\u0093 lr\u00E2\u0097\u00862 f (r) = e2.It will turn out to be convenient to eliminate the parameters l and e in favor of the single parametera 2 [0,2p), defined through the relationtana = H\u00E2\u009C\u0093 le\u00E2\u0097\u0086 , (4.1)where H =qL3 as before. The motivation for introducing the parameter a is that in the case M = 0it can be interpreted as an angle parametrizing the intersection of the horizon with q = p/2 (herewe use the term \u00E2\u0080\u009Chorizon\u00E2\u0080\u009D to mean the 2-surface living on a spacelike hypersurface T = constant,as opposed to the full three dimensional hypersurface). In the case M 6= 0, the interpretation of a ismore subtle, but it nevertheless holds true that the values a 2 [0,2p) parametrize the curve formedby the horizon\u00E2\u0080\u0099s intersection with q = p/2 (see figure 4.4 for examples of such curves). We willreturn to the interpretation of the parameter a in section 4.4.3.Rescaling the affine parameter l so that l ! \u00E2\u0087\u00A3l2 + e2H2\u00E2\u008C\u0098l and replacing l and e in favor of a , thegeodesic equations become dtdl = \u0000H cosaf (r) , (4.2)dfdl = \u0000sinar2 , (4.3)\u00E2\u009C\u0093 drdl \u00E2\u0097\u00862 +Ve f f (r) = H2, (4.4)where Ve f f (r) = sin2ar2 \u00E2\u009C\u00931\u0000 2Mr \u00E2\u0097\u0086 . (4.5)664.2. Null geodesic equationsThe equation for r(l ) has been written in effective potential form. From the effective potentialdiagram (figure 4.3) we see that for some values of a , light rays reaching r = \u00E2\u0080\u00A2 will have a turningpoint. We have two cases:(I) If sin2a \u00EF\u00A3\u00BF 27H2M2, then there are no turning points and drdl > 0 for all l .(II) If sin2a > 27H2M2, then there is a turning point at some l = l \u00E2\u0087\u00A4 and we have drdl < 0 for l < l \u00E2\u0087\u00A4and drdl > 0 for l > l \u00E2\u0087\u00A4 .We can use knowledge of the two cases above to tranform equation (4.4) into a first order equation,keeping in the mind the presence of the turning point at l = l \u00E2\u0087\u00A4 in the second case. This givessin2a \u00EF\u00A3\u00BF 27H2M2 =) drdl =qH2\u0000Ve f f (r) , (4.6)sin2a > 27H2M2 =) drdl =8><>:\u0000pH2\u0000Ve f f (r) ifl < l \u00E2\u0087\u00A4pH2\u0000Ve f f (r) ifl > l \u00E2\u0087\u00A4 (4.7)Alternatively, equation (4.4) can be transformed into a set of coupled first order equations:drdl = w, (4.8)dwdl = \u000012V 0e f f (r) . (4.9)The equations (4.2)-(4.3), combined with either (4.6)-(4.7) or (4.8)-(4.9) and appropriate initial con-ditions, form a complete set of equations that can be used to calculate the Schwarzschild coordinatesof a null geodesic in SdS spacetime. In this thesis, we will use either of these sets of equations, de-pending on the circumstances. First, however, they have to be transformed into equations whichgive the Schwarzschild coordinates (r,f) of a null geodesic as a function of LP coordinate time T .This will be done in the next section.674.2. Null geodesic equations4.2.2 LP coordinates and spherical coordinatesWe are interested in illustrating and analyzing the structure of the horizons. Although this canbe done in the LP coordinates developed in chapter 3, it will turn out that it is easier to insteaduse the following two step process. First, we restrict our attention to a single spacelike hypersur-face T = constant. Second, on this hypersurface we use the coordinates (r,f ,q), where r is theradial Schwarzschild coordinate. Notice that when restricting attention to a single hypersurfaceT = constant, the coordinate r is always a spacelike coordinate, even though it is a timelike coordi-nate for r < rb and r > rc when viewed as part of the Schwarzschild coordinates (t,r,f ,q). Usingthe coordinates (r,f ,q) on the spacelike hypersurfaces T = constant has two advantages. First, theline element instrinsic to these hypersurfaces takes the particularly simple form (3.39) in these co-ordinates. This greatly simplifies the analysis of the horizons. Second, as discussed in section 3.3.7,we can interpret the coordinates (r,f ,q) as spherical coordinates on the hypersurfaces and use thisinterpretation to illustrate the shape of the horizons. Notice that by \u00E2\u0080\u009Chorizons\u00E2\u0080\u009D here we mean thesurfaces which are formed by the intersection of a spacelike hypersurface T = constant with the fullnull hypersurface which is the boundary of the causal past of the observer. In other words, horizonsare the snapshots in time of the full spacetime horizon.As explained in section 3.3.7, the horizons can be visualized as being formed by the surface obtainedwhen rotating the curves such as those in figure 4.4. Therefore it suffices to use (r,f) as polarcoordinates in the Euclidean plane to analyze and illustrate the horizons. Since we will be usingthe coordinates (r,f) to analyze and illustrate the horizons on the hypersurfaces T = constant, wewant to find the equations for r(T ) and f(T ). Notice the slight abuse of notation since we haveused notation of the form r(\u00C2\u00B7) for both r(l ) and r(T ), even though these are different functions.r(T ) and r(l ) give the Schwarzschild coordinates of a null geodesic as a function of LP coordinatetime T or affine parameter l , respectively. If the 1-to-1 function T (l ) for a null geodesic is known,either one of these functions can be used to find the other (it is shown below that T (l ) is strictlyincreasing and therefore 1-to-1). The same abuse of notation applies to f(l ) and f(T ). In caseswhere they may be ambiguity as to which of these functions is involved in an equation, it will be684.2. Null geodesic equationsexplicitly mentioned.The equations for r(T ) and f(T ) are obtained by combining either the first order equations (4.6)-(4.7), or the coupled first order equations (4.8)-(4.9) for r(l ), and the equation (4.3) for f(l ) withthe equation for T (l ). The procedure is slighly different in the two cases, and so we consider themseparately. In either case, the equation for T (l ) is obtained by using the transformation lawdTdl = \u00E2\u0088\u0082T\u00E2\u0088\u0082 r drdl + \u00E2\u0088\u0082T\u00E2\u0088\u0082 t dtdl . (4.10)First order equationsSubstituting (4.2) and (4.6)-(4.7) for dt/dl and dr/dl into (4.10), and using (3.18)-(3.19) to obtainthe Jacobian coefficients \u00E2\u0088\u0082T/\u00E2\u0088\u0082 r and \u00E2\u0088\u0082T/\u00E2\u0088\u0082 t, we getdTdl = \u0000H f (re) 12 cosa\u0000 sgn\u0000 drdl \u0000( f (re)\u0000 f (r))1/2 sgn(r\u0000 re)pH2\u0000Ve f f (r)f (r) . (4.11)As will be shown in section 4.2.2 below, we always have dTdl > 0, so that T (l ) is a strictly increasingfunction. Because of this, for every value of T we have a unique value of l , and the first orderequations for r(T ) and f(T ) are obtained by dividing the first order equations for r(l ) and f(l ) in(4.6)-(4.7) and (4.3) by dTdl above. This yields the following equations for r(T ) and f(T ):drdT = \u0000 f (r)pH2\u0000Ve f f (r)sgn\u0000 drdT \u0000H f (re) 12 cosa +( f (re)\u0000 f (r)) 12 sgn(r\u0000 re)pH2\u0000Ve f f (r) , (4.12)dfdT = \u00E2\u009C\u0093sinar2 \u00E2\u0097\u0086 f (r)H f (re) 12 cosa + sgn\u0000 drdT \u0000( f (re)\u0000 f (r)) 12 sgn(r\u0000 re)pH2\u0000Ve f f (r) . (4.13)It will be useful to rewrite the above equations more succinctly asdrdT = G\u00C2\u00B1(r), (4.14)dfdT = K\u00C2\u00B1(r) . (4.15)694.2. Null geodesic equationswith the plus and minus signs corresponding to the cases drdT > 0 and drdT < 0, respectively. Theabove equations are most useful when performing analytical computations of the horizons. Forexample, they will be used in analyzing the late time behavior of the horizons in section 4.4.3, orwhen finding the merger point location in section 4.5. When solving the null geodesic equationsnumerically, these equations break down near the turning points, and so instead we will use thecoupled first order equations in (4.8)-(4.9). This system of equations are converted into equationsfor r(T ) and f(T ) instead of r(l ) and f(l ) in the next section.Coupled first order equationsWe once again substitute (4.2) for dt/dl and use (3.18)-(3.19) to obtain the Jacobian coefficients\u00E2\u0088\u0082T/\u00E2\u0088\u0082 r and \u00E2\u0088\u0082T/\u00E2\u0088\u0082 t in the equation for dT/dl , given by (4.10) above. The only difference is thatnow we use dr/dl = w, as given by the first of the two coupled equations (4.8)-(4.9) for r(l ), andobtain dTdl = \u0000H f (re) 12 cosa\u0000w( f (re)\u0000 f (r))1/2 sgn(r\u0000 re)f (r) .As before, the equations for r(T ) and f(T ) are obtained by simply dividing the equations for r(l )and f(l ) by the above expression for dT/dl . This givesdrdT = \u0000w f (r)H f (re) 12 cosa +w( f (re)\u0000 f (r))1/2 sgn(r\u0000 re) , (4.16)dwdT = 12V 0e f f (r) f (r)H f (re) 12 cosa +w( f (re)\u0000 f (r))1/2 sgn(r\u0000 re) , (4.17)dfdT = \u00E2\u009C\u0093sinar2 \u00E2\u0097\u0086 f (r)H f (re) 12 cosa +w( f (re)\u0000 f (r)) 12 sgn(r\u0000 re) . (4.18)The above set of three coupled equations will be solved numerically to find the coordinates of thenull geodesics, and these coordinates will be used to illustrate the shape of the horizons in section4.4. The details of the numerical integration process will be discussed in section 4.3.3.704.2. Null geodesic equationsProof that dTdl > 0Let N(r) be the numerator of the right hand side of (4.10):N(r) =\u0000H f (re) 12 cosa\u0000 sgn\u00E2\u009C\u0093 drdl \u00E2\u0097\u0086( f (re)\u0000 f (r))1/2 sgn(r\u0000 re)qH2\u0000Ve f f (r) . (4.19)We will show that r > rc ) N(r) < 0 , (4.20)rb < r < rc ) N(r) > 0 . (4.21)Combining the above with (see figure 2.6 for a plot of f (r))r > rc ) f (r) < 0 , (4.22)rb < r < rc ) f (r) > 0 , (4.23)in (4.6)-(4.7) and (4.3), it will follow that the right hand side of (4.10) is always positive, providedthat r > rb. As will be discussed in section 4.4.2, the null geodesics we are interested in satisfyr(T ) > rb for all T . We can thus be assured that the right hand side of (4.10) is always positive,as claimed, provided that (4.20)-(4.21) hold, and we are restricting attention to the null geodesicsconsidered in this thesis.Let us first prove (4.20). Throughout this part of the proof, it is understood that we are taking r > rc.First note thatsgn\u00E2\u009C\u0093 drdl \u00E2\u0097\u0086 = 1,sgn(r\u0000 re) = 1 .The first equation follows from the fact that there are no turning points with r > rc, which will beproven in section 4.4.2 below, while the second of these equations follows from the fact that rc > re,714.2. Null geodesic equationswhich can be seen in figure 2.6. Using the above, (4.19) becomesN(r) =\u0000H f (re) 12 cosa\u0000 ( f (re)\u0000 f (r))1/2qH2\u0000Ve f f (r) . (4.24)In addition, (4.22) implies that the following inequalities hold:( f (re)\u0000 f (r))1/2 > f (re) 12 , (4.25)rH2 cos2a\u0000 1r2 f (r)sin2a > H | cosa | . (4.26)Now we use the definition of the effective potential in equation (4.5) to convert (4.26) into thefollowing inequality:qH2\u0000Ve f f (r) > H | cosa | . (4.27)Applying (4.25) and (4.27) to (4.24), we obtainN(r) <\u0000H f (re) 12 (cosa+ | cosa |) .The right hand side above vanishes if cosa \u00EF\u00A3\u00BF 0 and is negative if cos> 0, so that we have establishedthat N(r) < 0 if r > rc.Next we prove (4.21). Throughout the proof, it is understood that we are taking rb < r < rc. As willbe shown in section 4.4.2, only null geodesics with cosa < 0 ever cross r = rc in order to satisfyrb < r(T ) < rc for some value of T . We can therefore assume that cosa < 0.First, we use (4.23) to obtain the inequalitiesrH2 cos2a\u0000 1r2 f (r)sin2a < H | cosa |,( f (re)\u0000 f (r))1/2 < f (re) 12 . (4.28)Using the definition of the effective potential in equation (4.5) and the fact that cosa < 0, the first724.3. Null geodesic calculationsinequality above becomesqH2\u0000Ve f f (r) <\u0000H cosa . (4.29)Multiplying the inequalities (4.28) and (4.29), we obtain\u0000H f (re) 12 cosa\u0000 ( f (re)\u0000 f (r))1/2qH2\u0000Ve f f (r) > 0 .The next step is to consider sgn\u0000 drdl \u0000sgn(r\u0000 re). Suppose sgn\u0000 drdl \u0000sgn(r\u0000 re) = \u00001. Then sincecosa < 0, both terms in N(r) are positive and N(r) > 0 follows immediately. If sgn\u0000 drdl \u0000sgn(r\u0000re) = \u00001, then N(r) > 0 follows from the above inequality. Thus we have shown that N(r) > 0when rb < r < rc. Since we have now shown that both (4.20) and (4.21) hold for the null geodesicsin this thesis, this completes the proof that dT/dl > 0.4.3 Null geodesic calculationsIn what follows, we outline the procedure used to solve the null geodesic equations (4.12)-(4.13)numerically.4.3.1 Compactification of the time variableFor the purposes of solving the null geodesic equations numerically, it is useful to compactify thetime variable T . We will denote the compactified variable T\u00CB\u0086 . Compactification serves two purposes.First, it is easier to implement the asymptotic requirements as T ! \u00E2\u0080\u00A2 once the variable T is com-pactified. Second, a wise choice of compactification of the time variable effectively introduces avariable step size in time. This is useful since the horizons undergo the largest change in shape andarea in a O( 1H ) time interval about the merger point, where H = pL/3 is the Hubble parameter.734.3. Null geodesic calculationsWith this in mind, we introduce the compactified time variableT\u00CB\u0086 = tanh\u00E2\u009C\u0093HT2\u00E2\u0097\u0086. (4.30)The infinte interval T 2 (\u0000\u00E2\u0080\u00A2,\u00E2\u0080\u00A2) has been compactified to the finite interval T\u00CB\u0086 2 (\u00001,1).Consider either the equations (4.14)-(4.15) or the coupled equations (4.16)-(4.18) for r(T ) andf(T ). Let us convert these into equations for r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ). Transforming (4.14)-(4.15) using thechange of variables (4.30), we get the following equations for r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ):drdT\u00CB\u0086 = 2H(1\u0000 T\u00CB\u0086 2)G\u00C2\u00B1 (r) , (4.31)dfdT\u00CB\u0086 = 2H(1\u0000 T\u00CB\u0086 2)K\u00C2\u00B1 (r) . (4.32)Similarly, applying the same change of variables to the coupled equations (4.16)-(4.18), we obtainthe following set of coupled equations for r(T\u00CB\u0086 ), w(T\u00CB\u0086 ) and f(T\u00CB\u0086 ):drdT\u00CB\u0086 = \u00E2\u009C\u0093 2H(1\u0000 T\u00CB\u0086 2)\u00E2\u0097\u0086 \u0000w f (r)H f (re) 12 cosa +w( f (re)\u0000 f (r))1/2 sgn(r\u0000 re) , (4.33)dwdT\u00CB\u0086 = \u00E2\u009C\u0093 2H(1\u0000 T\u00CB\u0086 2)\u00E2\u0097\u0086 12V 0e f f (r) f (r)H f (re) 12 cosa +w( f (re)\u0000 f (r))1/2 sgn(r\u0000 re) , (4.34)dfdT\u00CB\u0086 = \u00E2\u009C\u0093 2H(1\u0000 T\u00CB\u0086 2)\u00E2\u0097\u0086\u00E2\u009C\u0093sinar2 \u00E2\u0097\u0086 f (r)H f (re) 12 cosa +w( f (re)\u0000 f (r)) 12 sgn(r\u0000 re) . (4.35)Next we consider the initial conditions (or more precisely, limiting conditions as T\u00CB\u0086 ! 1\u0000) that willaccompany these equations.744.3. Null geodesic calculations4.3.2 Asymptotic requirementsWe impose the following asymptotic requirements on r(T ) and f(T ) as T ! \u00E2\u0080\u00A2:r(T ) ! \u00E2\u0080\u00A2, (4.36)f(T ) ! 0 .The first requirement is a consequence of the fact that the null rays are the boundary of the causalpast of an observer reaching future null/timelike infinity at r = \u00E2\u0080\u00A2, as shown in figure 2.10b. Thesecond requirement follows from the fact that the light rays remain a finite distance from our ob-server as T ! \u00E2\u0080\u00A2, and the fact that the observer moves along a trajectory satisfying f(t) = 0 for allt . These requirements were discussed in more detail in section 4.1, where they were shown to resultfrom the expected late time behavior of the null geodesic generators of the causal horizon.Expressed in terms of the compactified time variable (4.30), these conditions becomer\u00CB\u0086(T\u00CB\u0086 = 1\u0000) = \u00E2\u0080\u00A2, (4.37)f(T\u00CB\u0086 = 1\u0000) = 0 .The above can in some sense be thought of as providing initial conditions for the equations (4.31)-(4.32). The main difference between the above and proper initial conditions is that the above areactually limiting conditions on r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ) as T\u00CB\u0086 ! 1\u0000.Although the conditions (4.37) are necessary conditions for any light ray trajectory which is partof the horizon, they are insufficient for setting up initial conditions when finding the trajectoriesnumerically, since the right hand side of the equations (4.31)-(4.32) is ill defined at T\u00CB\u0086 = 1. Further-more, neither of these conditions capture the expected behavior of the horizon as T ! \u00E2\u0080\u00A2, whichis a simple closed curve parametrized by a 2 [0,2p), as shown in the last frame of figure 4.4, anddiscussed in section 4.1. Notice that the horizon in figure 4.4 is a simple closed curve instead of aclosed surface since we have set q = p/2 and supressed one of the spatial dimensions.754.3. Null geodesic calculationsTo set up initial conditions for the equations (4.31)-(4.32), and in order to understand the behaviorof the solutions of (4.12)-(4.13) as T ! \u00E2\u0080\u00A2, we will use a series expansion to obtain an approximatesolution for T ! \u00E2\u0080\u00A2. How this series expansion is used to set initial conditions for numerical pur-poses is explained in the next section. The derivation of the series solution will be deferred untilsection 4.4.3, where we will perform a detailed analysis of the late time shape of the horizons.4.3.3 Numerical methodsInitial conditionsThe initial conditions to (4.31)-(4.32) are set by using approximations to r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ) near T\u00CB\u0086 = 1.When presenting these approximations, it is useful to first define x > 0 asx = 1\u0000 T\u00CB\u0086 .In section 4.4.3, we will obtain an approximate series expansion for r(x ) and f(x ) in the limit thatx \u00E2\u008C\u00A7 1. These expansions are of the formr(x ) = 1x \u00001+ r1x + r2x 2 + r3x 3 + r4x 4 +O \u0000x 5\u0000\u0000 , (4.38)f(x ) = f1x +f2x 2 +f3x 3 +f4x 4 +O \u0000x 5\u0000 . (4.39)where r0, r1, r2 and r3 are given by (4.76) and f1, f2, f3 and f4 are given by (4.80)-(4.83). Theapproximations to r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ) near T\u00CB\u0086 = 1 are then found by making the substitution x = 1\u0000 T\u00CB\u0086in the above.Equations (4.31)-(4.32) are solved by integrating backwards from some initial value T\u00CB\u00860 = 1\u0000 x0for some small but finite value of x0, with the approximations to r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ) above providingthe initial conditions. The value of x0 should be chosen small enough that the error introduced by764.3. Null geodesic calculationsapproximating the initial conditions is negligible, but not so small that numerical round-off errorsdue to the finite number of digits cause a loss of significance either in evaluating the right handsides of (4.31)-(4.32), or in evaluating the expression for r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ) that are used to set theinitial conditions.Let T0 be the value of T corresponding to T\u00CB\u00860, as given by (4.30). A useful benchmark for determininga minimum size for the value of x0 is that we should have T0\u0000Tmerger \u0000 1H , where Tmerger is the timeat which horizon merger occurs. This benchmark is based on the expectation that the largest changein horizon area and shape takes place over an O( 1H ) time interval about Tmerger, where H =pL/3is Hubble parameter. Inverting (4.30) and using T\u00CB\u0086 = 1\u0000x , we haveT = 1H ln\u00E2\u009C\u00931+ T\u00CB\u00861\u0000 T\u00CB\u0086 \u00E2\u0097\u0086=1H ln\u00E2\u009C\u00932\u0000xx \u00E2\u0097\u0086=1H [\u0000 lnx + ln2+O(x )]\u00E2\u0087\u00A12.3H [\u0000 log10 x +0.3+O(x )] .For example, using the value x0 = 10\u000010 in the above gives T0 \u00E2\u0087\u00A1 23 1H \u0000 1H . Throughout this thesis,we use the value x0 = 10\u00004, unless otherwise indicated.Integration methodIn order to find the coordinates (r,f) as a function of LP coordinate time T for a null geodesic,one approach is to numerically integrate the null geodesic equations for r(T\u00CB\u0086 ) and f(T\u00CB\u0086 ), eitherin the uncoupled form (4.31)-(4.32) or the coupled form (4.33)-(4.35). Numerically integratingthe uncoupled equations (4.31)-(4.32) is problematic at the turning points, since it would requirean infinite number of time steps as one gets arbitrarily close to the turning point. To avoid thisdifficulty, we will use the the coupled equations (4.33)-(4.35) for trajectories with turning points,and the uncoupled equations (4.31)-(4.32) for null geodesic trajectories without turning points. The774.3. Null geodesic calculationsadvantage of the uncoupled equations is that there are two equations instead of three, allowing fora more efficient numerical integration. The uncoupled equations, either in the compactified form(4.31)-(4.32) or the raw form (4.14)-(4.15), will also be used in analytical calculations involvingthe null geodesics. For example, in section 4.4.3 they will be used to find the null geodesics at latetimes, and in section 4.5 they will be used to find the location of the merger point where the horizonsfirst touch.Equations (4.31)-(4.32) or (4.33)-(4.35) are solved subject to the initial conditions described in sec-tion 4.3.3 using MAPLE 14\u00E2\u0080\u0099s numeric dsolve procedure, with the integration preceding backwardsfrom the initial time T\u00CB\u00860. The MAPLE environment variable \u00E2\u0080\u009CDigits\u00E2\u0080\u009D is set to 15, instead of usingthe default value of 10, so that floating point calculations are done with 15 digits. This is equivalentto using double precision floating point calculations, and has the advantage of allowing the value ofx0 from section 4.3.3 to be, for example, x0 = 10\u000010 without creating a loss of significance due torounding of floating point numbers. Note that the system of equations (4.31)-(4.32) or (4.33)-(4.35)are both considered \u00E2\u0080\u009Cstiff\u00E2\u0080\u009D systems, in the sense that excessively small integration time steps arerequired to deal with the rapid variation in r(T\u00CB\u0086 ) near the initial time T\u00CB\u0086 = 1. For this reason, we usethe option stiff=\u00E2\u0080\u0099true\u00E2\u0080\u0099 when calling MAPLE\u00E2\u0080\u0099s dsolve procedure.For null geodesic trajectories with cosa < 0, both the numerator and denominator of the right handside of either (4.31) or (4.17) vanish for r = rc, and we encounter a potential difficulty during thenumerical integration. How this potential difficulty is handled is discussed in detail in section 4.3.3below.The equations are solved for a set of values for the parameter a appearing in the equations. Thedefinition of a is given by equation 4.1, and the interpretation of a is that the values a 2 [0,2p)parametrize the generators which make up the closed curve in the last frame of figure 4.4. Thisclosed curve is a representation of the late time horizon, where one spatial dimension has beensuppressed so that the horizon is a closed curve instead of a closed surface. If one considered thefull two dimensional closed surface, it would be parametrized by two angles a and b . However, therotational symmetry of the spacetime allows us to ignore the angle b . A more detailed interpretation784.3. Null geodesic calculationsof the parameter a can be found in section 4.4.3.The specific values of a used in a calculation depends on the problem at hand. For example, whenconsidering the area of the horizon, we will want to use values of a covering the entire interval[0,2p). Notice that by reflection symmetry, we can always limit ourselves to values a 2 [0,p)without any loss of generality.The sphere r = rcFor null geodesic trajectories with cosa \u00EF\u00A3\u00BF 0, the functions G+(r) and K+(r) in (4.31)-(4.32) areundefined when r = rc, since both numerator and denominator of the right hand side of (4.12)-(4.13)vanish at this point. The same difficulty occurs in the right hand sides of the coupled equations(4.33)-(4.35). Fundamentally, this problematic behavior stems from the fact that one of the nullgeodesic equations in Schwarzschild coordinates (equation (4.2)) contains a singularity, and thatalthough changing to LP coordinates eliminates this infamous coordinate singularity, it does so onlyby introducing a vanishing numerator to balance the vanishing denominator in the equations.For trajectories with cosa \u00EF\u00A3\u00BF 0 and which cross r = rc, this creates a problem when integrating theequations (4.31)-(4.32) or (4.33)-(4.35) numerically. Depending on the numerical algorithm andinitial conditions used, we have found that it is possible in some cases to integrate the equations(4.31)-(4.32) through r = rc to obtain (by brute force) a numerical solution satisfying r(Tc) = rc,where Tc is defined by r(Tc)= rc. However, the accuracy of such a solution is questionable, since it ispossible that the numerical algorithm would attempt to compute G+(r) or K+(r) with r sufficientlyclose to rc that a loss of significance would occur. This would be the result of attempting to computethe ratio of two floating point numbers sufficiently close to zero. The same potential loss of accuracyarises when dealing with the coupled equations (4.33)-(4.35). The strategy for dealing with the twocases is slightly different, and so we discuss them separately.794.3. Null geodesic calculationsUncoupled equations In the case of the uncoupled equations (4.31)-(4.32), our strategy for cir-cumventing this difficulty will be to approximate G+(r) and K+(r) in the vicinity of rc by using thefunctions g+(r) and k+(r), defined byg+(r) = 8><>:limr!rc G+(r) for rc\u0000D< r < rc +D ,G+(r) for r \u00EF\u00A3\u00BF rc\u0000D and r \u0000 rc +D , (4.40)k+(r) = 8><>:limr!rc K+(r) for rc\u0000D< r < rc +D ,K+(r) for r \u00EF\u00A3\u00BF rc\u0000D and rc \u0000 r+D . (4.41)The basic approximation used is that G+(r) and K+(r) are assumed to be constant for r 2 (rc\u0000D,rc +D), and equal to their limiting values as r ! rc. Notice that our approximation is equivalentto using a first order Taylor approximation to r(T ) for r 2 (rc\u0000D,rc+D), with the expansion takingplace about rc and the derivative r0(T ) being approximated by limr!rc G+(r). Equivalently, we areessentially taking the numerical method across (rc\u0000D,rc+D) to be one time step using the forwardEuler method.In the above, D is chosen to be small enough to limit the error introduced by this assumption, but notso small that rounding errors in the numerator and denominator in G+(r) cause a loss of significance(the very problem we are trying to avoid).The limit in (4.40) can be found using L\u00E2\u0080\u0099Hopital\u00E2\u0080\u0099s rule. First recall from (4.12) and (4.14) thedefinition of G+(r):G+(r) = \u0000 f (r)pH2\u0000Ve f f (r)H f (re) 12 cosa +( f (re)\u0000 f (r)) 12 sgn(r\u0000 re)pH2\u0000Ve f f (r) .804.3. Null geodesic calculationsProvided that cosa < 0, we havelimr!rc G+(r)=\u0000\u00E2\u009C\u0093limr!rcqH2\u0000Ve f f (r)\u00E2\u0097\u0086 limr!rc f (r)H f (re) 12 cosa +( f (re)\u0000 f (r)) 12 pH2\u0000Ve f f (r)! (4.42)= limr!rc H|cosa| f 0(r)12 ( f (re)\u0000 f (r))\u0000 12 f 0(r)pH2\u0000Ve f f (r)+( f (re)\u0000 f (r)) 12 12 (H2\u0000Ve f f (r))\u0000 12 V 0e f f (r)(4.43)= H cosa f 0(rc)12 f 0(re)\u0000 12 f 0(rc)H cosa + f (re) 12 12 1H cosaV 0e f f (rc) . (4.44)In going from (4.42) to (4.43) above, we have used Ve f f (rc) = H sina for the first limit andL\u00E2\u0080\u0099Hopital\u00E2\u0080\u0099s rule for the second limit. In going from (4.43) to (4.44) we have used cosa < 0 andf (rc) = 0 as well.We can simplify (4.44) by noticing thatVe f f (r) = sin2ar2 \u0000 f (r)+H2r2\u0000 , (4.45)which follows from (4.5) and (2.5). Differentiating the above and using f (rc) = 0, we obtainV 0e f f (rc) = f 0(rc)r2c sin2a .Substituting the above in (4.44) and simplifying, we getlimr!rc G+(r) = 2H2r2c f (re)1/2H2r2c + f (re)1/2 tan2a . (4.46)f (re) is given in terms of M and H in (2.7), and rc is the largest positive root of f (r). The aboveallows us to calculate limr!rc G+(r) for cosa < 0, given numerical values for M and H. For cosa =0, we simply have limr!rc G+(r) =G+(rc) = 0. Notice that as cosa ! 0, the above expression goesto zero, so that limr!rc G+(r) is continous as a function of a .814.3. Null geodesic calculationsBefore calculating limr!rc K+(r), recall the definition of K+(r) from (4.13) and (4.15):K+(r) = \u00E2\u009C\u0093sinar2 \u00E2\u0097\u0086 f (r)H f (re) 12 cosa + sgn\u0000 drdT \u0000( f (re)\u0000 f (r)) 12 sgn(r\u0000 re)pH2\u0000Ve f f (r) . (4.47)In the case that cosa < 0, limr!rc K+(r) can be calculated by noticing from the above that we havelimr!rc K+(r) = sinar2cpH2\u0000Ve f f (rc) limr!rc G+(r) .Using Ve f f (rc) = H sina , cosa < 0 and (4.46), the above simplifies tolimr!rc K+(r) = \u00002H2 f (re)1/2 tanaH2r2c + f (re)1/2 tan2a . (4.48)In the case that cosa = 0, (4.47) simplifies toK+(r) = \u00E2\u009C\u0093sgn(sina)r \u00E2\u0097\u0086 f (r)( f (re)\u0000 f (r)) 12 sgn(r\u0000 re)p f (r) ,where we have used |sina| = 1 and (4.45). Although the above is undefined at r = rc, a trivialsimplification reveals thatlimr!rc K+(r) = 0 .Let us summarize our strategy for dealing with the possible loss of accuracy when numericallyintegrating the uncoupled equations (4.31)-(4.32) across r = rc. For null geodesics with cosa \u00EF\u00A3\u00BF 0and r0(T ) > 0 at r = rc, the uncoupled equations (4.31)-(4.32) are replaced with the followingequations instead: drdT\u00CB\u0086 = 2H(1\u0000 T\u00CB\u0086 2)g+ (r) , (4.49)dfdT\u00CB\u0086 = 2H(1\u0000 T\u00CB\u0086 2)k+ (r) , (4.50)where the functions g+(r) and k+(r) are given by (4.40)-(4.41), and limr!rc G+(r) and limr!rc K+(r)are given by (4.46) and (4.48) for cosa < 0, and by limr!rc G+(r) = limr!rc K+(r) = 0 for cosa =824.3. Null geodesic calculations0.The coupled equations The difficulty that arises when numerically integrating the uncoupledequations (4.31)-(4.32) also occurs in the coupled equations (4.33)-(4.34). To see why, considernull geodesics with r0(T ) < 0 when r(T ) = rc. By combining equations (4.6)-(4.7) with (4.8), weget that, as long as r0(T ) > 0, we must havew =qH2\u0000Ve f f (r) .Now substituting the above into (4.34)-(4.35), these equations becomedrdT\u00CB\u0086 = \u00E2\u009C\u0093 2H(1\u0000 T\u00CB\u0086 2)\u00E2\u0097\u0086G+(r),dwdT\u00CB\u0086 = \u0000\u00E2\u009C\u0093 1H(1\u0000 T\u00CB\u0086 2)\u00E2\u0097\u0086 V 0e f f (r)pH2\u0000Ve f f (r)G+(r),dfdT\u00CB\u0086 = \u0000\u00E2\u009C\u0093 2H(1\u0000 T\u00CB\u0086 2)\u00E2\u0097\u0086 sinar2pH2\u0000Ve f f (r)!G+(r) .We see that the same difficulty in evaluating G+(r) that arose in the uncoupled equations also arisesin the above. The strategy for dealing with the coupled equations (4.33)-(4.34) is to replace themwith a modified version of the above equation when w < 0 and r 2 (rc \u0000 D,rc + D), where themodification that we make is to replace G+(r) with limr!rc G+(r) and set r = rc in the remainingterms. This is basically the same strategy that was used in (4.49)-(4.50) above, where we replacethe right hand sides of the equations with a constant for r 2 (rc\u0000D,rc +D). The main differencesin the case of the coupled equations are that we are dealing with three equations, and the right handsides of these equations are functions of both r and w.834.4. Horizon shapes4.4 Horizon shapes4.4.1 Numerical resultsWe begin our illustration and analysis of the shape and structure of the horizon by using numericalresults to plot the coordinates (r cosf , r sinf) at several instants of time T , as shown in figure4.4. The functions r(T ) and f(T ) are found by solving the equations (4.31)-(4.32) or (4.33)-(4.35)numerically using the procedure outlined in section 4.3. As discussed in sections 3.3.7 and 4.2.2, thehorizon can be thought of as a series of 2-surfaces, each embedded in one of the hypersurfaces T =constant, where T is the time coordinate from LP coordinates. These 2-surfaces can be visualizedby rotating the curves in 4.4 about the x-axis, where x = r cosf . Representing the horizon as a 2-surface in Euclidean space, or a curve in the Euclidean plane, has the advantage that it easily allowsus to identify key features of the horizon shape and structure. Some of these key features will bediscussed further in this section and section 4.4.2. The disadvantage of this representation is thatit introduces metrical distortions since the true geometry of the hypersurfaces T = constant is thatof a 3-cone, which is a curved manifold (see section 3.3.6 for a discussion of the geometry of thespacelike hypersurfaces).Consider the progression of frames in figure 4.4. Since integration runs backwards in time, considerfirst the last frame. In this frame, the horizon differs only slightly from the \u00E2\u0080\u009Cfinal shape\u00E2\u0080\u009D. As wasexplained in section 4.1, the horizon (viewed as a 2-surface) is expected to converge to a \u00E2\u0080\u009Cfinalshape\u00E2\u0080\u009D which is a closed surface surrounding the observer. In section 4.4.3 below, we will derivean explicit formula for the horizon at late times using a series expansion. This will confirm ourexpectation of the existence of such a final shape, and also allow us to precisely characterize theslight deviations from the final shape that occur at late times before the horizon finally settles down.Note that this final shape is not a circle (or not a sphere, if one considers the surface formed byrotating the curve). Instead it is prolate in the vertical direction, as can be deduced by looking atthe discrepancy between the horizontal and vertical range of the figure. We will return to a precisecharacterization of this shape in section 4.4.3. The non-sphericity of the late time horizon shape may844.4. Horizon shapesFigure 4.4: Six frames showing the progression from two distinct horizons at early times to a singlecosmological horizon at late times. Parameter values: M = 1 and L = 1/90. Times for the sixframes: T\u00CB\u0086 = 0, T\u00CB\u0086 \u00E2\u0087\u00A1 0.6, T\u00CB\u0086 \u00E2\u0087\u00A1 0.71, T\u00CB\u0086 \u00E2\u0087\u00A1 0.74, T\u00CB\u0086 \u00E2\u0087\u00A1 0.81, T\u00CB\u0086 \u00E2\u0087\u00A1 0.99854.4. Horizon shapesbe surprising, given that the spacetime geometry at late times approaches that of deSitter spacetime,as discussed in section 2.6, and given that the horizon surrounding an observer in deSitter spacetimeis spherical. The reason for the discrepancy is that the spacelike hypersurfaces T = constant ofour LP coordinate system do not coincide precisely with the usual planar constant time surfacesof the steady-state universe of deSitter spacetime. So although the full three dimensional horizon,when viewed as a null hypersurface embedded in spacetime, approaches at late times the samenull hypesurface as a horizon in deSitter spacetime, it has been sliced slightly differently in theLP coordinate system developed in this thesis, as compared to the usual planar deSitter steady-state universe slicing. In the limit that M = 0, the horizon shape in LP coordinates is spherical, asexpected from the fact that LP coordinates reduce to the planar coordinates of deSitter spacetime inthis limit, as was discussed in section 3.3.1.In the fifth frame, we begin to see a significant distortion in the horizon shape due to the influence ofthe black hole. In order to easily illustrate the change in shape that occurs due to the black hole, wehave used an unrealistically large value for the parameter characterizing the size of the black holerelative to the cosmological horizon. A realistic value based on the L-CDM model would be on theorder of e = 10\u000014 (see section 2.7). Also in the fifth frame, we see that the horizons have movedsignificantly closer to r = 0. This approach towards r = 0 occurs roughly as r \u00C2\u00B5 eHT for large valuesof T (r is decreasing since we are thinking of progressively smaller values of T ). This is relatedto the fact that moving forward in time, an observer caught up in the expansion of spacetime driftsaway with approximately r \u00C2\u00B5 eHT at late times, with the cosmological horizon roughly centered onthe observer\u00E2\u0080\u0099s position.In the fourth frame, we are just before (i.e. at an earlier time) the critical point where the horizonsmerge. Here we can see that we are dealing with two closely spaced but nevertheless disconnectedhorizons. With a clear separation into two horizons, we can define the black hole horizon as theinner horizon, and the cosmological horizon as the outer horizon. On the other hand, in the fifthand sixth frames, the horizon would be considered to be solely a cosmological horizon, althoughpossibly with significant distortions due to the black hole. At some time between the fourth andfifth frames, there is a critical moment in time where the horizons first touch. We call this time the864.4. Horizon shapesmerger of the horizons, and the precise location of this event in both space and time will be calledthe merger point. Determining the coordinates of the merger point will be the focus of section 4.5.As explained in section 4.1, the merger point is one of several caustic points where new generatorscan enter the horizon. The other caustic points are the points on the horizon which intersect theline f = p (or the line f = p, q = p/2 if one considers the horizons as 2-surfaces and not curves).Although it is difficult to see in figure 4.4, the horizon in the first four frames is not smooth alongthese points. These nonsmooth points on the horizon converge until they finally join together at themerger point. Prior to their merger, the green points in between the horizons are the locations ofgenerators which are not yet part of the horizons, but will later enter the caustic points and becomehorizon generators. These are similar to the green curves entering the horizon through the causticpoints along the \u00E2\u0080\u009Cinseam\u00E2\u0080\u009D of the \u00E2\u0080\u009Ctrousers\u00E2\u0080\u009D diagram shown in figure 4.1. In all six frames, the redpoints are the locations of the horizon generators.In the third frame, we see that the horizons are further apart, and there are more (green) generatorswhich have yet to join the horizon. In the second frame, there are even more green generators,and we see that these are moving towards the line f = 0 (when going backwards in time). Goingbackwards in time indefinitely, these green generators will rotate indefinitely, with three possiblebehaviors: either spiralling inwards into the black hole at r = rb, spiralling outwards into the cos-mological horizon r = rc, or spiralling into the light sphere at r = 3M. None of these generatorsever join the horizons, although all except those approaching r = 3M will get arbitrarily close tothe horizons as we go backwards in time. The generator (or family of generators if we considerthe horizon as a 2-surface and not a curve) that approaches r = 3M is the limiting case that sepa-rates the generators which approach either horizon, and this critical separator only occurs for onecritical value of the parameter a 2 [0,p), or two values of the parameter if a 2 [0,2p) (recall thata 2 [0,2p) parametrizes the generators of the sixth frame). These different qualitative behaviorsfor both the green and red generators will be analyzed in more detail in section 4.4.2, culminatingin a classification of different generators according to different qualitative behavior. For example,one important distinction is between those generators which at the very earliest times started on thehorizons, and those which at the very earliest times start not on the horizon, only to later join thehorizon. These are similar to the red and green points, respectively, of the first frame, except that874.4. Horizon shapesone must imagine the limit as T !\u0000\u00E2\u0080\u00A2. That is, one must imagine a very early frame. This dis-tinction between these two types of generators will be especially relevant in section 5.3.2, where wewill calculate the horizon area increase in time due to these two types of generators, and comparethem to the total horizon area increase.In the first frame, the horizon is very nearly that of a stationary observer: two nested spheres atr = rb and r = rc. This agrees with our expectations, since the observer drifting away from the blackhole starts at the equilibrium point r = re, and thus is close to the black hole for an arbitarily longtime. During this time the observer can receive signals from anywhere in the region rb < r < rc,much like an observer with rb < r(t) < rc for all t (see section 2.4 for a further discussion of thehorizons of such observers).An alternative way of depicting the merging of the black hole and cosmological horizons is touse a three dimensional plot, as we have done in figure 4.5. The vertical direction is time and thehorizontal directions are spatial, and as with the 2d plots, one spatial dimension has been suppressed.In this plot we can see the transition from two separate horizons into one horizon at later times, aswell as the non-smoothness of the horizons along the caustic points prior to merger. As with the\u00E2\u0080\u009Ctrousers\u00E2\u0080\u009D diagram of binary black hole merger (figure 4.1), these caustic points occur along an\u00E2\u0080\u009Cinseam\u00E2\u0080\u009D, although the diagram does not resemble a pair of pants (if it was a pair of pants, one pantleg would be inside the other). Note that unlike our 2d plot, on our 3d figure we have only plottedthe horizons, and not plotted the generators which eventually join through the caustic points alongthe inseam.The analysis of the light rays which make up the horizon is carried out in more detail in the nextsection. This analysis will serve two purposes. First, we will focus on certain key generators whichare crucial for further analysis that will be performed in the following sections. Second, a qualitativeanalysis of the behavior of the generators will allow us to gain further insight into the structure ofthe horizon, and confirm some of the numerical results presented above.884.4. Horizon shapesFigure 4.5: 3d plot of the merging of black hole and cosmological horizons. The vertical axis is timein LP coordinates and the horizontal axes are the coordinates x = r cosf and y = r sinf . The blueand crimson surfaces represent the black hole and cosmological horizons respectively. We clearlysee that the crimson and blue surfaces are not smooth prior to merger.894.4. Horizon shapesFigure 4.6: Schematic illustration of the possible behaviors for r(T ) as T !\u0000\u00E2\u0080\u00A2. The solid blackcircles are points where T =\u0000\u00E2\u0080\u00A2. The direction of the arrows are thought of as indicating the direc-tion of decreasing T , or equivalently, of decreasing l . The solid black curve is the effective potentialV (r)sin2a = 1r2 (1\u0000 2Mr ), and the three coloured curves indicate the different cases for the trajectories,based on the value of sina . Consider first the case where sina > 27e2, as shown by the green curve.If cosa \u0000 0, then we have T = \u0000\u00E2\u0080\u00A2 at rc, without a turning point in the behavior of r(T ). On theother hand, if cos(a) < 0, then we have T =\u0000\u00E2\u0080\u00A2 at rc also, but this time with a turning point in thebehavior of r(T ). For the cases where sin2a = 27e2 and sin2a < 27e2, if cosa \u0000 0 then T =\u0000\u00E2\u0080\u00A2at rc. If cosa < 0, then we have r(T =\u0000\u00E2\u0080\u00A2) = 3M and r(T =\u0000\u00E2\u0080\u00A2) = rb for the cases sin2a = 27e2and sin2a < 27e2, respectively.4.4.2 Classification of null generatorsAs discussed in section 4.1, the horizon can be thought of as a null surface generated by a familyof light rays, known as the null generators. We can gain further insight into the overall structureof the horizon by classifying these generators into different categories, based on the behavior ofthe generators at early times. This classification will be done by using the parameter a , whichparametrizes the generators such that each generator has precisely one value of a . The interpretationof a was briefly discussed in section 4.2.1, and will be revisited in more detail in section 4.4.3. Even904.4. Horizon shapesthough the coordinates of the generators depend on the parameter a as well as the time T , so thatwe could write r = r(T,a), we will suppress the dependence on a and simply write r = r(T ).The first classification of generators is based on the limiting behavior of the radial Schwarzschildcoordinate r(T ) as T !\u0000\u00E2\u0080\u00A2. As can be seen in the first frame of figure 4.4, generators appear to startfrom either the sphere r = rb or the sphere r = rc. We will confirm this by combining the effectivepotential diagram (figure 4.3) with an understanding of the behavior of T (l ), as given by (4.10).Note that in everything that follows we are considering null geodesics satisfying r(l = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2.These can be visualized as moving to the right with increasing l for large values of r in figure 4.3.Since we are interested in the behavior of r(T ) as T ! \u0000\u00E2\u0080\u00A2, let us start by considering equation(4.10) and looking for finite values l0 such that T (l+0 ) =\u0000\u00E2\u0080\u00A2. To identify such values, notice thatdTdl \u0000\u0000\u0000\u0000l+0 = \u00E2\u0080\u00A2() T (l+0 ) =\u0000\u00E2\u0080\u00A2 ,which follows from the fact that T 0(l ) > 0, as was shown in section 4.2.2. Thus it suffices tofind values of l0 where the right hand side of (4.11) diverges. This requires that the numerator isnonvanishing and the denominator vanishes. The vanishing of the denominator leads tof (r(l0)) = 0 =) r(l0) = rb or r(l0) = rc , (4.51)which can be seen from figure 2.6. The condition that the numerator be nonvanishing leads to thefollowing requirement:sgn (r\u0000 re) drdl \u0000\u0000\u0000\u0000l0 cosa!\u0000 0 . (4.52)Putting together the previous two requirements, we conclude thatr(l0) = rb and sgn drdl \u0000\u0000\u0000\u0000l0 cosa!\u00EF\u00A3\u00BF 0=) T (l+0 ) =\u0000\u00E2\u0080\u00A2=) r(T =\u0000\u00E2\u0080\u00A2) = rb , (4.53)914.4. Horizon shapesand r(l0) = rc and sgn drdl \u0000\u0000\u0000\u0000l0 cosa!\u0000 0=) T (l+0 ) =\u0000\u00E2\u0080\u00A2=) r(T =\u0000\u00E2\u0080\u00A2) = rc , (4.54)where we have used the fact that sgn(rb\u0000re) =\u00001 and sgn(rc\u0000re) = 1, respectively, as can be seenfrom figure 2.6. It will turn out that there are only two generators (i.e. two values of a) that do notsatisfy either of the above conditions, so that r(T =\u0000\u00E2\u0080\u00A2) = rb or r(T =\u0000\u00E2\u0080\u00A2) = rc for all except twonull generators. To classify the values of a according to either r(T =\u0000\u00E2\u0080\u00A2) = rc or r(T =\u0000\u00E2\u0080\u00A2) = rb,let l1 and l2 be defined such that r(l1) 2 {rb,rc} and r(l2) 2 {rb,rc}, with l1 < l2. From theeffective potential diagram 4.3, we see that for any trajectory with r(l = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2, there are threepossibilities for r(l1) and r(l2). These aresin2a < 27H2M2 =) r(l1) = rb, r(l2) = rc , (4.55)sin2a > 27H2M2 =) r(l1) = rc, r(l2) = rc , (4.56)sin2a = 27H2M2 =) l1 does not exist, r(l2) = rc . (4.57)Furthermore, the signs of r0(l1) and r0(l2) are as follows:sin2a < 27H2M2 =) r0(l1) > 0 and r0(l2) > 0 , (4.58)sin2a > 27H2M2 =) r0(l1) < 0 and r0(l2) > 0 , (4.59)sin2a = 27H2M2 =) r0(l2) > 0 . (4.60)Combining (4.55), (4.58) and (4.53), we conclude thatcosa < 0 and sin2a < 27H2M2 (4.61)=) r(l1) = rb and T (l+1 ) =\u0000\u00E2\u0080\u00A2=) r(T =\u0000\u00E2\u0080\u00A2) = rb .924.4. Horizon shapesSimilarly, combining (4.56), (4.59) and (4.54), we havecosa < 0 and sin2a > 27H2M2 (4.62)=) r(l1) = rc and T (l+1 ) =\u0000\u00E2\u0080\u00A2=) r(T =\u0000\u00E2\u0080\u00A2) = rc .Lastly, combining (4.55)-(4.57), (4.58)-(4.58) and (4.54), we getcosa \u0000 0 =) r(l2) = rc and T (l+2 ) =\u0000\u00E2\u0080\u00A2 =) r(T =\u0000\u00E2\u0080\u00A2) = rc . (4.63)The three cases (4.61), (4.62) and (4.63) give the limiting behavior of r(T ) as T ! \u00E2\u0080\u00A2 for all valuesof a except those for whichcosa < 0 and sin2a = 27H2M2 .From figure 4.6 we see that the above corresponds to the critical case where the light ray trajectoryapproaches the unstable equilibrium at r = 3M and satisfiesr(l =\u0000\u00E2\u0080\u00A2) = 3M .Since T 0(\u0000\u00E2\u0080\u00A2) > 0, as shown in section 4.2.2, we have T (l = \u0000\u00E2\u0080\u00A2) = \u0000\u00E2\u0080\u00A2 and the above impliesthatcosa < 0 and sin2a = 27H2M2=) r(l =\u0000\u00E2\u0080\u00A2) = 3M and T (l =\u0000\u00E2\u0080\u00A2) =\u0000\u00E2\u0080\u00A2 (4.64)=) r(T =\u0000\u00E2\u0080\u00A2) = 3M .Finally, we can put together (4.61), (4.62), (4.63) and (4.64), so that we have the behavior of r(T )934.4. Horizon shapesas T !\u0000\u00E2\u0080\u00A2 for all values of a:cosa < 0 and sin2a < 27H2M2 =) r(T =\u0000\u00E2\u0080\u00A2) = rb ,cosa < 0 and sin2a = 27H2M2 =) r(T =\u0000\u00E2\u0080\u00A2) = 3M ,cosa \u0000 0 or sin2a > 27H2M2 =) r(T =\u0000\u00E2\u0080\u00A2) = rc .The above results are summarized in figure 4.6. They confirm the basic separation of generatorsinto those which start either near r = rb or r = rc, as was discovered numerically and can be seen inthe first frame of figure 4.4. Furthermore, we have given precise values of the parameter a wherethis separation occurs. Notice that the above result is not merely stating that the horizon at earlytimes consists of the spheres r = rb and r = rc, as could easily be deduced from the Penrose diagramin figure 3.1b. Instead, our result is that all generators except those with sin2a 6= 27H2M2 andcosa < 0, including those which do not start on the horizon, approach r = rb or r = rc as T !\u0000\u00E2\u0080\u00A2.A further subdivision separates null generators into the following two categories: those which starton either the black hole or cosmological horizon, as opposed to those which only join these horizonslater. This distinction is best understood by looking at figure 4.7, where the color of the generatorindicates its origin. Red and blue generators start on the cosmological and black hole horizons,respectively, whereas green generators do not start on either horizon, but instead join the horizonsby entering through the caustic points along f = p . Based on the last frame in this figure, we seethat we can introduce a1 2 [0,p) and a2 2 [0,p) such that generators satisfying a1 \u00EF\u00A3\u00BF a \u00EF\u00A3\u00BF a2 ora1 \u00EF\u00A3\u00BF 2p\u0000a \u00EF\u00A3\u00BF a2 will be those which did not begin on either horizon (i.e. green generators).Suppose we assume that the horizons have the basic shape as shown in figure 4.4, with causticpoints on the horizon occuring if and only if f = p . This claim can be proven analytically througha detailed analysis of the properties of the function f(T ;a), but we will not pursue this here. Withthis basic assumption it follows from the fact that generators can only enter through caustic points,as discussed in section 4.1, that we havea1 \u00EF\u00A3\u00BF a \u00EF\u00A3\u00BF a2 or a1 \u00EF\u00A3\u00BF 2p\u0000a \u00EF\u00A3\u00BF a2 () |f(T =\u0000\u00E2\u0080\u00A2)|\u0000 p . (4.65)944.4. Horizon shapesFigure 4.7: Three frames showing the merging of horizons, with a color scheme to indicate theorigin of each null generator. Blue and red are used to indicate null generators which begin onthe black hole and cosmological horizons, respectively. Green is used to indicate generators whichbegin on neither horizon. This color scheme is unlike the one in figure 4.4, where we used the colorsto indicate if a null generator was currently part of either the cosmological horizon, the black holehorizon, or neither.954.4. Horizon shapesThis above condition will be important in section 5.3.2, where a calculation of a1 and a2 for e \u00E2\u008C\u00A7 1will be used to find the relative area increase due to the three types of generators, in the limit thate ! 0.4.4.3 Horizon at late timesFor sufficiently large values of LP coordinate time T , we can see from the effective potential diagram(figure 4.6) that drdT > 0, so that r(T\u00CB\u0086 ) obeys the following equation:drdT\u00CB\u0086 = 2H 11\u0000 T\u00CB\u0086 2G+ (r) , (4.66)where G+(r) is the right hand side of (4.12) for the case drdT > 0. Let x = 1\u0000 T\u00CB\u0086 replace T\u00CB\u0086 as theindependent variable. The equation (4.66) above becomes the following equation for r(x ):drdx = \u00001Hx (1\u0000x/2)G+ (r) . (4.67)We will look for an approximate asymptotic solution of the formr(x ) = 1x n\u00C3\u0082i=0rix i +O(x n+1) , (4.68)where the coefficients ri are allowed to depend on M, H and a . Notice that x ! 0+ correspondsto T ! \u00E2\u0080\u00A2, and that (4.68) automatically incorporates the asymptotic requirement (4.36). In section4.4.3, we briefly discuss the movitation behind choosing an expansion of the form (4.68). Wethen find the expansions for r(x ) and f(x ), which will be (or already have been) used for severalpurposes. In section 4.4.1, they were used as initial conditions when solving the null geodesicequations from section 4.3.1 numerically. In section 4.4.3 below, we will use r(x ) and f(x ) todescribe and analyze the shape of the horizon as T ! \u00E2\u0080\u00A2. In section 4.4.3, we discuss the validity ofthe above expansion.964.4. Horizon shapesThe coefficient r0Although we have allowed r0 to depend on a for the sake of generality, we can see that if theshape of the horizon at late times is to be a closed surface at a finite distance from the observer (asdiscussed in section 4.1), then r0 must be independent of a . The reasoning is as follows. Supposer0(a1) 6= r0(a2), so that r0 does depend on a . Then if r(x ;a) is the solution to equation (4.67),with the expansion given by (4.68), we have that k(r(x ;a1),f(x ;a1))\u0000 (r(x ;a2),f(x ;a2))k! \u00E2\u0080\u00A2as e ! 0, where the norm k \u00C2\u00B7 k is, for example, the geodesic distance along the hypersurface T =constant, with the metric (3.39) providing the measure of distance, and we have set q = p/2 forsimplicity. In other words, if r0 depends on a , the distance between two geodesics with differentvalues of r0 would become infinite as T ! \u00E2\u0080\u00A2 (i.e. x ! 0+), and this would clearly contradict ourrequirement and expectation that the null geodesics form a closed surface a finite distance from theobserver as T ! \u00E2\u0080\u00A2. This expectation was discussed in 4.1, and is motivated by the fact that asT ! \u00E2\u0080\u00A2, the observer is in a region of spacetime well approximated by deSitter spacetime.As can be deduced from (4.72) below, the coefficient r0 in the expansion (4.68) is essentially arbi-trary, in the sense that changing the value of r0 simply amounts to a shift in the time variable T . Thisis unlike the coefficients r1, r2 and r3 in the expansion, which will be determined by substituting theexpansion (4.68) into the equation (4.67). Since changing the value of r0 corresponds to a simpleshift of the time variable T , we will use the following convenient value:r0 = 2H .Motivation for the expansionTo understand the expansion (4.68), let us first rewrite it asr(x ) = r0x + r1 + n\u00C3\u0082k=2rkx k +O(x n+1) . (4.69)974.4. Horizon shapesConsider the first term on the right hand side, which characterizes the lowest order asymptoticbehavior of r(x ): r(x )\u00E2\u0087\u00A0 r0x . (4.70)To understand this lowest order behavior, we invert the relation (4.30) and use x = 1\u0000 T\u00CB\u0086 to geteHT = 2x \u00001 . (4.71)Using the above in (4.69), this gives r(T ) = r02eHT +O(1) . (4.72)As discussed in section 4.4.3, r0 is independent of a . Provided that we have f \u00E2\u0087\u00A0 x , which can bededuced from (4.77) below, it follows from the above that the family of light rays with a 2 [0,2p)are all a finite distance away from each other. Here the measurement of distance is made using theproper length of spacelike geodesics on the hypersurfaces T = constant. Now compare the abovewith the r coordinate as a function of proper time for our drifting observer, which can be deducedfrom the r! \u00E2\u0080\u00A2 limit of equation (2.12):robs(T ) = AeHT +O(1) . (4.73)The above has precisely the same form as (4.72), so that the light rays with r(T ) given by (4.72) canbe interpreted as not only a finite distance from one another, but a finite distance from the observeras well. As with the constant r0, changing the constant A in the above amounts to shifting the timevariable T by a constant amount. Provided that we choose A = r0/2, the observer\u00E2\u0080\u0099s trajectory willbe precisely centered on the family of light rays with coordinates given by (4.72). Changing theconstant r0 amounts to shifting between the different observer trajectory curves in figure 4.2a, andchanging the constant A amounts to shifting between the different light cones shown in figure 4.2b.By choosing A = r0/2, we are requiring that the observer\u00E2\u0080\u0099s trajectory (one of the curves in figure4.2a) is correctly matched with one of the light cones from figure 4.2b.The fact that (4.72) and (4.73) have the same exponential dependence on T can be seen as a conse-984.4. Horizon shapesquence of the way we constructed LP coordinates. By choosing timelike geodesic trajectories as thetimelike coordinate curves, and by constructing spacelike coordinate curves which are orthogonal tothe timelike curves, we have ensured that the coordinates of the null generators at late times followthe coordinates of the observer. This is similar to the way that the cosmological horizon is a finitedistance from an observer in deSitter spacetime, where the distance is measured along the flat planarslices of the deSitter universe.Another way of understanding the asymptotic behavior (4.70) is to consider the equation (4.12) forlarge values of r, so that it becomes drdT = Hr+O(1) .The above has the approximate asymptotic solution (4.72), which then becomes (4.70) after using(4.71).Next consider the second term on the right hand side of (4.69). This term is O(1), and we can writer(x )\u0000 r0x \u00E2\u0087\u00A0 r1 .Notice that the right hand side above is the lowest order term which depends on a , and thereforewill be the largest term in the series which contributes to the shape of the horizon as T ! \u00E2\u0080\u00A2. Thefact that this term is also independent of x suggests that it will play a role in determining a finalshape of the horizon. In section 4.4.3 below, we show that there is indeed a final shape which isindependent of T , and we will use the O(1) and O(x ) terms of r(x ) and f(x ), respectively, toderive an anlalytical formula for this shape. This final shape can also be seen in the last frame offigure 4.4.Finally, consider the terms which are O(x ) and smaller in (4.69). These terms will play a rolein determining the slight deformations that occur in the final shape for large values of T (i.e. forx \u00E2\u008C\u00A7 1). An example of such a deformation can be seen in the second to last frame of figure 4.4.994.4. Horizon shapesWe have motivated the form of the series (4.69), and we will confirm its accuracy by comparingit to numerical solutions in section 4.4.3. However, as will be discussed in section 4.4.3, we havegood reason to believe that the series (4.69) is not an exact solution but instead is only an asymptoticseries.The expansion coefficients r1, r2 and r3To find the unkown coefficients r1, r2 and r3, we first expand the function G+(r):G+(r) = Hr\u0000 f (re)1/2 cosa +\u00E2\u009C\u0093 f (re)cos2a\u000012H \u00E2\u0097\u0086 1r (4.74)+ \u0000 f (re)\u0000 12\u0000 f (re)1/2 cosa sin2a +MHH2 ! 1r2 +O\u00E2\u009C\u0093 1r3\u00E2\u0097\u0086 .From (4.68) we have1r = 1r0 x \u0000 r1r20 x 2 +O(x 3), (4.75)1r2 = x 2r20 +O(x 3) .Substituting the above and (4.68) into (4.74) and collecting terms of like powers, we obtainG+(x ) = Hr0 1x +Hr1\u0000 f (re)1/2 cosa +\u00E2\u009C\u0093 f (re)cos2a\u000012Hr0 +Hr2\u00E2\u0097\u0086x+ Hr3\u0000 f (re)cos2a\u000012H \u00E2\u009C\u0093 r1r20\u00E2\u0097\u0086+ \u0000 f (re)\u0000 12\u0000 f (re)1/2 cosa sin2a +MHH2r20 !x 2 +O(x 3) .Substituting the above and the expansion11\u0000x/2 = 1+ x2 + x 24 + x 38 +O(x 4)1004.4. Horizon shapesinto (4.67) and collecting like powers again, we obtaindrdx =\u0000r0 1x 2 \u0000 r02 + r1\u0000 f (re)1/2 cosaH ! 1x\u0000 r04+r12\u0000f (re)1/2 cosa2H + f (re)cos2a\u000012H2r0 + r2!\u0000 r08+r14\u0000f (re)1/2 cosa4H + f (re)cos2a\u000014H2r20 (r0\u00002r1)+ r22 + r3+\u0000 f (re)\u0000 12\u0000 f (re)1/2 cosa sin2a +MHH3r20 !x +O(x 2) .From (4.68), we also know that drdx =\u0000 r0x 2 + r2 +2r3x +O(x 2) .Equating like powers of x in our previous two expressions, we can systematically solve for r1, r2and r3 to getr1 = f (re)1/2 cosa\u00001H ,r2 = 1\u0000 f (re)cos2a8H , (4.76)r3 = 124H \u00EF\u00A3\u00BF\u00E2\u009C\u0093 f (re)\u0000 14\u00E2\u0097\u0086 f (re)1/2 cos3a\u0000 32 f (re)cos2a\u0000 34 f (re)1/2 cosa + 32\u0000\u0000 M12 ,where we have used r0 = 2/H. This process can be continued ad infinitum to obtain all higherorder coefficients in the series. More practically, one can use a computer algebra program suchas MAPLE to find the coefficients. For example, one can apply the \u00E2\u0080\u009Cseries\u00E2\u0080\u009D method of the dsolveroutine of MAPLE 14 to equation (4.67) to determine the higher order coefficients in the expansion.1014.4. Horizon shapesExpansion for f(x )We can use the expansion for r(x ) to obtain an expansion for f(x ), valid for x \u00E2\u008C\u00A7 1. For this purposeit is useful to first use r as the independent variable instead of x . It is possible to do this for x \u00E2\u008C\u00A7 1since r(x ) is a monotonic function. The equation for f(r) can be found by dividing the equation forf(T ) by the equation for r(T ) in (4.12). This givesdfdr = \u0000sinar2 1(H2\u0000Ve f f (r))1/2 .Expanding the right hand side for large r givesdfdr = \u0000sinaH \u00E2\u009C\u0093 1r2 + sin2a2H2 1r4 \u0000M sin2aH2 1r5\u00E2\u0097\u0086+O\u00E2\u009C\u0093 1r6\u00E2\u0097\u0086 .Now integrate with respect to r to getf(r) = sinaH \u00E2\u009C\u00931r + 13 sin2a2H2 1r3 \u0000M sin2a4H2 1r4\u00E2\u0097\u0086+O\u00E2\u009C\u0093 1r5\u00E2\u0097\u0086 . (4.77)(4.68) implies the following expansions:1r = xr0 \u00E2\u009C\u00931\u0000 r1r0 x + r21\u0000 r2r0r20 x 2 + 2r1r2r0\u0000 r3r20\u0000 r31r30 x 3 +O(x 4)\u00E2\u0097\u0086 ,1r3 = x 3r30 \u00E2\u009C\u00931\u0000 3r1r0 x +O(x 2)\u00E2\u0097\u0086 ,1r4 = x 4r40 +O(x 5) .1024.4. Horizon shapesSubstituting the above into (4.77) and collecting like powers of x , we obtain the following seriesfor f(x ): f(x ) = sinaH xr0 \u00EF\u00A3\u00BF1\u0000 r1r0 x + 1r20 \u00E2\u009C\u0093r21\u0000 r2r0 + sin2a6H2 \u00E2\u0097\u0086x 2 (4.78)+1r30 \u00E2\u009C\u00932r1r2r0\u0000 r3r20\u0000 r31\u0000 r1 sin2a2H2 \u0000M sin2a4H2 \u00E2\u0097\u0086x 3 +O(x 4)\u0000 ,where r1, r2 and r3 are given by (4.76). Writing the above asf(x ) = f1x +f2x 2 +f3x 3 +f4x 4 +O \u0000x 5\u0000 , (4.79)we find the following explicit formulas for the coefficients of the expansion:f1 = sina2 , (4.80)f2 = \u0000sina4 \u00E2\u0087\u00A3 f (re)1/2 cosa\u00001\u00E2\u008C\u0098 , (4.81)f3 = sina24 \u00E2\u009C\u009314 (9 f (re)\u00001)cos2a\u00006 f (re)1/2 cosa + 32 f (re)+ 52\u00E2\u0097\u0086 , (4.82)f4 = sina16 \u00EF\u00A3\u00BF23\u00E2\u009C\u009314 \u0000 f (re)\u00E2\u0097\u0086 f (re)1/2 cos3a + 14\u00E2\u009C\u00939 f (re)+ MH2 \u00001\u00E2\u0097\u0086cos2a (4.83)\u0000\u00E2\u009C\u0093 f (re)+ 52\u00E2\u0097\u0086 f (re)1/2 cosa + 32 f (re)+ 524MH + 12\u0000 .where we have used r0 = 2/H.Validity of the expansionOur choice for the form of the series (4.68) was not motivated by any rigorous mathematical theoremor technique. However, it is nevertheless accurate, as can be seen in figure 4.8. To understand ourseries solution further, it is useful to make a change of variables and define r\u00C2\u00AF(x ) asr\u00C2\u00AF(x ) = x r(x )\u0000 r0 . (4.84)1034.4. Horizon shapesFigure 4.8: The green curve is the numerical solution to (4.85), and the two red curves are the seriessolution to (4.85) at third order and eleventh order. Notice the remarkable accuracy of the eleventhorder series solution, especially for 0 < x < 1. Parameter values used are: M = 1, H = (6p3)\u00001,a = p/4. The initial condition leading to the numerical solution is set by using r\u00C2\u00AF(x0) = r\u00C2\u00AF(2)(x0),where x0 = 10\u000010 and r\u00C2\u00AF(2)(x ) is the second order series solution.1044.4. Horizon shapesThe equation (4.67) for r(x ) can then be transformed into the following equation for r\u00C2\u00AF(x ):dr\u00C2\u00AFdx = 1x \u00E2\u009C\u0093 \u00002xH (2\u0000x )G+\u00E2\u009C\u0093r0 + r\u00C2\u00AFx \u00E2\u0097\u0086+ r0 + r\u00C2\u00AF\u00E2\u0097\u0086 . (4.85)In figure 4.8, we plot the numerical solution to the above equation, which for our purposes can beconsidered the exact solution, along with the series solution to the above at two different orders. Theseries solution is obtained using MAPLE\u00E2\u0080\u0099s dsolve procedure with the series method, and leads to thesame answer as would be obtained by substituting (4.68)-(4.76) into (4.84). We see the increasedaccuracy with higher order, as well as the remarkable accuracy of the higher order solution, espe-cially for values 0 < x < 1 (notice that x ! 2 corresponds to T !\u0000\u00E2\u0080\u00A2). This agreement with thenumerical solution increases our confidence in the series expansion, especially for smaller values ofx .The difficulty in applying traditional mathematical techniques and theorems to equation (4.85) liesin the fact that it is ill-defined at x = 0. Although the right hand side of (4.85) is undefined at x = 0,it approaches a finite limit as x ! 0 provided thatr\u00C2\u00AF(x )\u00E2\u0087\u00A0 x .Based on this we can construct the approximate solutionr\u00C2\u00AF(x ) = r1x +o(x ) ,where o(\u00C2\u00B7) is little-o notation, and r1 is given by (4.76). Continuing this process, we can constructthe following series: r\u00C2\u00AF(x ) = n\u00C3\u0082i=1rix i +o(x n) , (4.86)where the coefficients ri are determined iteratively using the procedure outlined in section (4.4.3).The above is an approximation to the solution of (4.85) in the sense of being an asymptotic series.However, we cannot be assured that it approaches an exact solution of (4.85) as n ! \u00E2\u0080\u00A2. In fact itis unlikely to be an exact solution. For example, if it is an exact solution then we ought to be ableto use it to find the asymptotic behavior of the horizon area at late times. With this in mind, let us1054.4. Horizon shapessubstitute the power series (4.68) into the formula for the area element dA in (5.4), which will bederived in chapter 5. Doing so, and collecting terms of like powers in x , we finddAp = |sina|H2 +o\u0000x 4\u0000 , (4.87)where we have attached the subscript p to dA to remind ourselves that we are calculating dA in thecontext of approximating r(x ) by a power series. Integrating the above to obtain the cosmologicalhorizon area, as in equation (5.7), we getAp(x ) = 4p\u00E2\u009C\u0093 1H\u00E2\u0097\u00862 +o\u0000x 4\u0000 , (4.88)where the subscript p attached to A(x ) has the same meaning as before. By choosing the specificvalues M = 1 and H = (6p3)\u00001 and using MAPLE to find the series for r(x ) and A(x ), we canextend this result and calculate that the fifth order term vanishes as well. Beyond fifth order theMAPLE calculations become prohibitively time consuming. If this vanishing of terms extends toall orders, then using a power series for r(x ) leads to the result that the horizon area is constant intime, which is clearly false. This provides a hint that the power series (4.68) is not an exact solution,and that either it must be a divergent series, or there must be additional terms in the solution to(4.85) which are not positive integer powers of x . The idea that the exact solution to (4.85) cannotbe expressed as a series in positive integer powers of x is actually a natural expectation, given ifthere was such an exact solution, then the area at late times would be of the formA(x ) = 4p\u00E2\u009C\u0093 1H\u00E2\u0097\u00862\u0000ax n +o(x n) ,for some a > 0 and natural number n. Then using (4.71) to return to the original time variable T ,the above becomes A(T ) = 4p\u00E2\u009C\u0093 1H\u00E2\u0097\u00862\u0000a2n e\u0000nHT +o(x n) .However, there is no reason to think that as T ! \u00E2\u0080\u00A2 the area A(T ) of horizons should have theasymptotic behavior A(T )\u0000A(\u00E2\u0080\u00A2)\u00E2\u0087\u00A0 e\u0000nHT1064.4. Horizon shapesfor some natural number n, and we once again have a hint that a power series cannot be an exactsolution.Although the preceding discussion suggests that (4.86) does not converge to the exact solution of(4.85) as n! \u00E2\u0080\u00A2, we have found that numerical evidence strongly suggests that the series in (4.86)does converge as n!\u00E2\u0080\u00A2. In figure 4.9, we plot the value of the series solution at two different valuesof x , for different values of n in (4.86). Suppose we are correct and the series does converge, butdoes not converge to the exact solution. Let r\u00C2\u00AFe(x ) be the exact solution to (4.85) and r\u00C2\u00AFp(x ) be theasymptotic power series solution, and define the difference function d(x ) asd(x ) = r\u00C2\u00AFe(x )\u0000 r\u00C2\u00AFp(x ) .Now suppose we describe the change in shape of the horizon using a transverse deformation tensorB\u00CB\u009Cab and decompose it into a trace part and two trace free parts in the usual manner [25]:B\u00CB\u009Cab = qdab +sab +wab .Given that the horizon consists of a family of null generators which form a null geodesic congruencewhich is hypersurface orthogonal, it follows that the rotation wab for the congruence vanishes [25],so that we are only left with the expansion term containing q and the shear term sab in the above.From the result (4.87), it appears that the asymptotic power series solution r\u00C2\u00AFp(x ) only describes achange in shape of the horizon, without affecting its area, and thus the difference d(x ) must containthe dependence on x which probably accounts for the change in area of the horizons, as capturedby the expansion parameter q . On the other hand, the power series r\u00C2\u00AFp(x ) describes a dependenceon x which decribes a shearing of the horizon. That is, it describes a change in shape which doesnot affect the area and which does not cause any rotation.The final issue which must be addressed is the question of the uniqueness of the solution to (4.85).Because the right hand side is not defined at x = 0, we are not aware of any theorems that can beapplied to ensure the uniqueness of a solution to the equation (4.85) with the \u00E2\u0080\u009Cinitial condition\u00E2\u0080\u009D1074.4. Horizon shapesFigure 4.9: Plot of the series solution (4.86) (blue points) vs order n for x = 0.5 (left) and x = 1.5(right). The horizontal red line is the numerically determined solution to (4.85), evaluated at x =0.5 (left) or x = 1.5 (right). The blues points appear to converge to a limiting value, suggestingthat the series in (4.86) does converge to a final value as n ! \u00E2\u0080\u00A2. Although it appears that theseries solution (blue points) converges to the numerical solution (red line), we believe that thereis a residual difference between the two that cannot be eliminated, even in the limit as n ! \u00E2\u0080\u00A2.This difference is too small to be seen on the plots, however. Parameter values used: M = 1,H = (3p6)\u00001, a = p/4. Initial conditions for the numerical solution as set using the n = 2 seriessolution at x = 10\u000010.r\u00C2\u00AF(0) = 0 (more precisely, because it is not possible to define a solution for x = 0, we can only definea limiting condition on r\u00C2\u00AF(e), which should be written r\u00C2\u00AF(0+)= 0). However, one can define the initialcondition r\u00C2\u00AF(x0) = r\u00C2\u00AF0 for some x0 > 0. The Picard-Lindelof theorem can then be used to ensure aunique solution for some interval about x0. To ensure a unique solution satisfying r\u00C2\u00AF(0+) = 0, onmust show that there is a unique value of r\u00C2\u00AF0 such that r\u00C2\u00AF(0+) = 0. Doing this is beyond the scope ofthis thesis. However, we have plotted the phase portrait associated with (4.85) and have confirmednumerically that there is a single trajectory satisfying r\u00C2\u00AF(0+) = 0.1084.4. Horizon shapesThe shape of the horizon at late timesAs can be seen in figure 4.4, the horizon at late times settles down to a final shape. As discussed insection 4.4.1, this shape is not spherical (i.e. not circular when one dimension is suppressed). Thismay seem suprising, but is simply an artifact of the choice of time slicing. Although the horizon atlate times is in a part of SdS spacetime that is well approximated by deSitter spacetime, the slicesT = constant in this part of the spacetime do not coincide with the usual constant time slices ofdeSitter spacetime in the so-called planar coordinates. The unusual choice of time slices obscuresthe fact that the horizon, when viewed as a null hypersurface, does indeed approach the horizon ofdeSitter spacetime.In this section, we combine the expansions for r(x ) and f(x ) (equations (4.68) and (4.78)) todescribe this final shape of the horizon at late times analytically.In order to describe this final shape, it is useful to introduce the variables r and y as follows:r = \u00E2\u0087\u00A3(r cosf \u0000 ravg)2 +(r sinf)2\u00E2\u008C\u00981/2 ,y = arctan\u00E2\u009C\u0093 r sinfr cosf \u0000 ravg\u00E2\u0097\u0086 .where ravg =(r(x ;a = 0)+r(x ;a = p))/2. r andy can be thought of as polar coordinates centeredat the point with coordinates r = ravg and f = 0, where ravg is the midpoint of the two parts of thehorizon which intersect the line f = 0. Substituting the expansions (4.68) and (4.78) for r(x ) andf(x ) into the above, we can obtain the expansions for r(x ) and y(x ). Doing so, and keeping onlythe lowest order term in each expansion, we obtainr = 1Hq f (re)cos2a + sin2a +O(x ),y = arctan 1p f (re) tana!+O(x ) .1094.5. Merger pointTruncating at the lowest order and eliminating a , we can obtain r(y) in the limit that x ! 0:r(y) = 1Hs1+ f (re)\u000011+ f (re) tan2y . (4.89)Plotting the above gives the same shape as the last frame in figure 4.4, with r a maximum aty = \u00C2\u00B1p/2. Notice that in the limit that the black hole mass vanishes, i.e., for M = 0, we havef (re) = 1 and therefore r(y) = 1/H, so that the horizon shape is circular (or spherical if the qvariable is restored). This is precisely what we would expect, since the case M = 0 corresponds todeSitter spacetime, and the cosmological horizon shape in deSitter spacetime is spherical. Noticealso that for M = 0 the spacelike hypersurfaces T = constant of the LP coordinates used in thisthesis are the same as the constant time slices of the conventional planar coordinates of deSitterspacetime (as was discussed in section 3.3.1).4.5 Merger pointIn this section we find the location in spacetime of the point where the black hole and cosmologicalhorizons first touch (as in the fourth frame of figure 4.4). More precisely, we find the value of theSchwarzschild coordinate r for this event in spacetime, in the limit that e ! 0 (i.e. the limit of aninfinitesimally small black hole). Due to the Killing symmetry discussed in section 3.3.5, it is notpossible to assign an unambiguous value for the time coordinate T of this merger point. Althoughwe introduce the merger time Tp below, only its derivative is meaningful, since it is always possibleto add a constant to Tp.Although we are finding the value the Schwarzschild coordinate r, and thus dealing with a par-ticular coordinate system, the result is essentially coordinate independent. This is due to the factthat the coordinate r has a simple geometric interpretation which can be implemented in any othercoordinate system.1104.5. Merger point4.5.1 Defining equationsLet (Tp,rp) be the LP coordinates of the merger point and let ap be the value of the parameter afor the unique generator that goes through the merger point (we use the subscript p instead of mto indicate the merger point). Note that this generator is really only unique when suppressing theq coordinate by setting q = p/2; otherwise it unique up to rotations about the \u00E2\u0080\u009Cx-axis\u00E2\u0080\u009D shown infigure 4.4.There are three equations relating the three unknowns Tp, rp and ap. These equations aref(Tp;ap) = p , (4.90)r(Tp;ap) = rp , (4.91)\u00E2\u0088\u0082T\u00E2\u0088\u0082a \u0000\u0000\u0000\u0000f=p,a=ap = 0 , (4.92)where f(T ;a) and r(T ;a) are the solutions (4.13) and (4.12), respectively. The function T (f ;a)in the third equation above is simply the inverse of f(T ;a), with a as a passive variable not playingany role in the inversion process. The inverse T (f ;a) is guaranteed to exist by virtue of the fact thatdf/dT > 0 for sina 6= 0, which will be the case for values of a in a sufficiently small neighborhoodabout ap.(4.90) comes from the fact that we expect the merger point to be located along the \u00E2\u0080\u009Cx-axis\u00E2\u0080\u009D (seefigure 4.4), and so the f coordinate of the merger point is f = p . This expectation is based on thenumerical results presented in section 4.4.1, and is not a logical necessity. (4.91) simply followsfrom the definition of Tp, rp and ap. To understand (4.92), consider the function Tp(a) = T (f =p;a). The equation Tp(a) = T0 has no roots for T0 > Tp, two roots for T0 < Tp and precisely oneroot for T0 = Tp. Furthermore, by the definitions of ap and Tp, we clearly have Tp(ap) = Tp. Henceit follows that Tp(a) has a local maximum at ap, and we havedTpda \u0000\u0000\u0000\u0000a=ap = 0 .1114.5. Merger pointThe above equation is precisely (4.92), written using the notation Tp(a) = T (f = p;a).Below we derive two coupled equations relating rp and ap. The first of these equations combines(4.90) and (4.91) into one equation by eliminating Tp. The second of these equations will followfrom (4.92).4.5.2 First equationIn order to derive the equation resulting from (4.90)-(4.91), it will be useful to first consider a familyof spacetime points (Tp ,rp) parametrized by a , and defined byf(Tp ;a) = p , (4.93)r(Tp ;a) = rp . (4.94)Once the above are coupled to (4.92), then we will set a =ap and the above will become the desiredequations (4.90)-(4.91). The reason for not setting a = ap right away is that it will be necessary toleave a arbibtrary when computing dTpda in the next section.For the moment, the only restrictrion we place on the values of a is that they belong to a smallneighborhood about ap. This will ensure that (4.93)-(4.94) has a solution, and that the generatorsunder consideration have a turning point, in the sense that drdT > 0 for T > T \u00E2\u0087\u00A4 and drdT < 0 for T < T \u00E2\u0087\u00A4,where T \u00E2\u0087\u00A4 is the time of the turning point. We only consider generators which have a turning pointsince it turns out this is a necessary condition for (4.92) to have a solution.In order to utilize the condition (4.93), we first define the functions f\u0000(r) and f+(r) asf\u00C2\u00B1(r)\u00E2\u008C\u0098 f(T\u00C2\u00B1(r)) ,where f(T ) is the solution to (4.13) and T\u0000(r) and T+(r) are the inverses of r(T ) for T < T \u00E2\u0087\u00A4 and1124.5. Merger pointT > T \u00E2\u0087\u00A4, respectively. These inverses are guaranteed to exist since drdT > 0 for T > T \u00E2\u0087\u00A4 and drdT < 0for T < T \u00E2\u0087\u00A4.The equations obeyed by f\u00C2\u00B1(r) are obtained by dividing equation (4.13) by (4.12) and taking drdT > 0or drdT < 0 for f+(r) and f\u0000(r), respectively. This givesdf\u00C2\u00B1dr =\u00E2\u008C\u00A5 sinar2 (H2\u0000Veff(r))1/2 . (4.95)We integrate f\u0000(r) from rp to the turning point r\u00E2\u0087\u00A4 and f+(r) from r\u00E2\u0087\u00A4 to r = \u00E2\u0080\u00A2. This gives thefollowing equation relating a , rp and r\u00E2\u0087\u00A4:0\u00CB\u0086p df = r\u00E2\u0087\u00A4\u00CB\u0086rp sinar2 (H2\u0000Veff(r))1/2 dr\u0000 \u00E2\u0080\u00A2\u00CB\u0086r\u00E2\u0087\u00A4 sinar2 (H2\u0000Veff(r))1/2 dr , (4.96)where the lower and upper limits of integration on the left hand side follow from (4.93)-(4.94) andf(r = \u00E2\u0080\u00A2) = 0, respectively. Manipulating the limits of integration, we can rewrite the above moresuccinctly as p =0@2 \u00E2\u0080\u00A2\u00CB\u0086r\u00E2\u0087\u00A4 \u0000 \u00E2\u0080\u00A2\u00CB\u0086rp 1A sinar2 (H2\u0000Veff(r;a))1/2 dr , (4.97)where we have written Ve f f (r) = Ve f f (r;a) to emphasize the dependence on a . Since r\u00E2\u0087\u00A4 is theturning point for the trajectory with parameter a , we have from (4.4) the following equation relatingr\u00E2\u0087\u00A4 and a: Ve f f (r\u00E2\u0087\u00A4;a) = H2 . (4.98)The above can be used to eliminate a from (4.97), so that it becomes an equation simply relating rpand r\u00E2\u0087\u00A4. In this way r\u00E2\u0087\u00A4 replaces the role of a in (4.97).Next, we define up and u\u00E2\u0087\u00A4 as up = Mrp , (4.99)u\u00E2\u0087\u00A4 = Mr\u00E2\u0087\u00A4 . (4.100)1134.5. Merger pointWritten in terms of u\u00E2\u0087\u00A4, the turning point condition (4.98) becomesu2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4) = e2sin2a . (4.101)The two integrals on the right hand side of (4.97) can be viewed as functions of up and u\u00E2\u0087\u00A4. Theseintegrals are best recast using the new variables of integrationv = r\u00E2\u0087\u00A4r ,u = Mr .Using these new variables of integration, and replacing the dependence on a , rp and r\u00E2\u0087\u00A4 in favor ofa dependence on u\u00E2\u0087\u00A4 and up instead, (4.97) becomes2I(u\u00E2\u0087\u00A4)\u0000 I1(u\u00E2\u0087\u00A4,up) = p , (4.102)where I(u\u00E2\u0087\u00A4) = 1\u00CB\u00860dv(1\u00002u\u00E2\u0087\u00A4 \u0000 v2(1\u00002u\u00E2\u0087\u00A4v))1/2 , (4.103)I1(u\u00E2\u0087\u00A4,up) = up\u00CB\u00860du(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000u2(1\u00002u))1/2 . (4.104)The equation (4.102), along with the definition of the integrals defined above, gives us an equationrelating u\u00E2\u0087\u00A4 and up . Equivalently, this equation can be tought of as an equation relating rp and a ,with these related to up and u\u00E2\u0087\u00A4 by (4.99) and (4.101), respectively. Similarly, by setting a = apand rp = rp, we obtain an equation relating ap and rp. In the next section we use (4.92) to deriveanother equation relating ap and rp.1144.5. Merger point4.5.3 Second equationThe second equation relating rp and ap will be derived using the following steps: First, we findT 0p(a). Second, we simplify T 0p(a) by taking the limit Tf ! \u00E2\u0080\u00A2, where Tf will be introduced below.Third, we simplify T 0p(a) by taking the limit u\u00C2\u00AF! u\u00E2\u0087\u00A4, where u\u00C2\u00AF will also be introduced below. In thefourth and final step, we find the equation T 0p(a) = 0.First step: Calculation of T 0p(a)The second equation relating rp and ap will be derived from the condition (4.92), and thus requiresus to first find an expression for the function T (f = p;a) \u00E2\u008C\u0098 Tp(a). We start by integrating thedifferential dT from the time Tp(a) to some arbitrary final time Tf , with the restrictions that Tf > T \u00E2\u0087\u00A4and that Tf is independent of a . This givesTf\u00CB\u0086Tp dT = T \u00E2\u0087\u00A4\u00CB\u0086Tp dT + Tf\u00CB\u0086T \u00E2\u0087\u00A4 dT=) Tf \u0000Tp(a) = r\u00E2\u0087\u00A4\u00CB\u0086rp dT\u0000dr dr+ r f\u00CB\u0086r\u00E2\u0087\u00A4 dT+dr dr . (4.105)where r f = r(Tf ;a), and T\u0000(r;a) and T+(r;a) are the inverses of r(T ;a) for T < T \u00E2\u0087\u00A4 and T > T \u00E2\u0087\u00A4,respectively, with a held fixed throughout the inversion process. In going from the first to the secondequation above, we have used (4.94) in changing the limit of integration from Tp to rp .Note that in the above we have assumed that the generator under consideration has a turning point, asin the previous section. As was mentioned, this assumption is based on the fact that it is a necessarycondition for the equations (4.90)-(4.92) to have a solution.It should also be noted that since drdT !0 as T ! T \u00E2\u0087\u00A4, the integrands in the integrals on the right hand1154.5. Merger pointside above are divergent. However, the integrals are convergent, in keeping with the fact that it takesa finite time for a generator to reach the turning point.Since r(T ;a) is given by (4.14), with drdT < 0 for T < T \u00E2\u0087\u00A4 and drdT > 0 for T > T \u00E2\u0087\u00A4, dT\u0000dr and dT+dr canbe obtained by simply inverting the right hand side of (4.14):dT\u0000dr = (G\u0000(r))\u00001 ,dT+dr = (G+(r))\u00001 .(4.105) now becomes Tf \u0000Tp(a) = r f\u00CB\u0086r\u00E2\u0087\u00A4 drG+(r;a) \u0000 rp\u00CB\u0086r\u00E2\u0087\u00A4 drG\u0000(r;a) , (4.106)where we have written G\u00C2\u00B1(r) = G\u00C2\u00B1(r;a) to emphasize the dependence on a , and we have flippedthe limits of integration in one of the integrals. As when deriving the first equation in the previoussection, it is useful to introduce a new variable of integration:u = Mr . (4.107)(4.106) now reads: Tf \u0000Tp(a) = M0B@ u\u00E2\u0087\u00A4\u00CB\u0086u f duu2G+(Mu ;a) \u0000 u\u00E2\u0087\u00A4\u00CB\u0086up duu2G\u0000(Mu ;a)1CA , (4.108)where up , u\u00E2\u0087\u00A4 and u f are given by (4.99), (4.100) and u f = M/r f , respectively. For technical reasonswhich will be discussed below, it will be useful to split the two integrals above as follows:Tf \u0000Tp(a) = M0@\u0000 u f\u00CB\u00860duu2G+(Mu ;a) + u\u00C2\u00AF\u00CB\u00860 duu2G+(Mu ;a) + u\u00E2\u0087\u00A4\u00CB\u0086u\u00C2\u00AF duu2G+(Mu ;a)+up\u00CB\u00860duu2G\u0000(Mu ;a) \u0000 u\u00C2\u00AF\u00CB\u00860 duu2G\u0000(Mu ;a) \u0000 u\u00E2\u0087\u00A4\u00CB\u0086u\u00C2\u00AF duu2G\u0000(Mu ;a)1A , (4.109)1164.5. Merger pointwhere u\u00C2\u00AF is a constant independent of a and satifiesup < u\u00C2\u00AF < u\u00E2\u0087\u00A4 . (4.110)Such a constant u\u00C2\u00AF is guaranteed to exist, provided one restricts values of a to a sufficiently smallinterval about ap.Grouping the integrals with the same limits of integration in (4.109), we getTf \u0000Tp(a) = M0@ u\u00E2\u0087\u00A4\u00CB\u0086u\u00C2\u00AF 1u2 1G+(Mu ;a) \u0000 1G\u0000(Mu ;a)!du+ u\u00C2\u00AF\u00CB\u00860 1u2 1G+(Mu ;a) \u0000 1G\u0000(Mu ;a)!du+up\u00CB\u00860duu2G\u0000(Mu ;a) \u0000 u f\u00CB\u00860 duu2G+(Mu ;a)1A . (4.111)When differentiating the above with respect to a , one encounters potential singularities at the twoproblematic points u\u00E2\u0087\u00A4 and uc, where uc \u00E2\u008C\u0098 rc/M. The first integral contains u\u00E2\u0087\u00A4 in its domain ofintegration and the second and third integrals contain uc in their domains of integration. The fourthintegral contain neither. This follows from the inequalities (4.110) and u f < uc < up . This latterinequality can be deduced from rp < rc < r f , where r f > rc assumes that Tf has been chosen tobe sufficiently large, and where rp < rc is most easily deduced by looking at the effective potentialdiagram (figure 4.6), while keeping in mind that Tp < T \u00E2\u0087\u00A4 and r(T =\u0000\u00E2\u0080\u00A2) = rc.By splitting the domains of integration as we have done, we can deal with these singularities sepa-rately. The first singularity occurs since differentiating the first integral in (4.111) with respect to aforces us to evaluate the integrand at u = u\u00E2\u0087\u00A4, where it is undefined. We can obviate this difficulty bymaking a change of variables to a new integration variable v defined as follows:v = uu\u00E2\u0087\u00A4 . (4.112)1174.5. Merger pointUsing the change of variables (4.112), (4.111) becomesTf \u0000Tp(a) = M0B@ 1\u00CB\u0086u\u00C2\u00AF/u\u00E2\u0087\u00A4 1u\u00E2\u0087\u00A4v2 1G+( Mu\u00E2\u0087\u00A4v ;a) \u0000 1G\u0000( Mu\u00E2\u0087\u00A4v ;a)!dv+u\u00C2\u00AF\u00CB\u008601u2 1G+(Mu ;a) \u0000 1G\u0000(Mu ;a)!du+ up\u00CB\u00860 duu2G\u0000(Mu ;a) \u0000 u f\u00CB\u00860 duu2G+(Mu ;a)1A . (4.113)It turns out that if one makes this same change of variables in the second and third integrals above,then when differentiating the integrand with respect to a , one obtains an integrand with a (u\u00E2\u0087\u00A4v\u0000uc)\u00001 divergence at v = uc/u\u00E2\u0087\u00A4. However, if one retains the integration variable u, this difficulty doesnot arise. This is the technical reason for splitting the integrals that we alluded to earlier. The factthat the choice of integration variables can affect the ability to calculate T 0p(a) may seem surprising.It is a consequence of the fact that if one does not choose integration variables carefully, then T 0p(a)is only defined in the sense of having two diverging terms cancelling. This is reminiscent of animproper integral whose value is only defined when one considers the Cauchy principal value ofthis integral.An alternative method to the one outlined in (4.109)-(4.113) would be to replace the upper limitsof integration in the two integrals in (4.108) with a family of functions u\u00CB\u0086(a;d ), such that u\u00CB\u0086(a;0) =u\u00E2\u0087\u00A4(a) and u\u00CB\u00860(a;0) = u0\u00E2\u0087\u00A4(a), and then take the limit d ! 0 only after the differentiation of theintegrals with respect to a . In general such a method is difficult to implement numerically, sincethe \u00E2\u0080\u009Cprincipal value\u00E2\u0080\u009D one obtains when taking d ! 0 depends on the cancellation of two divergingterms, one of which is an integral that cannot be evaluated analytically. This leads to the subtractionof two very large numbers and a loss of significance in the numerical calculations. However, bysheer luck it turns out that one can manipulate the expressions involved and analytically cancel thetwo diverging quantities, so that the limit d ! 0 can be implemented numerically.As an equivalent method to the one described in the previous paragraph, one can take the limitu\u00C2\u00AF! u\u00E2\u0087\u00A4 once the expression for T 0p(a) is obtained. This is what will be done below.1184.5. Merger pointThe final steps before differentiating (4.113) is to find an explicit expression for G\u00C2\u00B1(Mu ,a) in termsof the integration variable u, and to use (4.101) in order to eliminate the parameter a in favor of theturning point u\u00E2\u0087\u00A4, as was done in the previous section. Recall from (4.14) that G\u00C2\u00B1(r;a) is given byG\u00C2\u00B1(r;a) = \u0000 f (r)pH2\u0000Ve f f (r)\u00C2\u00B1H f (re) 12 cosa +( f (re)\u0000 f (r)) 12 sgn(r\u0000 re)pH2\u0000Ve f f (r) , (4.114)where Ve f f (r;a) is given by (4.5). Using (4.101) and (4.107), we can replace a and r in Ve f f (r;a)in favor of u\u00E2\u0087\u00A4 and u to giveVe f f \u00E2\u009C\u0093Mu ;a(u\u00E2\u0087\u00A4)\u00E2\u0097\u0086= e2u2u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)M2 (1\u00002u) , (4.115)where we have written a = a(u\u00E2\u0087\u00A4) to emphasize the dependence on u\u00E2\u0087\u00A4. Also using (4.107), we havecosa =\u0000s1\u0000 e2u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4) , (4.116)where we have used cosa < 0, which is a necessary condition for generators with a turning point(discussed in more detail in section 4.4.2). Let us define F\u00C2\u00B1(u,u\u00E2\u0087\u00A4) as follows:F\u00C2\u00B1(u,u\u00E2\u0087\u00A4) = 1G\u00C2\u00B1 \u0000Mu ,a(u\u00E2\u0087\u00A4)\u0000 . (4.117)Now we substitute (4.107), (4.115) and (4.116) into (4.114), and substitute the resulting expressionfor G\u00C2\u00B1(Mu ;a(u\u00E2\u0087\u00A4)) into the above to obtainF\u00C2\u00B1(u,u\u00E2\u0087\u00A4) = 11\u00002u\u0000 e2u2 0@\u00C2\u00B1(1\u00003e 23 ) 12 \u0000u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2\u0000 12(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000u2(1\u00002u)) 12 \u0000\u00E2\u009C\u00932u+ e2u2 \u00003e 23\u00E2\u0097\u0086 12 sgn(ue\u0000u)1A ,(4.118)where we have used the result (2.6) for re, the definitions (2.5) and (2.13) for f (r) and e , respec-tively, and we have also defined ue = re/M. Written in terms of F\u00C2\u00B1(u,u\u00E2\u0087\u00A4), (4.113) becomesTf \u0000Tp(a) = M0B@ 1\u00CB\u0086u\u00C2\u00AF/u\u00E2\u0087\u00A4 H(u\u00E2\u0087\u00A4v,u\u00E2\u0087\u00A4)u\u00E2\u0087\u00A4v2 dv+ u\u00C2\u00AF\u00CB\u00860 H(u,u\u00E2\u0087\u00A4)u2 du+ up\u00CB\u00860 F\u0000(u,u\u00E2\u0087\u00A4)u2 du\u0000 u f\u00CB\u00860 F+(u,u\u00E2\u0087\u00A4)u2 du1CA ,(4.119)1194.5. Merger pointwhere it is understood that u\u00E2\u0087\u00A4 = u\u00E2\u0087\u00A4(a) and up = up(a), and we have defined H(u,u\u00E2\u0087\u00A4) asH(u,u\u00E2\u0087\u00A4) = F+(u,u\u00E2\u0087\u00A4)\u0000F\u0000(u,u\u00E2\u0087\u00A4) . (4.120)Using (4.118), we have the following explicit formula for H(u,u\u00E2\u0087\u00A4):H(u,u\u00E2\u0087\u00A4) = 2(1\u00003e 23 ) 12 \u0000u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2\u0000 12\u00E2\u0087\u00A31\u00002u\u0000 e2u2\u00E2\u008C\u0098(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000u2(1\u00002u)) 12 . (4.121)We are now in a position to differentiate (4.119) and obtain T 0p(a). Recalling that Tf and u\u00C2\u00AF havebeen chosen to be independent of a , we havedTpda =\u0000M0B@ 1\u00CB\u0086u\u00C2\u00AF/u\u00E2\u0087\u00A4 1v2 ddu\u00E2\u0087\u00A4 \u00E2\u009C\u0093H(u\u00E2\u0087\u00A4v,u\u00E2\u0087\u00A4)u\u00E2\u0087\u00A4 \u00E2\u0097\u0086dv+ H(u\u00C2\u00AF,u\u00E2\u0087\u00A4)u\u00E2\u0087\u00A4u\u00C2\u00AF + u\u00C2\u00AF\u00CB\u00860 1u2 \u00E2\u0088\u0082H\u00E2\u0088\u0082u\u00E2\u0087\u00A4 du+F\u0000(up ,u\u00E2\u0087\u00A4)u2p dupdu\u00E2\u0087\u00A4 + up\u00CB\u008601u2 \u00E2\u0088\u0082F\u0000\u00E2\u0088\u0082u\u00E2\u0087\u00A4 du1A du\u00E2\u0087\u00A4da \u0000 F+(u f ,u\u00E2\u0087\u00A4)u2f \u00E2\u0088\u0082u f\u00E2\u0088\u0082a \u0000 du\u00E2\u0087\u00A4da u f\u00CB\u008601u2 \u00E2\u0088\u0082F+\u00E2\u0088\u0082u\u00E2\u0087\u00A4 du , (4.122)where we have utilized the chain rule in order to differentiate the term in brackets with respect tou\u00E2\u0087\u00A4. Let us define the expressions E1, E2 and E3 as follows:E1(u\u00C2\u00AF) = 1\u00CB\u0086u\u00C2\u00AF/u\u00E2\u0087\u00A4 1v2 ddu\u00E2\u0087\u00A4 \u00E2\u009C\u0093H(u\u00E2\u0087\u00A4v,u\u00E2\u0087\u00A4)u\u00E2\u0087\u00A4 \u00E2\u0097\u0086dv ,E2(u\u00C2\u00AF) = H(u\u00C2\u00AF,u\u00E2\u0087\u00A4)u\u00E2\u0087\u00A4u\u00C2\u00AF + u\u00C2\u00AF\u00CB\u008601u2 \u00E2\u0088\u0082H\u00E2\u0088\u0082u\u00E2\u0087\u00A4 du , (4.123)E3(u f ) = F+(u f ,u\u00E2\u0087\u00A4)u2f \u00E2\u0088\u0082u f\u00E2\u0088\u0082a + du\u00E2\u0087\u00A4da u f\u00CB\u008601u2 \u00E2\u0088\u0082F+\u00E2\u0088\u0082u\u00E2\u0087\u00A4 du .In the next step we show that E3 vanishes in the limit Tf ! \u00E2\u0080\u00A2. We then consider simplifications ofE1(u\u00C2\u00AF) and E2(u\u00C2\u00AF) as u\u00C2\u00AF! u\u0000\u00E2\u0087\u00A4 .1204.5. Merger pointSecond step: Simplification of T 0p(a) as Tf ! \u00E2\u0080\u00A2Since the constant Tf was arbitrary, we can take the limit Tf ! \u00E2\u0080\u00A2, which leads to u f ! 0. Let usshow that taking this limit eliminates the last two terms in (4.122). Consider the last term first. From(4.118), we can compute:\u00E2\u0088\u0082F\u00C2\u00B1\u00E2\u0088\u0082u\u00E2\u0087\u00A4 =\u00E2\u008C\u00A5 \u00E2\u0087\u00A31\u00003e 23\u00E2\u008C\u0098 12 u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)u2(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2) 12 (u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000u2(1\u00002u)) 32 . (4.124)The limiting behavior of the above as u! 0 is\u00E2\u0088\u0082F\u00C2\u00B1\u00E2\u0088\u0082u\u00E2\u0087\u00A4 \u00E2\u0087\u00A0 u2 .Hence it follows thatlimu f!0 u f\u00CB\u008601u2 \u00E2\u0088\u0082F+\u00E2\u0088\u0082u\u00E2\u0087\u00A4 du = 0 , (4.125)so that the last term in (4.122) vanishes in the limit that Tf ! \u00E2\u0080\u00A2. Note that multiplication of theabove by du\u00E2\u0087\u00A4/da does not affect this conclusion, since differentiating (4.101) and isolating du\u00E2\u0087\u00A4/da ,we have du\u00E2\u0087\u00A4da = e2 cosau\u00E2\u0087\u00A4(3u\u00E2\u0087\u00A4 \u00001)sin3a ,from which it follows that0 3M (see figure 4.6), we have 0 < u\u00E2\u0087\u00A4 < 1/3 and thus A(u) > 0 for 0 < u \u00EF\u00A3\u00BF u\u00E2\u0087\u00A4. Thisensures that the integrand above has the following behavior as u! u\u0000\u00E2\u0087\u00A4 :(2u\u00001+4u\u00E2\u0087\u00A4)\u0000A\u00002 +A\u00001A\u00001\u00E2\u0087\u00A4 +A\u00002\u00E2\u0087\u00A4 \u0000AA\u00E2\u0087\u00A4\u00E2\u009C\u0093A\u0000 32 +A\u0000 32\u00E2\u0087\u00A4 \u00E2\u0097\u0086(u\u00E2\u0087\u00A4 \u0000u) 12 \u00E2\u0087\u00A0 1(u\u00E2\u0087\u00A4 \u0000u) 12 ,so that (4.135) is a convergent integral as u\u00C2\u00AF ! u\u0000\u00E2\u0087\u00A4 . The second integral in (4.131) is divergent asu\u00C2\u00AF! u\u0000\u00E2\u0087\u00A4 , but can be evaluated analytically:u\u00C2\u00AF\u00CB\u00860du(2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 (u\u00E2\u0087\u00A4 \u0000u) 32 = 2(2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 1(u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12 \u0000 1u 12\u00E2\u0087\u00A4 ! .1244.5. Merger pointUsing the above and (4.135), we can now write (4.131) as the sum of three terms, the first two ofwhich are convergent as u\u00C2\u00AF! u\u00E2\u0087\u00A4:u\u00C2\u00AF\u00CB\u00860du(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000u2(1\u00002u)) 32 = u\u00C2\u00AF\u00CB\u00860 (2u\u00001+4u\u00E2\u0087\u00A4)\u0000A\u00002 +A\u00001A\u00001\u00E2\u0087\u00A4 +A\u00002\u00E2\u0087\u00A4 \u0000AA\u00E2\u0087\u00A4\u00E2\u009C\u0093A\u0000 32 +A\u0000 32\u00E2\u0087\u00A4 \u00E2\u0097\u0086(u\u00E2\u0087\u00A4 \u0000u) 12 du\u00002u 12\u00E2\u0087\u00A4 (2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 + 2(2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 (u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12 .Substituting the above into (4.130), we get the following expression:E2(u\u00C2\u00AF) = 2\u00E2\u0087\u00A31\u00003e 23\u00E2\u008C\u0098 12 0@ u\u00C2\u00AF\u0000u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2\u0000 12u\u00E2\u0087\u00A4 (u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)) 12\u00002u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2) 12 (2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 (u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12 !\u00002\u00E2\u0087\u00A31\u00003e 23\u00E2\u008C\u0098 12 u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2) 12\u00E2\u0087\u00A50BB@u\u00C2\u00AF\u00CB\u00860(2u\u00001+4u\u00E2\u0087\u00A4)\u0000A\u00002 +A\u00001A\u00001\u00E2\u0087\u00A4 +A\u00002\u00E2\u0087\u00A4 \u0000AA\u00E2\u0087\u00A4\u00E2\u009C\u0093A\u0000 32 +A\u0000 32\u00E2\u0087\u00A4 \u00E2\u0097\u0086(u\u00E2\u0087\u00A4 \u0000u) 12 du\u0000 2u 12\u00E2\u0087\u00A4 (2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 1CCA . (4.136)The first term in large brackets is a subtraction of two divergent terms as u\u00C2\u00AF! u\u0000\u00E2\u0087\u00A4 . Let us now showthat this term in fact vanishes as u\u00C2\u00AF! u\u0000\u00E2\u0087\u00A4 . First we rewrite it asu\u00C2\u00AF\u0000u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2\u0000 12u\u00E2\u0087\u00A4 (u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)) 12 \u0000 2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2) 12 (2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 (u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12=\u0000u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2\u0000 12u\u00E2\u0087\u00A4 u\u00C2\u00AF(u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)A(u\u00C2\u00AF) 12 \u0000 u\u00E2\u0087\u00A4(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2)A 12\u00E2\u0087\u00A4 ! 1(u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12 . (4.137)We can see we are clearly dealing with a 0/0 limit, and so we apply L\u00E2\u0080\u0099Hopital\u00E2\u0080\u0099s rule:limu\u00C2\u00AF!u\u0000\u00E2\u0087\u00A4 u\u00C2\u00AF(u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)A(u\u00C2\u00AF) 12 \u0000 u\u00E2\u0087\u00A4(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2)A 12\u00E2\u0087\u00A4 ! 1(u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12= limu\u00C2\u00AF!u\u0000\u00E2\u0087\u00A4 \u00002(u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12 ddu\u00C2\u00AF u\u00C2\u00AF(u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)A(u\u00C2\u00AF) 12 ! . (4.138)1254.5. Merger pointWe can compute:ddu\u00C2\u00AF u\u00C2\u00AF(u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)A(u\u00C2\u00AF) 12 !=\u0000u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2\u0000A(u\u00C2\u00AF)\u0000 u\u00C2\u00AF\u0000(2u\u00C2\u00AF\u00006u\u00C2\u00AF2)A(u\u00C2\u00AF)+ \u0000u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2\u0000 12A0(u\u00C2\u00AF)\u0000(u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)2A(u\u00C2\u00AF) 32 .From (4.132)-(4.133), we see that A(u\u00C2\u00AF) and A0(u\u00C2\u00AF) are bounded as u\u00C2\u00AF ! u\u0000\u00E2\u0087\u00A4 . Furthermore, the de-nominator is non-vanishing as u\u00C2\u00AF! u\u0000\u00E2\u0087\u00A4 since (4.101) implies that u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4) > e2 and A(u) > 0 for0 < u\u00EF\u00A3\u00BF u\u00E2\u0087\u00A4 as discussed previously. Therefore we have\u0000\u0000\u0000\u0000\u0000limu\u00C2\u00AF!u\u0000\u00E2\u0087\u00A4 ddu\u00C2\u00AF u\u00C2\u00AF(u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)A(u\u00C2\u00AF) 12 !\u0000\u0000\u0000\u0000\u0000< \u00E2\u0080\u00A2 ,and solimu\u00C2\u00AF!u\u0000\u00E2\u0087\u00A4 \u00002(u\u00E2\u0087\u00A4 \u0000 u\u00C2\u00AF) 12 ddu\u00C2\u00AF u\u00C2\u00AF(u\u00C2\u00AF2(1\u00002u\u00C2\u00AF)\u0000 e2)A(u\u00C2\u00AF) 12 != 0as expected. Combining the above with (4.138), we have that the expression in (4.137) vanishes asu\u00C2\u00AF! u\u0000\u00E2\u0087\u00A4 , so that (4.136) finally becomeslimu\u00C2\u00AF!u\u0000\u00E2\u0087\u00A4 E2(u\u00C2\u00AF) = 2\u00E2\u0087\u00A31\u00003e 23\u00E2\u008C\u0098 12 u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)(u2\u00E2\u0087\u00A4(1\u00002u\u00E2\u0087\u00A4)\u0000 e2) 12\u00E2\u0087\u00A5 2u 12\u00E2\u0087\u00A4 (2u\u00E2\u0087\u00A4(1\u00003u\u00E2\u0087\u00A4)) 32 \u0000 u\u00E2\u0087\u00A4\u00CB\u00860 (2u\u00001+4u\u00E2\u0087\u00A4)\u0000A\u00002 +A\u00001A\u00001\u00E2\u0087\u00A4 +A\u00002\u00E2\u0087\u00A4 \u0000AA\u00E2\u0087\u00A4\u00E2\u009C\u0093A\u0000 32 +A\u0000 32\u00E2\u0087\u00A4 \u00E2\u0097\u0086(u\u00E2\u0087\u00A4 \u0000u) 12 du1CCA . (4.139)Fourth step: The equation T 0(a) = 0Taking the limits Tf !\u00E2\u0080\u00A2 (i.e. u f ! 0) and u\u00C2\u00AF! u\u0000\u00E2\u0087\u00A4 and using (4.125), (4.128) and (4.129) in (4.122),we get dTpda =\u0000M0@ limu\u00C2\u00AF!u\u0000\u00E2\u0087\u00A4 E2(u\u00C2\u00AF)+ F\u0000(up ,u\u00E2\u0087\u00A4)u2p dupdu\u00E2\u0087\u00A4 + up\u00CB\u008601u2 \u00E2\u0088\u0082F\u0000\u00E2\u0088\u0082u\u00E2\u0087\u00A4 du1A du\u00E2\u0087\u00A4da ,1264.5. Merger pointwhere limu\u00C2\u00AF!u\u00E2\u0087\u00A4 E2 is given by (4.139). As shown in the steps leading up to (4.126), we have 0 0 and therefore have substituted|sinf | = sinf in the expression for dA above.5.1.2 Numerical approximationsFor the numerical calculations in section 5.2, it suffices to restrict our attention to the case wherethe line element is given by (5.1). The integral (5.7) is then approximated using Simpson\u00E2\u0080\u0099s method.Let I(a) be the integrand in the above integrals. That is:I(a) = 2pr | sinf | 1f (re)\u00E2\u009C\u0093 drda\u00E2\u0097\u00862 + r2\u00E2\u009C\u0093 dfda\u00E2\u0097\u00862!1/2 . (5.8)Suppose r(a) and f(a) are known at a sequence of evenly spaced grid points (a0, ... ,an), witha0 = 0, an = p and n and even number. Then by Simpson\u00E2\u0080\u0099s method we haveAch = \u00CB\u0086 p0I(a)da \u00E2\u0087\u00A1 Da3n2\u00001\u00C3\u0082k=0 (I(a2k)+4 I(a2k+1)+ I(a2k+2)) , (5.9)where Da is the spacing between neighboring values of ai (i.e. Da = ai+1\u0000ai). Computing theabove requires knowledge of r0(ai) and f 0(ai), both of which can be approximated using centraldifferences: drda \u0000\u0000\u0000\u0000ai \u00E2\u0087\u00A1 r(ai+1)\u0000 r(ai\u00001)2Da , (5.10)dfda \u0000\u0000\u0000\u0000ai \u00E2\u0087\u00A1 f(ai+1)\u0000f(ai\u00001)2Da .1365.1. Horizon area calculationsThe endpoints a0 = 0 and an = p can be dealt with by exploiting r(\u0000a) = r(a) and f(\u0000a) =\u0000f(a) so that the central differences becomedrda \u0000\u0000\u0000\u0000a0 = drda \u0000\u0000\u0000\u0000an = 0 , (5.11)dfda \u0000\u0000\u0000\u0000a0 \u00E2\u0087\u00A1 f(a1)Da ,dfda \u0000\u0000\u0000\u0000an \u00E2\u0087\u00A1 p\u0000f(an\u00001)Da .Using (5.10)-(5.11) in (5.8)-(5.9), we can estimate (5.7) using only knowledge of r(a) and f(a) atthe sequence of points (a0, ... ,an). The method used for computing r(a) and f(a) was describedin section 4.3.To estimate the integrals in (5.5), let us first define ai1 and ai2 such thatai1\u00001 < ac < ai1 < ... < ai2 < ab < ai2+1 .We then separate each integral into two pieces:\u00CB\u0086 ac0I(a)da = \u00CB\u0086 ai1\u0000p0+\u00CB\u0086 acai1\u0000p I(a)da , (5.12)\u00CB\u0086 pab I(a)da = \u00CB\u0086 ai2+lab +\u00CB\u0086 pai2+l I(a)da ,where p = 1 if i1 is odd and p = 2 if i1 is even. Similarly, l = 1 if i2 is odd and l = 2 if i2 is even.The reason for separating the above integrals into two pieces is that one of the pieces is most easilydealt with using Simpson\u00E2\u0080\u0099s method, whereas the other piece can be most easily approximated usingthe trapezoid method. Using Simpson\u00E2\u0080\u0099s method, we have\u00CB\u0086 ai1\u0000p0I(a)da \u00E2\u0087\u00A1 Da3i1\u0000p2 \u00001\u00C3\u0082k=0 (I(a2k)+4 I(a2k+1)+ I(a2k+2)) , (5.13)\u00CB\u0086 pai2+l I(a)da \u00E2\u0087\u00A1 Da3 n2\u00001\u00C3\u0082k= i2+l2 \u00001(I(a2k)+4 I(a2k+1)+ I(a2k+2)) .1375.1. Horizon area calculationsNotice that p 2 {1,2} and l 2 {1,2} are always chosen so that i1\u0000 p and i2 + l are even. We thushave a sum over an odd number of integers in the above, as required by Simpson\u00E2\u0080\u0099s method. Usingthe trapezoid method, we approximate the remaining integrals in (5.12). This gives\u00CB\u0086 acai1\u0000p I(a)da \u00E2\u0087\u00A1 (p\u00001) I(ai1\u00002)+ I(ai1\u00001)2 Da + I(ai1\u00001)+ I(ac)2 (ac\u0000ai1\u00001) , (5.14)\u00CB\u0086 pai2+l I(a)da \u00E2\u0087\u00A1 (l\u00001) I(ai2+1)+ I(ai2+2)2 Da + I(ab)+ I(ai2+1)2 (ai2+1\u0000ab) .If p = 1 in the above, the first term vanishes and we have are applying the trapezoid method usingthe two grid points {ai1\u00001,ac}. By contrast, if p = 2 then we have two terms in the above andwe are using the three grid points {ai1\u00002,ai1\u00001,ac} in the trapezoid method. A similar distinctionapplies to the l = 1 and l = 2 cases. The above formulae require knowledge of ab and ac, which arenot known in general. However, using the fact thatf(ab) = f(ac) = pand knowledge of f(a) at the grid points ai1\u00001 and ac (or ai2 and ab ), we can use a linear inter-polant between these grid points and approximate ac and ab asac = ai1\u00001 + p\u0000f(ai1\u00001)f(ai1)\u0000f(ai1\u00001)Da , (5.15)ab = ai2 + f(ai2)\u0000pf(ai2)\u0000f(ii2+1)Da .Using (5.15) in (5.14) and approximating the integrals in (5.12) using (5.13)-(5.14), we arrive atan approximation of the integrals in (5.5), which requires only knowledge of r(a) and f(a) atthe uniform grid of point (a0, ... ,an). The method for computing r(a) and f(a) was outlined insection 4.3.1385.2. Numerical resultsFigure 5.1: Total area of the horizons vs LP coordinate time T . The area has been normalized tothe limiting value of the area as T ! \u00E2\u0080\u00A2. The horizontal blue line is the limiting value as T !\u0000\u00E2\u0080\u00A2.Parameter values used are: M = 1, L= 1/18, n = 400 and x0 = 10\u00004.5.2 Numerical resultsWe begin by discussing the general features of the area vs time graph in section 5.2.1 below. Wethen focus on three aspects of the horizon area, as revealed by numerical computations. In section5.2.2, we analyze the location of the inflection point in the area vs time graph, and present numericalevidence that it coincides precisely with the time of merger. In section 5.2.3, we analyze the horizonarea in the limit e ! 0, where recall that e = HM. Specifically, we compute the horizon area at thetime of merger for e \u00E2\u008C\u00A7 1. What we find is that for e \u00E2\u008C\u00A7 1, the area at the time of merger differsfrom the final area by an amount which is vanishingly small, with an O(e) dependence on e (thiswill be true when the area is measured in dimensionless units). Also, we analyze the influence ofdifferent horizon generators for e \u00E2\u008C\u00A7 1, and find that in this limit, the horizon area increase can beattributed as being due in equal parts to new generators joining the horizons, and existing generatorsexpanding on the horizon.1395.2. Numerical results5.2.1 Horizon area vs time graph: general featuresUsing the procedure outlined in section 4.3.3, we can compute the coordinates of the null generatorsof the horizon. We then use the procedure described in section 5.1.2 to compute the total area of thehorizons numerically. The resulting graph is shown in figure 5.1.The early time behavior of the area is as expected. That is, as T !\u0000\u00E2\u0080\u00A2, the area approches the valueA(\u0000\u00E2\u0080\u00A2) = 4p \u0000r2b + r2c\u0000 (5.16)which is simply the sum of the areas of the spheres with radii r = rb and r = rc. This agrees withthe fact that at early times, the horizons are simply the concentric spheres at r = rb and r = rc, aswas discussed in section 2.5, and deduced in section 4.4.2.As T ! \u00E2\u0080\u00A2, the area approaches the valueA(\u00E2\u0080\u00A2) = 4p\u00E2\u009C\u0093 1H\u00E2\u0097\u00862 . (5.17)The above is what one would expect for the area of a spherical cosmological horizon in deSitterspacetime with a Hubble radius 1/H. This agrees with the fact that in the limit r!\u00E2\u0080\u00A2, SdS spacetimeis well approximated by deSitter spacetime, as was discussed in section 2.6. The above value isobtained numerically when integrating foward in time, after setting the initial conditions at somelate time T0, as described in section 4.3.3. It can also be obtained analytically by calculating the areaof the final horizon shape found in section 4.4.3. As discussed in section 4.4.3, the LP coordinatesused in this thesis lead to a non-spherical shape for the horizon. However, the area is still whatwould be obtained if the slicing was chosen such that the horizon is a sphere with radius 1/H. Thisis in keeping with the fact that for a stationary horizon, the area of the horizon is independent of thechoice of slicing.The overall shape of the graph is also as expected: it is monotonically increasing, with an inflection1405.2. Numerical resultspoint at the moment where the rate of area increase is greatest. The monotonicity of the area graphis hardly surprising, in light of the famous area theorem of Hawking. However, it should be notedthat the area theorem, as originally proven by Hawking [16], only applies to proper event horizons,and not the causal horizons like the one considered in this thesis. Nevertheless, it is not difficult tosee how the original proof could be extended to causal horizons, given that the original proof restson the focusing theorem, along with the observation that horizon generators cannot leave an eventhorizon through a future caustic point. The focusing theorem clearly applies to the congruence ofnull geodesics which are the null generators of a causal horizon. Furthermore, the presence of afuture caustic point on a causal horizon would seem irreconcilable with the definition of a causalhorizon as the boundary of the causal past of an observer\u00E2\u0080\u0099s trajectory. Although the generalizationof the area theorem to causal horizons seems straight forward enough, to our knowledge, a proofhas not been published at this time. The existence of an unpublished proof has been alluded to in[17] however.5.2.2 Inflection point in the horizon area vs time graphThe location and structure of the inflection point in the graph turn out to be quite intriguing. First,we have found numerically that the time at which the inflection point occurs coincides with the timeof merger of the horizons. This can be seen in figure 5.2b, where the green vertical line has beendrawn at the time of horizon merger. We see that A00(T ) changes sign precisely at the merger time,in keeping with the observation that the merger time coincides with the time where the inflectionpoint occurs. The graphs of A00(T ) also reveals that we do not have A00(T ) = 0 at the inflection point,unlike what we would expect if A(T ) was a twice differentiable function. Instead A00(T ) diverges atthe inflection point. This immediately suggest a hypothesis regarding the reason for the inflectionand merger times coinciding: perhaps there is something about the merger of the horizons whichcauses A00(T ) to diverge as the merger time is approached. For example, we might hypothesize thatthe presence of the caustic prior to merger causes A00(T ) to diverge right before merger time.1415.2. Numerical results(a)(b)Figure 5.2: Examples plots of A0(T ) and A00(T ) for merging black hole and cosmological horizons.Both plots reveal a discontinuity in A00(T ) at the critical merger time Tp. This is most apparent inthe plot of A00(T ), where we see a divergence just prior to the merger time (indicated by a verticalgreen line). Furthermore, this plot reveals that A00(T ) goes from positive to negative at the mergerpoint. Parameter values used: M = 1, L= 1/18, n = 400, x0 = 10\u00004.1425.2. Numerical resultsWe thus have two intriguing hypotheses about the horizon area as a function of time. The first resultis that A00(T ) diverges right before the merger point, and the second is that the merger time coincidesprecisely with the inflection point in the horizon area vs time graph. The first of these hypotheseswill be shown to hold in a broad class of coordinate systems in section 5.3.1 below.5.2.3 Horizon area in the limit e ! 0Area increase prior to mergerThe parameter e = MH characterizes the size of the black hole relative to the size of the cosmolog-ical horizon. As discussed briefly at the end of section 2.7, for a typical supermasive black hole ina universe with a cosmological constant given by the current value according to the L-CDM model,we have e \u00E2\u0087\u00A1 10\u000014, which is an exceedingly small number. For this reason, it is interesting andrelevant to consider questions regarding the horizon area for small values of e . When consideringquestions related to the total area increase, it is useful to introduce the dimensionless area A\u00CB\u0086(T ) asA\u00CB\u0086(T ) = A(T )\u0000A(\u0000\u00E2\u0080\u00A2)A(\u00E2\u0080\u00A2)\u0000A(\u0000\u00E2\u0080\u00A2) , (5.18)where A(T ) is the area at LP coordinate time T , and A(\u00C2\u00B1\u00E2\u0080\u00A2) are given by (5.16) and (5.17). Noticethat A\u00CB\u0086(\u0000\u00E2\u0080\u00A2) = 0 and A\u00CB\u0086(\u00E2\u0080\u00A2) = 1. The above can also be written asA\u00CB\u0086(T ) = DA(T )DA , (5.19)where DA(T ) = A(T )\u0000A(\u0000\u00E2\u0080\u00A2) is the area increase after time T , and DA = A(\u00E2\u0080\u00A2)\u0000A(\u0000\u00E2\u0080\u00A2) is thetotal area increase. Another useful quantity to introduce is a normalized value of e . Recall fromsection 2.7 that SdS spacetime only has a black hole and cosmological horizon provided that e 0 .The second result which would be needed to complete the proof is A00(t+p ) < 0. However, it isnot possible to show that A00(t+p ) < 0 for an arbitrary coordinate system since the sign of A00(t)after merger ultimately depends on the choice of time coordinate. Since showing that A0(t) has a1535.3. Analytical resultsmaximum at the time of merger requires us to show that both A00(t\u0000p )> 0 and A00(t+p )< 0 for a givenchoice choice of time coordinate, our proof is somewhat incomplete. Despite the incompleteness ofour proof, we can motivate that A00(t\u0000p ) > 0 and A00(t+p ) < 0 should occur for a family of coordinatesystems. The reasoning is as follows. If we think of taking a coordinate system which is in somesense close to LP coordinates, then by continuity the fact that A00(t+p ) < 0 for LP coordinates (ascan be seen from figure 5.2b) implies that A00(t+p ) < 0 must hold for any other sufficiently \u00E2\u0080\u009Cnear\u00E2\u0080\u009Dcoordinate system. Since LP coordinates belong to the class of coordinate systems consideredbelow, this other nearby coordinate system can be constrained be within this class, from which itfollows that A00(t\u0000p ) > 0 for this other coordinate system as well. This establishes that A00(t\u0000p ) > 0and A00(t+p ) < 0 hold for coordinate systems which are sufficiently \u00E2\u0080\u009Cnear\u00E2\u0080\u009D to LP coordinates, andwhich satisfy the constraints of the coordinate systems considered below.Second derivative prior to mergerConsider a family of coordinate systems such that the line element on the constant time hypersur-faces is given by (5.2). From (5.5), we know that the area before merger is given by the followingexpression: A(t) = 2p \u00CB\u0086 ac0r | sinf | F(r,t)\u00E2\u009C\u0093 drda\u00E2\u0097\u00862 + r2\u00E2\u009C\u0093 dfda\u00E2\u0097\u00862!1/2 da+2p \u00CB\u0086 pab r | sinf | F(r,t)\u00E2\u009C\u0093 drda\u00E2\u0097\u00862 + r2\u00E2\u009C\u0093 dfda\u00E2\u0097\u00862!1/2 da , (5.30)where a 2 [0,2p) parametrizes the curve which is the intersection of the surface St and q = p/2(see section). In other words, a parametrizes the curves such as those shown in figure 4.4. Insteadof using a to parametrize these curves, it is useful instead to use the arclength along the curve as aparameter. The relationship between arclength l and a isdl2 = F(r,t)\u00E2\u009C\u0093 drda\u00E2\u0097\u00862 + r2\u00E2\u009C\u0093 dfda\u00E2\u0097\u00862!da2 .1545.3. Analytical resultsSince arclength is a strictly increasing function of a we can use it as an integration variable, so thatthe integrals in (5.30) becomeA(t) = 2p \u00CB\u0086 lc0r sinf dl + 2p \u00CB\u0086 Llb r sinf dl ,where lc and lb are the arc length values corresponding to ac and ab, respectively, and L is the totalarc length of the curve. Note that lc = lc(t), lb = lb(t) and L = L(t) are all functions of t , and thatthere is also a dependence on t through r = r(t, l) and f = f(t, l). Differentiating the area formulaabove with respect to t , we get12p A0(t) = r (t, lc(t))sin(f (t, lc(t))) l0c(t)+\u00CB\u0086 lc0 \u00E2\u0088\u0082\u00E2\u0088\u0082t (r sinf) dl+ r (t,L(t))sin(f (t,L(t)))L0(t)\u0000 r (t, lb(t))sin(f (t, lb(t))) l0b(t)+\u00CB\u0086 Llb \u00E2\u0088\u0082\u00E2\u0088\u0082t (r sinf) dl .Differentiating with respect to t again and simplifying slightly, we obtain the following expressionfor A00(t):12p A00(t) = \" \u00E2\u0088\u0082 r\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lc l0c(t)+2 \u00E2\u0088\u0082 r\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lc#sin(f (t, lc(t))) l0c(t)+\" \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lc l0c(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lc#r (t, lc(t))cos(f (t, lc(t))) l0c(t)+ r (t, lc(t))sin(f (t, lc(t))) l00c (t)+\u00CB\u0086 lc0\u00E2\u0088\u0082 2\u00E2\u0088\u0082t2 (r sinf) dl +\u00EF\u00A3\u00BF \u00E2\u0088\u0082 r\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=L L0(t)+2 \u00E2\u0088\u0082 r\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000\u0000sin(f (t,L(t)))L0(t)+\u00EF\u00A3\u00BF \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=L L0(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=L\u0000r (t,L(t))cos(f (t,L(t)))L0(t)+ r (t,L(t))sin(f (t,L(t)))L00(t)\u0000\" \u00E2\u0088\u0082 r\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lb l0b(t)+2 \u00E2\u0088\u0082 r\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lb#sin(f (t, lb(t))) l0b(t)\u0000\" \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lb l0b(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lb#r (t, lb(t))cos(f (t, lb(t))) l0b(t)\u0000 r (t, lb(t))sin(f (t, lb(t))) l00b (t)+\u00CB\u0086 Llb \u00E2\u0088\u0082 2\u00E2\u0088\u0082t2 (r sinf) dl . (5.31)We want to find A00(t) in the limit t ! t\u0000p . To do so, first consider the functions lc(t) and lb(t).Recall that at any time t < tp prior to merger, lc(t) and lb(t) are the values of the parameter l1555.3. Analytical resultsfor which (r(t, lc),f(t, lc)) and (r(t, lb),f(t, lb)) are the locations of the caustic points where newgenerators enter the horizon (these are the nonsmooth points on the horizon in figure 4.4). Byspherical symmetry, we can assume without loss of generality that these caustic points occur alongf = p , so that lc(t) and lb(t) are given implicitly by the following equation:f(t, l) = p . (5.32)That is, we have f(t, lc(t)) = p ,f(t, lb(t)) = p ,for all values t < tp. Similarly, L(t) is defined as the value of l > 0 for which f(t, l) = 0. Thisis half of the total arc length of any of the curves shown in figure 4.4, since the top half of thesecurves starts at f = 0 for l = 0 and ends at f = 0 for l = L. Using f(t,L) = 0 and the above, we canimmediately conclude that the terms involving sin(f(t, lc(t))), sin(f(t, lb(t))) or sin(f(t,L(t)))in (5.31) vanish for all values t < tp, and we are left with12p A00(t) = \" \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lc l0c(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lc#r (t, lc(t))cos(f (t, lc(t))) l0c(t)\u0000\" \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lb l0b(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lb#r (t, lb(t))cos(f (t, lb(t))) l0b(t)+\u00EF\u00A3\u00BF \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=L L0(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=L\u0000r (t,L(t))cos(f (t,L(t)))L0(t)+\u00CB\u0086 lc0\u00E2\u0088\u0082 2\u00E2\u0088\u0082t2 (r sinf) dl +\u00CB\u0086 Llb \u00E2\u0088\u0082 2\u00E2\u0088\u0082t2 (r sinf) dl . (5.33)Next we consider the asymptotic behavior of lc(t) and lb(t) for t < tp and t near tp. For anyvalue of t < tp, (5.32) yields two solutions for l, which are lc(t) and lb(t). For t > tp, horizonmerger has occured and (5.32) no longer has any roots (as in the fifth frame in figure 4.4). Lettinglp = lc(tp) = lb(tp), this implies that f(t, l) has the following Taylor expansion about (tp, lp):f(t, l) = p + \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp (t\u0000 tp)+ \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp (l\u0000 lp)2 +O \u0000(t\u0000 tp)2 +(l\u0000 lp)3\u0000 , (5.34)1565.3. Analytical resultswhere \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000tp,lp < 0 and \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000tp,lp < 0. From the above and (5.32) we can deduce that lc(t) and lb(t)have the following asymptotic behavior:lc(t) = lp\u000024 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp! \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp!\u0000135 12 (tp\u0000 t) 12 +O \u00E2\u0087\u00A3(t\u0000 tp) 32\u00E2\u008C\u0098 , (5.35)lb(t) = lp +24 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp! \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp!\u0000135 12 (tp\u0000 t) 12 +O \u00E2\u0087\u00A3(t\u0000 tp) 32\u00E2\u008C\u0098 . (5.36)Differentiating the above, we get the asymptotic behavior of the first derivatives of lc(t) and lb(t):l0c(t) = 12 24 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp! \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp!\u0000135 12 (tp\u0000 t)\u0000 12 +O \u00E2\u0087\u00A3(t\u0000 tp) 12\u00E2\u008C\u0098 ,l0b(t) = \u000012 24 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp! \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp!\u0000135 12 (tp\u0000 t)\u0000 12 +O \u00E2\u0087\u00A3(t\u0000 tp) 12\u00E2\u008C\u0098 .From (5.34) and (5.35)-(5.36), we can also deduce the following:\u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lc = \u00002 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp! 12 (tp\u0000 t) 12 +O (t\u0000 tp) ,\u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lb = 2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp! 12 (tp\u0000 t) 12 +O (t\u0000 tp) ,\u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lc = \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp +O (t\u0000 tp) ,\u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lb = \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp +O (t\u0000 tp) .1575.3. Analytical resultsUsing the results above, we can now find the asymptotic behavior of the first two terms in (5.33):\" \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lc l0c(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lc#r (t, lc(t))cos(f (t, lc(t))) l0c(t) = (5.37)2432 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp!2 \u0000 \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp!\u0000 12 r(tp, lp)35(tp\u0000 t)\u0000 12 +O(1) (5.38)\" \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=lb l0b(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=lb#r (t, lb(t))cos(f (t, lb(t))) l0b(t) = (5.39)\u00002432 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp!2 \u0000 \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp!\u0000 12 r(tp, lp)35(tp\u0000 t)\u0000 12 +O(1) (5.40)Now consider the third term in (5.33). Since L(t) is defined by f(t,L(t)), the implicit functiontheorem implies that we have:L(t) = Lp + \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,Lp \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000tp,Lp!\u00001 (t\u0000 tp)+O \u0000(t\u0000 tp)2\u0000where Lp = L(tp). Using the above, we find that the third term in (5.33) has the following asymp-totic behavior:\u00EF\u00A3\u00BF \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000l=L L0(t)+2 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000l=L\u0000r (t,L(t))cos(f (t,L(t)))L0(t)= 3 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,Lp!2 \u00E2\u0088\u0082f\u00E2\u0088\u0082 l \u0000\u0000\u0000\u0000tp,Lp!\u00001 +O(t\u0000 tp) = O(1) (5.41)Lastly, consider the two integrals in (5.33). Since the first and second partial derivatives of r(t, l)and f(t, l) with respect to t are O(1), we have:\u00CB\u0086 lc0\u00E2\u0088\u0082 2\u00E2\u0088\u0082t2 (r sinf) dl +\u00CB\u0086 Llb \u00E2\u0088\u0082 2\u00E2\u0088\u0082t2 (r sinf) dl = O(1) (5.42)1585.3. Analytical resultsUsing (5.37)-(5.39), (5.41) and (5.42), we can deduce that our expression for A00(t) in (5.33) hasthe following asymptotic behavior:12p A00(t) = 243 \u00E2\u0088\u0082f\u00E2\u0088\u0082t \u0000\u0000\u0000\u0000tp,lp!2 \u0000 \u00E2\u0088\u0082 2f\u00E2\u0088\u0082 l2 \u0000\u0000\u0000\u0000tp,lp!\u0000 12 r(tp, lp)35(tp\u0000 t)\u0000 12 +O(1)From the above it follows that:limt!t\u0000p 12p A00(t) = \u00E2\u0080\u00A25.3.2 Horizon area increase and horizon generatorsIn this section we derive approximate formulas for DAng(\u00E2\u0080\u00A2) and DAeg(\u00E2\u0080\u00A2), valid for e \u00E2\u008C\u00A7 1, whereDAng(T ) and DAeg(T ) were defined in section 5.2.3. We then use these formulas to prove the mainresult of this section, which is that:DAng(\u00E2\u0080\u00A2)DAeg(\u00E2\u0080\u00A2) ! 1 as e ! 0 (5.43)That is, we prove that in the limit e ! 0, the total area increase due to the joining of new generatorsis equal to the total area increase due to the expansion of generators. This result was discussed insection 5.2.3, where we provided numerical evidence hinting at its validity.We prove (5.43) in three steps. In section (5.3.2), we find approximate formulas for a1 and a2, validfor e \u00E2\u008C\u00A7 1, where a1 and a2 were defined in section 5.2.3. In sections 5.3.2 and 5.3.2, we find thearea increase due to new generators and expanding generators, respectively, for e \u00E2\u008C\u00A7 1. That is, wefind approximate formulas for DAng(\u00E2\u0080\u00A2) and DAeg(\u00E2\u0080\u00A2) , valid for e \u00E2\u008C\u00A7 1. Finally, we combine theresults from sections 5.3.2 and 5.3.2 to find the ratio of DAng(\u00E2\u0080\u00A2) to DAeg(\u00E2\u0080\u00A2), in the limit that e ! 0.1595.3. Analytical resultsCalculation of a1 and a2 for e \u00E2\u008C\u00A7 1Recall from section 5.2.3 that we defined a! and a2 such that generators with a1 \u00EF\u00A3\u00BF a \u00EF\u00A3\u00BF a2 ora1 \u00EF\u00A3\u00BF 2p \u0000a \u00EF\u00A3\u00BF a2 are the new generators which join the horizon at some time. We wish to findformulas for a1(e) and a2(e), viewed as functions of e , in the limit that e \u00E2\u008C\u00A7 1. This is the limit inwhich the Schwarzschild radius of the black hole is much smaller than the Hubble radius associatedwith the cosmological constant. We will do this in three steps. First we find the defining equationsfor a1(e) and a2(e), then we find the formula for a1(e), and finally we find the formula for a2(e).The formula for a1(e) and a2(e) will be used in sections 5.3.2 and 5.3.2 to find the area increasedue to expanding generators and new generators.Defining equations We start by defining f0(a;e) as:f0(a;e) = limT!\u0000\u00E2\u0080\u00A2f(T,a;e)provided such a limit exists. Next notice thatf0(a1;e) = p, (5.44)f0(a2;e) = p (5.45)which follows from the fact that null generators must enter the horizon through the caustic point,together with the fact that the caustic is located at f = p . Also note that f(a1;e) and f(a2;e) areguaranteed to exist since as T !\u0000\u00E2\u0080\u00A2, the null generators with a1 and a2 asymptotically approachthe null generators of the horizon for a stationary observer, which are know to have f = constant.(5.44)-(5.45) implicitly define the functions a1(e) and a2(e). To make further progress, we mustintegrate the null geodesics equations (4.3) and (4.6)-(4.7) in order to find integral expressions forf0(a1;e) and f0(a2;e). From (4.3) and (4.6)-(4.7), we have that null geodesics, when thought of as1605.3. Analytical resultsdescribed by functions f(r), must satisfy one of the two equations:Ifa 2 [0,a\u00E2\u0087\u00A4) : dfdr =8>><>>: \u0000sinar2pH2\u0000Ve f f (r) forr > r\u00E2\u0087\u00A4sinar2pH2\u0000Ve f f (r) forr < r\u00E2\u0087\u00A4 (5.46)Ifa 2 [a\u00E2\u0087\u00A4,p) : dfdr = \u0000sinar2pH2\u0000Ve f f (r) (5.47)In the first case above, there is a turning point of the motion, with r\u00E2\u0087\u00A4 the Schwarzschild coordinateof the turning point. a\u00E2\u0087\u00A4 and r\u00E2\u0087\u00A4 are given by (as discussed in section 4.4.2):sin2a\u00E2\u0087\u00A4 = 27M2H2, a\u00E2\u0087\u00A4 2 \u00E2\u0087\u00A3p2,p\u00E2\u008C\u0098Ve f f (r\u00E2\u0087\u00A4) = H2, r\u00E2\u0087\u00A4 > 3M (5.48)We hypothesize that the null generator with a = a1 has a turning point of the motion, whereas thenull generator with a = a2 does not have a turning point. This hypothesis will then be vindicated byour calculations of a1 and a2 under this assumption. Using our hypothesis, we have that f0(a1;e)and f0(a2;e) will be found by integrating (5.46) and (5.47), respectively. Before integrating (5.46)-(5.47), we need appropriate limits of integration. These can be obtained by considering r(T ) andf(T ) as T ! \u00C2\u00B1\u00E2\u0080\u00A2. Based on our results from section 4.4.2, the null generator with a = a1 has thefollowing limiting behavior: r(T =\u0000\u00E2\u0080\u00A2) = rc (5.49)r(T = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2f(T =\u0000\u00E2\u0080\u00A2) = f0(a1;e)f(T = \u00E2\u0080\u00A2) = 01615.3. Analytical resultsSimilarly, the null generator with a = a2 obeys:r(T =\u0000\u00E2\u0080\u00A2) = rb (5.50)r(T = \u00E2\u0080\u00A2) = \u00E2\u0080\u00A2f(T =\u0000\u00E2\u0080\u00A2) = f0(a2;e)f(T = \u00E2\u0080\u00A2) = 0Using (5.49)-(5.50) as the limits of integration, we can integrate (5.46)-(5.47) to obtain integralexpressions for f0(a1;e) and f(a2;e). These are:f0(a1;e) = \u00E2\u0080\u00A2\u00CB\u0086r\u00E2\u0087\u00A4 sina1r2pH2\u0000Ve f f (r)dr+ rc\u00CB\u0086r\u00E2\u0087\u00A4 sina1r2pH2\u0000Ve f f (r)drf0(a2;e) = \u00E2\u0080\u00A2\u00CB\u0086rb sina2r2pH2\u0000Ve f f (r)drCombining the above with (5.44)-(5.45), we obtain:p = \u00E2\u0080\u00A2\u00CB\u0086r\u00E2\u0087\u00A4 sina1r2pH2\u0000Ve f f (r)dr+ rc\u00CB\u0086r\u00E2\u0087\u00A4 sina1r2pH2\u0000Ve f f (r)drp = \u00E2\u0080\u00A2\u00CB\u0086rb sina2r2pH2\u0000Ve f f (r)drThe above implicitly define the functions a1(e) and a2(e). The depedence of a1 and a2 on e canbe more easily seen by first making the following change of variables in the above integrals:u = Mr1625.3. Analytical resultsThe integrals become:p = 1M u\u00E2\u0087\u00A4\u00CB\u00860sina1qH2\u0000Ve f f \u0000Mu \u0000du+ 1M u\u00E2\u0087\u00A4\u00CB\u0086uc sina1qH2\u0000Ve f f \u0000Mu \u0000du (5.51)p = 1M ub\u00CB\u00860sina2qH2\u0000Ve f f \u0000Mu \u0000du (5.52)where u\u00E2\u0087\u00A4 = M/r\u00E2\u0087\u00A4, uc = M/rc and ub = M/rb. Substituting the explicit expression for Ve f f (r):Ve f f (r) =Ve f f \u00E2\u009C\u0093Mu \u00E2\u0097\u0086= sinaM2 u2 (1\u00002u) (5.53)into (5.51)-(5.52), and manipulating the limits of integration in (5.51), we obtainp = 2 u\u00E2\u0087\u00A4(a1,e)\u00CB\u008601r\u00E2\u0087\u00A3 esina1\u00E2\u008C\u00982\u0000u2(1\u00002u)du\u0000 uc(e)\u00CB\u00860 1r\u00E2\u0087\u00A3 esina1\u00E2\u008C\u00982\u0000u2(1\u00002u)du (5.54)p = ub(e)\u00CB\u008601r\u00E2\u0087\u00A3 esina2\u00E2\u008C\u00982\u0000u2(1\u00002u)du (5.55)where we have written u\u00E2\u0087\u00A4 = u\u00E2\u0087\u00A4(a1,e), uc = uc(e) and ub = ub(e) to emphasize the depedendenceon a1 or e . In the above we clearly see that we are dealing with implicit relations for a1(e) anda2(e). Our task is now to extract the limiting behavior of a1(e) and a2(e) for e \u00E2\u008C\u00A7 1. This will bedone by expanding the above integrals for e \u00E2\u008C\u00A7 1. First define:I(a1,e) = u\u00E2\u0087\u00A4(a1,e)\u00CB\u008601r\u00E2\u0087\u00A3 esina1\u00E2\u008C\u00982\u0000u2(1\u00002u)du, (5.56)I1(a1,e) = uc(e)\u00CB\u008601r\u00E2\u0087\u00A3 esina1\u00E2\u008C\u00982\u0000u2(1\u00002u)du,I2(a2,e) = ub(e)\u00CB\u008601r\u00E2\u0087\u00A3 esina2\u00E2\u008C\u00982\u0000u2(1\u00002u)du1635.3. Analytical resultsso that (5.54)-(5.55) become:2I\u0000 I1 = p, (5.57)I2 = p (5.58)Calculation of a1 Consider (5.57) first. For the integrals I and I1, it will be useful to make thesubstitutions v = u/u\u00E2\u0087\u00A4 and w = u/uc, respectively, so that the integrals become:I(a1,e) = 1\u00CB\u00860u\u00E2\u0087\u00A4r\u00E2\u0087\u00A3 esina1\u00E2\u008C\u00982\u0000 (u\u00E2\u0087\u00A4v)2(1\u00002u\u00E2\u0087\u00A4v)dv (5.59)I1(a1,e) = 1\u00CB\u00860ucr\u00E2\u0087\u00A3 esina1\u00E2\u008C\u00982\u0000 (ucw)2(1\u00002ucw)dw (5.60)Recall that by definition, r\u00E2\u0087\u00A4 and rc satisfy:Ve f f (r\u00E2\u0087\u00A4) = H2,f (rc) = 0so that u\u00E2\u0087\u00A4 and uc satisfy Ve f f \u00E2\u009C\u0093Mu\u00E2\u0087\u00A4\u00E2\u0097\u0086 = H2,f \u00E2\u009C\u0093Muc\u00E2\u0097\u0086 = 0Using Ve f f (r) = sin2ar2 (1\u0000 2Mr ) and f (r)\u00001\u0000 2Mr \u0000H2r2 , the above two equations become(u\u00E2\u0087\u00A4)2(1\u00002u\u00E2\u0087\u00A4) = \u00E2\u009C\u0093 esina1\u00E2\u0097\u00862 , (5.61)u2c(1\u00002uc) = e2 ) uc = e +O(e2) (5.62)1645.3. Analytical resultsUsing (5.61) in (5.59)-(5.60) to eliminate a1 and e in favor of u\u00E2\u0087\u00A4, the integrals become:I(u\u00E2\u0087\u00A4) = 1\u00CB\u008601p1\u00002u\u00E2\u0087\u00A4 \u0000 v2(1\u00002u\u00E2\u0087\u00A4v)dv (5.63)I1(u\u00E2\u0087\u00A4,uc) = ucu\u00E2\u0087\u00A4 1\u00CB\u008601q1\u00002u\u00E2\u0087\u00A4 \u0000 \u0000 ucu\u00E2\u0087\u00A4 \u00002w2(1\u00002ucw)dw (5.64)where we have written I = I(u\u00E2\u0087\u00A4) and I1 = I1(u\u00E2\u0087\u00A4,uc) to emphasize that these can be viewed as func-tions of u\u00E2\u0087\u00A4 and uc instead of a1 and e . For the purpose of expanding the above integrals for e \u00E2\u008C\u00A7 1,it is easier to treat u\u00E2\u0087\u00A4 \u00E2\u008C\u00A7 1 as the small parameter, and furthermore view uc = uc(u\u00E2\u0087\u00A4) as a function ofu\u00E2\u0087\u00A4, whose behavior is defined implicitly by (5.57). That is:2I(u\u00E2\u0087\u00A4)\u0000 I1(u\u00E2\u0087\u00A4,uc) = p (5.65)Once uc(u\u00E2\u0087\u00A4) for u\u00E2\u0087\u00A4 \u00E2\u008C\u00A7 1 is determined using the above, the behavior of a1(e) for e \u00E2\u008C\u00A7 1 can then inturn be obtained from (5.61)-(5.62). We start by expanding uc(u\u00E2\u0087\u00A4) in a Taylor series:uc(u\u00E2\u0087\u00A4) = uc(0)+u0c(0)u\u00E2\u0087\u00A4+ 12u00c (0)(u\u00E2\u0087\u00A4)2 +O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)3\u00E2\u008C\u0098From (5.61)-(5.62), we see that uc(0) = 0, so that the above simplifies touc(u\u00E2\u0087\u00A4) = u0c(0)u\u00E2\u0087\u00A4+ 12u00c (0)(u\u00E2\u0087\u00A4)2 +O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)3\u00E2\u008C\u0098Let us expand the integral in (5.64) to lowest order in u\u00E2\u0087\u00A4. We have:1\u00002u\u00E2\u0087\u00A4 \u0000\u00E2\u0087\u00A3ucu\u00E2\u0087\u00A4\u00E2\u008C\u00982w2(1\u00002ucw) = 1\u0000u0c(0)2w2 +O(u\u00E2\u0087\u00A4))r1\u00002u\u00E2\u0087\u00A4 \u0000\u00E2\u0087\u00A3ucu\u00E2\u0087\u00A4\u00E2\u008C\u00982w2(1\u00002ucw) = 1\u0000 12u0c(0)2w2 +O(u\u00E2\u0087\u00A4))\u00E2\u009C\u00931\u00002u\u00E2\u0087\u00A4 \u0000\u00E2\u0087\u00A3ucu\u00E2\u0087\u00A4\u00E2\u008C\u00982w2(1\u00002ucw)\u00E2\u0097\u0086\u0000 12 = 1+ 12u0c(0)2w2 +O(u\u00E2\u0087\u00A4))1\u00CB\u008601q1\u00002u\u00E2\u0087\u00A4 \u0000 \u0000 ucu\u00E2\u0087\u00A4 \u00002w2(1\u00002ucw)dw = 1+ 16u0c(0)+O(u\u00E2\u0087\u00A4)1655.3. Analytical resultsTherefore, to lowest order in u\u00E2\u0087\u00A4, (5.64) isI1(u\u00E2\u0087\u00A4,uc) = 0@u0c(0)u\u00E2\u0087\u00A4+O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)2\u00E2\u008C\u0098u\u00E2\u0087\u00A4 1A\u00E2\u009C\u00931+ 16u0c(0)+O(u\u00E2\u0087\u00A4)\u00E2\u0097\u0086= u0c(0)\u00E2\u009C\u00931+ 16u0c(0)\u00E2\u0097\u0086+O(u\u00E2\u0087\u00A4)To lowest order in u\u00E2\u0087\u00A4, (5.63) is I(u\u00E2\u0087\u00A4) = I(0)+O (u\u00E2\u0087\u00A4)Using the above expansions for I and I1 in (5.65), this equation becomes, to lowest order2I(0)+u0c(0)\u00E2\u009C\u00931+ 16u0c(0)\u00E2\u0097\u0086+O(u\u00E2\u0087\u00A4) = p (5.66)Computing I(0) = 1\u00CB\u008601p1\u0000 v2 dv = p2and substituting into (5.66), we find that equating lowest order terms gives u0c(0) = 0. Our Taylorexpansion for uc(u\u00E2\u0087\u00A4) now reads:uc(u\u00E2\u0087\u00A4) = 12u00c (0)(u\u00E2\u0087\u00A4)2 +O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)3\u00E2\u008C\u0098We now consider (5.66) at first order in u\u00E2\u0087\u00A4. To first order in u\u00E2\u0087\u00A4, (5.64) isI1(u\u00E2\u0087\u00A4,uc) = \u00E2\u009C\u0093u0c(0)+ 12u00c (0)u\u00E2\u0087\u00A4+O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)2\u00E2\u008C\u0098\u00E2\u0097\u0086\u00E2\u009C\u00931+ 16u0c(0)+O(u\u00E2\u0087\u00A4)\u00E2\u0097\u0086=\u00E2\u009C\u009312u00c (0)u\u00E2\u0087\u00A4+O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)2\u00E2\u008C\u0098\u00E2\u0097\u0086(1+O(u\u00E2\u0087\u00A4))=12u00c (0)u\u00E2\u0087\u00A4+O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)2\u00E2\u008C\u0098and to first order in u\u00E2\u0087\u00A4, (5.63) isI(u\u00E2\u0087\u00A4) = I(0)+ I0(0)u\u00E2\u0087\u00A4+O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)2\u00E2\u008C\u00981665.3. Analytical resultsUsing the above expansions for I and I1 in (5.57), we get:2I(0)+2I0(0)u\u00E2\u0087\u00A4 \u0000 12u00c (0)u\u00E2\u0087\u00A4+O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)2\u00E2\u008C\u0098= pEquating terms that are first order in u\u00E2\u0087\u00A4, we get:u00c (0) = 4I0(0)I0(0) can be computed to be: I0(0) = 1\u00CB\u008601\u0000 v3(1\u0000 v2) 32 dv = 2so that we finally arrive at our Taylor expansion for uc(u\u00E2\u0087\u00A4):uc(u\u00E2\u0087\u00A4) = 2I0(0)(u\u00E2\u0087\u00A4)2 +O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)3\u00E2\u008C\u0098= 4(u\u00E2\u0087\u00A4)2 +O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)3\u00E2\u008C\u0098We now have the lowest order behavior of uc(u\u00E2\u0087\u00A4). From it we will obtain the lowest order behaviorof a1(e). First we substitute the above in (5.62) to get:16(u\u00E2\u0087\u00A4)4 +O \u00E2\u0087\u00A3(u\u00E2\u0087\u00A4)6\u00E2\u008C\u0098= e2Let u\u00C2\u00AF = (u\u00E2\u0087\u00A4)2 (5.67)so that the above becomes:16u\u00C2\u00AF2 +O \u0000u\u00C2\u00AF3\u0000= e2 (5.68)Substituting a Taylor expansion for u\u00C2\u00AF(e):u\u00C2\u00AF(e) = u\u00C2\u00AF0(0)e +O(e2) (5.69)into (5.68), we get, to leading order:16u\u00C2\u00AF0(0)2 = 11675.3. Analytical resultsSo that (5.69) becomes: u\u00C2\u00AF(e) = 14e +O(e2)Going to back to (5.67) and solving for u\u00E2\u0087\u00A4 gives the lowest order behavior of u\u00E2\u0087\u00A4(e):u\u00E2\u0087\u00A4(e) = 12e 12 +O \u00E2\u0087\u00A3e 32\u00E2\u008C\u0098Substituting the above in (5.61) and solving for sina1, we get, to leading order:sina1 = 2e 12 +O (e)Solving for a1(e) in the above and retaining only leading order terms, we finally get:a1(e) = p\u00002e 12 +O (e) (5.70)Calculation of a2 Next we turn to finding a2(e) for e \u00E2\u008C\u00A7 1. It will be useful to introduce thefunction s2(e), defined as: s2(e) = \u00E2\u009C\u0093 esina2\u00E2\u0097\u00862 (5.71)The integral (5.56) can now be viewed as a function of s2 and e instead of a2 and e . That is:I2(s2,e) = ub(e)\u00CB\u008601ps2\u0000u2(1\u00002u)duEquation (5.58) now reads: I2(s2,e) = p (5.72)The above implicitly defines the function s2(e). Substituting a Taylor expansion for s2(e):s2(e) = s2(0)+O(e) (5.73)1685.3. Analytical resultsinto (5.72) and using ub(e) = 12 +O(e), we find, to leading order:12\u00CB\u008601ps2(0)\u0000u2(1\u00002u)du = pThe above can be solved numerically to give:s2(0) = 0.05033384242... (5.74)Substituting (5.73) into (5.71) and solving for sina2, we get, to leading order:sina2 = \u00E2\u009C\u0093 1s2(0)1/2\u00E2\u0097\u0086e +O(e2)so that to lowest order, a2(e) is given by:a2(e) = p\u0000a 02(0)e +O(e2) (5.75)where a 02(0) is given by a 02(0) = 1s2(0)1/2 = 4.457280417...The value of s2(0) in the above was obtained from (5.74).Horizon area increase from new generatorsUsing the formulae for a1(e) and a2(e) in (5.70) and (5.75), we can find an approximate formulafor Ang(\u00E2\u0080\u00A2), where Ang(T ) is the area at time T due to the joining of new horizon generators (thiswas defined in section 5.2.3). That is, according to our definitions of a1 and a2 from section 5.2.3,Ang(T ) is the area of the portion of the horizon made up of generators with parameter a satisfyinga1 \u00EF\u00A3\u00BF a \u00EF\u00A3\u00BF a2 or a1 \u00EF\u00A3\u00BF 2p\u0000a \u00EF\u00A3\u00BF a2.We want to find the Ang(T ) in the limit that T ! \u00E2\u0080\u00A2, and thus will first need the coordinates r(T ;a)1695.3. Analytical resultsand f(T ;a) of the generators in this limit. The formula (5.4) can then be used to find the areaelement, and the area element can be integrated from a1 to a2 to find Ang(T ) in the limit thatT ! \u00E2\u0080\u00A2. Recall from (4.69) and (4.79) that we have:r(e) = 2H 1e +O(1)f(e) = sina2e +O(e2)where the limit e ! 0 corresponds to T ! \u00E2\u0080\u00A2. Substituting the above into (5.4), we obtain thefollowing lowest order behavior of the area element, in the limit T ! \u00E2\u0080\u00A2:dA\u00E2\u0087\u00A0 sinaH2 dadb (5.76)Integrating the above over the intervals a1 \u00EF\u00A3\u00BF a \u00EF\u00A3\u00BF a2 and a1 \u00EF\u00A3\u00BF 2p\u0000a \u00EF\u00A3\u00BF a2, as well as 0\u00EF\u00A3\u00BF b < p ,we obtain Ang(\u00E2\u0080\u00A2): Ang(\u00E2\u0080\u00A2) = 2pH2 (\u0000cosa2 + cosa1)Now we substitute the formulas for a1(e) and a2(e) in (5.70) and (5.75) into the above to get:Ang(\u00E2\u0080\u00A2) = 2pH2 hcos\u0000a 02(0)e +O(e2)\u0000\u0000 cos\u00E2\u0087\u00A32e 12 +O(e)\u00E2\u008C\u0098iExpanding the above and keeping only the lowest order terms, we get:Ang(\u00E2\u0080\u00A2) = 4pH2 \u00E2\u0087\u00A3e +O(e3/2)\u00E2\u008C\u0098 (5.77)It follows from the definition of new generators that Ang(\u0000\u00E2\u0080\u00A2) = 0, so that the above is also the totalarea increase due to the new generators. That is:DAng(\u00E2\u0080\u00A2) = Ang(\u00E2\u0080\u00A2)\u0000Ang(\u0000\u00E2\u0080\u00A2)=4pH2 \u00E2\u0087\u00A3e +O(e3/2)\u00E2\u008C\u0098 (5.78)1705.3. Analytical resultsHorizon area increase from expansion of existing generatorsRecall from section 5.2.3 that Aeg(T ) was defined to be the area of the horizon due to generatorsexisting on the horizon for all times. That is, Aeg(T ) is the area of the portion of the horizonmade up of generators which are not new generators, and therefore with parameter a not satisfyinga1 \u00EF\u00A3\u00BF a \u00EF\u00A3\u00BF a2 or a1 \u00EF\u00A3\u00BF 2p \u0000a \u00EF\u00A3\u00BF a2. The two contributions Ang(\u00E2\u0080\u00A2) and Aeg(\u00E2\u0080\u00A2) must add up to thetotal area at late times A(\u00E2\u0080\u00A2). That is, we must have:A(\u00E2\u0080\u00A2) = Ang(\u00E2\u0080\u00A2)+Aeg(\u00E2\u0080\u00A2) (5.79)A(\u00E2\u0080\u00A2) can be found be integrating (5.76) over 0\u00EF\u00A3\u00BF a < 2p and 0\u00EF\u00A3\u00BF b < p . This gives:A(\u00E2\u0080\u00A2) = 4pH2This is precisely what one would expect, since it is the area of a sphere with radius 1/H, whichis the expected horizon shape at late times. Using the above and (5.77) in (5.79), we can obtainAeg(\u00E2\u0080\u00A2): Aeg(\u00E2\u0080\u00A2) = 4pH2 \u00E2\u0087\u00A31\u0000 e +O(e3/2)\u00E2\u008C\u0098 (5.80)At early times, we can also separate the area into two contributions and write:A(\u0000\u00E2\u0080\u00A2) = Ang(\u0000\u00E2\u0080\u00A2)+Aeg(\u0000\u00E2\u0080\u00A2) (5.81)From section 4.4.2 we know that all generators which make up the horizon start at either r = rb orr = rc, so that the horizon at early times consists of the spheres r = rb and r = rc. Therefore wehave: A(\u0000\u00E2\u0080\u00A2) = 4p \u0000r2b + r2c\u0000Furthermore, it follows from the definition of new generators that Ang(\u0000\u00E2\u0080\u00A2) = 0. Using this and theabove in (5.79), we get: Aeg(\u0000\u00E2\u0080\u00A2) = 4p \u0000r2b + r2c\u00001715.3. Analytical resultsUsing the above and (5.80), we can calculate the total area increase due to the expansion of existinggenerators:DAeg(\u00E2\u0080\u00A2) = Aeg(\u00E2\u0080\u00A2)\u0000Aeg(\u0000\u00E2\u0080\u00A2)=4pH2 \u00E2\u0087\u00A31\u0000 (Hrb)2\u0000 (Hrc)2\u0000 e +O(e3/2)\u00E2\u008C\u0098=4pH2 \u00E2\u0087\u00A3e +O(e3/2)\u00E2\u008C\u0098where we have used used (2.15) in the last step. Combining the above with (5.78), we see that wehave:limT!\u00E2\u0080\u00A2 DAeg(\u00E2\u0080\u00A2)DAng(\u00E2\u0080\u00A2) = 1The above is the main result of this section.172Chapter 6Concluding Remarks6.1 Summary and discussion of resultsBy constructing a new coordinate system for SdS spacetime, and then solving the equations forthe null generators of the horizon, we have been able to reveal that the spacetime can provide arare example of an analytically known spacetime which has merging horizons. Such an examplecan then be used as a mathematical laboratory to investigate various questions about the structureand area merging horizons. Here we have focused on three mathematical aspects of the merging ofhorizons: the location and structure of the merger point, the shape of the horizon at late times, andthe horizon area.Our motivation for studying the location and structure of the merger point stems from the fact thatby using an analytical metric, we have a unique opportunity to make analytical claims about thelocation and structure of the merger point. Furthermore, many of these claims can be formulatedin a coordinate independent manner. We were able to find an analytical formula for the location ofthe caustic in the limit of infinitesimally small black hole mass (i.e. the limit e ! 0). Although ithas long been known from numerical simulations [23] that such a caustic point exists (for example,in binary black hole mergers) our study is only the second one which is analytical (the first being[14]), and the first one where the spacetime is known exactly. Furthermore, by focusing on thelimit e ! 0 and by using coordinate systems which have a simple geometric interpretation, we were1736.1. Summary and discussion of resultsable to formulate results in a coordinate independent manner. This is something that would not bepossible without the luxury of having analytical knowledge of the spacetime, as was the case in thisthesis.Our initial motivation for studying the late time behavior of the horizons was that it provided theinitial conditions for the equations describing the coordinates of the null geodesic generators thatmake up the horizon. We were able to develop an asymptotic series for the late time behaviorof the generators. Although this series is very accurate, it is unlikely that it is an exact solution,as was discussed in section 4.4.3. We hypothesized that the discrepancy between our asymptoticsolution and the exact solution is a correction that is vanishingly small at all orders, and whosemain importance is in determining the behavior at late times of the horizon area. If this hypothesisis true, it raises the natural question of whether it could be extended to other spacetimes where ahorizon setttles down to a final shape, or whether it is specific to SdS spacetime. Therefore, althoughwe initially analyzed the behavior of null generators at late times in order to accurately calculatethe coordinates of these generators, we may have accidently stumbled upon a hidden mathematicalrichness in the behavior of these generators.The third aspect of the mathematics of horizons that we considered was the horizon area. Thisis the most physically motivated category of mathematical questions, given that horizon area isdirectly proportional to horizon entropy, as well as being proportional to mass squared in the caseof a Schwarzschild black hole. Our first main result involves an intriguing connection between themerger time and the time at which the rate of change of area is maximal. We provided compellingevidence that these times coincide, and that this coincidence is not merely numerical, nor an artifactof our choice of coordinates. Instead it stems from the fact that the caustic structure before mergerresults in a diverging positive second derivative of area with respect to time. When this is combinedwith a negative second derivative after merger, then it necessarily implies that the rate of changeof area with respect to time must be maximal at merger. It seems to be the case that this secondderivative is negative for a broad class of coordinate systems, so that the result is quite general. Ourresult reveals a deep connection between the rate of change of horizon area and the presence of ahorizon caustic during the merger process. Notice that this connection is not immediately obvious,1746.1. Summary and discussion of resultssince the area increase during the merging process occurs everywhere along the horizon (i.e. allthe generators already on the horizon are expanding). Furthermore, the location of local maximalarea increase does not occur along the caustic, so that this connection cannot be reduced to simplyanalyzing the area increase in the vicinity of the caustic. Nevertheless, the diverging discontinuityin the second derivative of A(T ) ensures that we can firmly link the time of maximal area increasewith the merger time.The second main result regarding the horizon area states that in the limit e ! 0, the contributionto the area increase from expanding generators is precisely equal to the contribution to the areaincrease from the joining of new generators not previously on the horizon. This result is coordinateindependent in the sense that there are canonical constant time slicings that one can use to measurearea at early and late times. That is, the result holds in the broad class of coordinate systems whichuse these canonical slicings at early and late times. This canonical slicing at late times is the onewhich approximates the slicing of deSitter spacetime (recall from the discussion in section 2.6 thatSdS spacetime reduces to deSitter spacetime in the r!\u00E2\u0080\u00A2 limit). As was discussed in section 4.4.3,in general our slicing does not approach this deSitter slicing at late times. However, in the limitthat e ! 0, our slicing does reduce to the deSitter slicing. This can be seen as a consequence ofthe 3-cone geometry reducing to an essentially flat geometry in the limit that e ! 0 (see section3.3.6 for a discussion of the 3-cone geometry). Whether our result regarding the equal contributionsof generators to the area is a mathematical coincidence or is connected to a deeper principle is stillunclear. One way to begin to answer this question would be to investigate this same question in otherspacetimes with merging horizons, such as extreme-mass ratio binary black holes. This possibly isdiscussed below.The third and final set of results regarding the area of horizons are those regarding the behavior ofthe A(T ) graph in the limit e ! 0. More specifically, we investigated three questions in this limit: (i)What is the fractional area increase before merger? (ii) What is the maximal rate of area increase?(iii) What is the average rate of area increase over some suitable time interval?Regarding (i), we were able to present numerical evidence that in the limit e ! 0, all of the area1756.2. Applications and future directionsincrease occurs before merger. Roughly speaking, this result is essentially stating that at mergertime, the area of the two horizons is nearly equal to the area of a spherical cosmological horizonin the absence of a black hole. This suggests that in the limit e ! 0, at the time of merger thesmall distortions that occur near the black hole are of negligible importance relative to the areacontribution from the rest of the horizon. In light of this interpretation of the result, it is naturalto expect that it would hold in other coordinate systems as well. That is, any coordinate system inwhich the slightly distorted cosmological horizon has area close to 4pH\u00002 would yield the sameresult. In this sense we expect our result to be quite general and not restricted to the LP timeslicing chosen in this thesis. In terms of applications, our result could be useful when calculatingcosmological horizon entropy.Questions (ii) and (iii) yield the same qualitative answer: the maximum rate of change of area andthe average rate of change of area both scale as the black hole mass. The constant of proportionalityin the two cases is different of course, but the scaling is the same. Whether this scaling result holdsin other coordinates is unclear. Suppose that it can be shown that in the limit e ! 0, the area increaseis concentrated along a specific part of the horizon. In this case it is conceivable that any coordinatesystem with constant time slices that cross this part of the horizon in a non-pathological mannerwould yield the same scaling relation. The specific details of the choice of time slices would onlyaffect the proportionality constant in the scaling. We suspect that our result is generic in this sense,but the only way to definitively answer the question would be to find an analytical description ofthe horizon area in the limit e ! 0. Such a study is a question we are currently investigating. Aswith question (i) above, questions (ii) and (iii) could have applications in calculating cosmologicalhorizon entropy, or more specifically, the rate of change of cosmological horizon entropy.6.2 Applications and future directionsIn this section we discuss possible applications of the results in this thesis, as well as directions forfuture research.1766.2. Applications and future directions6.2.1 Mathematical questionsOne of the main themes of this thesis has been to use SdS spacetime as a mathematical laboratoryfor exploring questions related to the merging of horizons. By working with a spacetime where thegeometry is known analytically, we have the opportunity to approach many of these questions usinganalytical methods instead of resorting to numerical methods. Furthermore, when using numericalmethods we are solving a family of ordinary differential equations, and are able to confidentlycontrol the errors involved. Despite these advantages, we resorted to numerical methods insteadof analytical methods on several occasions. In addition, these numerical results often had residualcomputational errors which will require further work to eliminate.Numerical issuesThe first issue one could address would be to reduce the magnitude of these computational errors.This could be done in one of three ways: either by improving the numerical accuracy of the solutionsto the differential equations used in this thesis, or by using more null generators to approximate thehorizon, or by improving the algorithm used to compute the area of the horizons. In order controlthe accuracy of the solution to the differential equations in this thesis, the key ingredient is to usean algorithm that can deal with the behavior of the generators near the \u00E2\u0080\u009Cinitial\u00E2\u0080\u009D time T = \u00E2\u0080\u00A2 (orT\u00CB\u0086 = 1\u0000 in compactified coordinates). We chose to solve for the function r(T\u00CB\u0086 ), and use the built-inalgorithm of the MAPLE 14 software, with the option \u00E2\u0080\u009Cstiff=true\u00E2\u0080\u009D to indicate that we are dealingwith a stiff system near T\u00CB\u0086 = 1\u0000. We found that this provided a more accurate solution than solvingfor r\u00CB\u0086(T\u00CB\u0086 ), where r\u00CB\u0086 is a compactification of the Schwarzschild radius r. It also provided a moreefficient means of computing than solving for r(T ), since compactification of the time variableis essential to reducing the runtime of the integration. However, there may be better choices ofvariables and/or algorithms, in the sense that they would provide more efficient and/or accuratesolutions. This is a possibility which we are currently exploring.1776.2. Applications and future directionsIn addition to the choices made for both the algorithm and variables, there are two additional pos-sible soures of error in computing the solution r(T ). The first is due to the fact that the differentialequations are singular at T\u00CB\u0086 = 1, so that we are forced to set initial conditions at T\u00CB\u00860 = 1\u0000e0 for somee \u00E2\u008C\u00A7 1. Such initial conditions must be set using an analytical approximation to the solution, andhere we have used an asymptotic series truncated after a finite number of terms. There is obviouslya choice that must be made regarding the number of terms in the series and the value of the constante0. Here we have chosen to keep eleven terms in the series and use the value e0 = 10\u000010. However,it may be possible to use a smaller value of e0 and keep more terms in the series. The number ofterms in the series is limited only by the computation time involved in calculating the series usinga symbolic manipulation software such as MAPLE. On the other hand, the value of e0 is limitedto sufficiently large values for more fundamental reasons. Using a value of e0 which is too smallwould result in a loss of significance due to rounding errors. A careful choice of variables can helppartly eliminate this problem. For example, compactifying the Schwarzschild radius r such thatr 2 (0,\u00E2\u0080\u00A2) becomes r\u00CB\u0086 2 (0,1) would result in a loss of significance when rounding errors cause r\u00CB\u0086\u00E2\u0087\u00A1 1to become r\u00CB\u0086 = 1. On the other hand, compactifying variables such that r = \u00E2\u0080\u00A2 gets mapped to d = 0would not have such a problem. In this thesis, we have chosen not to compactify the Schwarzschildradius r, for reasons which are discussed in the next paragraph. For such an uncompactified choiceof variables, the initial condition for r(T\u00CB\u0086 ) is:r(T\u00CB\u00860) = r0e0 + r1 + r2e0 + ...As we can see, the difficulty with the above is that for sufficiently small values of e0, rounding dueto finite digits will cause the second and third term in the series to be ignored. Again, compactifyingvariables such that r = \u00E2\u0080\u00A2 gets mapped to d = 0 would not have such a problem. Determining theideal choice of variables in order to minimize these types of rounding errors is an issue we arecurrently working on.The second additional source of error in computing the solution r(T ) comes from the compactifi-cation of the time variable T . It is necessary to compactify the time variable in order to reduce theruntime of the computation. The disadvantage of the compactification comes when attempting to1786.2. Applications and future directionsreturn to the original time variable T by applying the inverse of the mapping used to compactifythe time variable. Such a mapping takes small differences in the compactified time variable T\u00CB\u0086 andmagnifies them greatly. Similarly, small rounding errors in the variable T\u00CB\u0086 get magnified to largeerrors in the original variable T . This poses a problem when calculating derivatives with respect toT , since the error in a small interval of time DT\u00CB\u0086 becomes a large error in the interval DT , which canaffect the accuracy of the denominator in a discretized formula for the derivative (e.g. the derivativeof area with respect to T ). The same problem arises when compactifying the variable r and for thisreason, we chose to only compactify the time variable. One possible solution to these difficulties isto always work with the compactified variables, and never return to the original variables. This is apossibility which we are currently exploring.As mentioned above, a simpler way of achieving more accuracy in our approximation of the hori-zons would be to simply use more horizon generators. Apart from the issues discussed above inchoosing an algorithm used to calculate the horizon generators, this comes down to our choice ofprogramming language, as well as sheer computational power. Here we have used the MAPLEsoftware, but there are clearly more efficient programming languages or software, such as Matlab.As well, we typically chose to compute four hundred or so horizon generators, but could computemany more by using a computer cluster instead of a personal computer.The last improvement which could be made to our numerical computations would be a more ac-curate calculation of the horizon area. The horizon area formula can be well approximated andcomputed using Simpson\u00E2\u0080\u0099s rule for integration, provided one has enough mesh points for integra-tion (i.e. enough horizon generators). Thus the main source of error in the area computation isrelated to the number of generators used, as well as the error in the calculation of the horizon gener-ators. By choosing more horizon generators, we would improve the area calculations. Furthermore,by choosing to compute more horizon generators at points on the horizon where the area is chang-ing rapidly we would also achieve more accuracy. In this thesis we have used a uniform mesh ofhorizon generators, where the uniformity is over the parameter a 2 [0,2p). However, ideally wewould use a non-uniform mesh with more generators near important points on the horizon, such asnear the black hole event horizon or near the merger and caustic points. This possibility of using a1796.2. Applications and future directionsnon-uniform mesh is an issue we are currently exploring.Caustic structureOne of the avenues that was not fully explored analytically in this thesis is the caustic structure nearthe merger point of the horizons. Although we were able to illustrate this caustic structure usingfigure 4.5 and confirm its basic shape, we have not determined the quantitative properties of themerger point geometry, nor have we analytically proven that is has the shape shown in figure 4.5.Ideally one would want the description of the caustic structure to have as little dependence on thechoice of coordinates as possible. One coordinate independent quantity that one can calculate isthe curvature of the spacelike curve of caustic points (the \u00E2\u0080\u009Cinner seam\u00E2\u0080\u009D of the surface in figure 4.5).To have a full description of the horizon structure near the merger point, one would ideally want todescribe the surface in the neighborhood of the singular point in terms of a canonical normal formrepresentation, as commonly used in the singularity theory of surfaces. Although such a normalform representation would inevitably have coordinate dependent aspects, the ambiguity associatedwith the choice of coordinates could be partly eliminated by using a canonical coordinate system,such as Riemann normal coordinates.Horizon areaIn terms of horizon area, the main direction for future work would be to confirm many of thenumerical results in this thesis using analytical methods. Two results which may be amenable toanalytical methods are the late time behavior of the horizon area and the horizon area in the limite ! 0. As was discussed in section 4.4.3, finding an analytical description of the horizon area atlate times most likely involves uncovering vanishingly small terms in the series approximation forthe late time behavior of the horizon generators.1806.2. Applications and future directionsIn order to support our numerical results regarding the horizon area in the limit e ! 0 presented insection 5.2.3, we would need an approximate analytical solution to the equations for the null gen-erators in the limit e ! 0. It is possible that such a solution could be obtained using a perturbationseries. Such a perturbation series would most likely be a singular asymptotic series, since the caseof e = 0 (i.e. no black hole) the behavior of some of the generators is qualitatively different thanin the case where e 6= 0. Regardless of how small the black hole is, provided that e 6= 0, therewill always be generators that are strongly lensed by the black hole. Whether it is necessary tofind an asymptotic expansion for the behavior of these strongly lensed generators depends partlyon whether or not they make a significant contribution to the horizon area. With such an analyticalformula, we could confirm many of the results from section 5.2.3, such as the idea that in the limite ! 0, all of the area increase occurs prior to merger. Furthermore, such results could potentiallybe generalized to other coordinate systems, provided one knows the explicit mapping between theseother coordinates and the coordinates used in this thesis. Lastly, if the area increase occurs at aspecific location on the horizon (ex. near the caustic points), then it may be possible to state thequalitative aspects of the results in a coordinate independent manner.6.2.2 Binary black hole mergersOne of the primary motivations for the study of merging horizons undertaken in this thesis is its po-tential for stimulating questions about the merging horizons that occur in binary black hole mergers.It has long been known that the horizons in binary black hole mergers have the familiar \u00E2\u0080\u009Ctrousers\u00E2\u0080\u009Dshape (see figure 4.1). In this thesis, we have created a similar picture (see figure 4.5). There aretwo important differences between our picture and the one for binary black holes. The first is thatin the case of the cosmological horizon merging with a black hole, one horizon is contained withinthe other. Despite this difference, the two cases are similar topologically, in the sense that in bothcases we have two disconnected surfaces of spherical topology joining to later form a single surfaceof spherical topology. The second main difference is that in the merging horizons considered inthis thesis, the spacetime is known analytically. This is the crucial difference for the purpose of thisthesis. Since the spacetime is known analytically, the only source of error comes directly from the1816.2. Applications and future directionscalculation of the horizon, and not from the knowledge of the spacetime. This error is only limitedby the accuracy with which one can solve the ordinary differential equations for null generators, aswell as the accuracy with which one can calculate quantities based on these coordinates, such as thearea of the horizons. For both of these types of computations, the errors are well understood andeasily controlled. This is in contrast to a case where one does not know the spacetime analytically,such as in the case of binary black holes. In these cases there are two methods for approximating thespacetime. One can use the initial value formulation of Einstein\u00E2\u0080\u0099s equation, where one numericallysolves a set of nonlinear coupled partial differential equations for the spacetime metric components.Alternatively, one can approximate the spacetime of an extreme mass ratio binary black hole by us-ing a perturbation method. In both such computations one faces the difficulty of having to choose anappropriate gauge (i.e. coordinate system), and of controlling the sources of error in the spacetimemetric.Caustic locationAll of the results about the caustic obtained in this thesis lead naturally to similar questions in binaryblack hole mergers. For example, we obtained the formular \u00E2\u0087\u00A1 5.25M (6.1)for the Schwarzschild radius of the merger point in the limit e ! 0. The same method that wasused to find this formula could also be used to find a similar formula in the case of an extrememass ratio binary black hole merger. The key to performing such a calculation is to realize thatby the equivalence principle, the event horizon of the larger black hole can be approximated asa Rindler horizon. Thus, if we focus attention on the merger point, the merging of a black holewith the Rindler horizon of an accelerated observer has the same local caustic structure as for abinary black hole merger in the extreme-mass ratio limit. Furthermore, to lowest order we canignore the tidal effects of the larger black hole and treat the spacetime as being only curved dueto the smaller black hole. For the head-on of collision of non-rotating black holes, this leads to1826.2. Applications and future directionsconsidering the Rindler horizon of an accelerated observer in Schwarzchild spacetime. Using thisRindler approximation, we have performed the calculation leading to the location of the mergerpoint and found the Schwarzschild radius to be:r \u00E2\u0087\u00A1 3.52M (6.2)One might have expected this value for the Schwarzshild radius to be the same as (6.1). Thisexpectation would be based on the fact that one can use the equivalence principle to argue that thecosmological horizon is locally a Rindler horizon, much like in the case of extreme mass ratio binaryblack holes. Despite this similarity, we have the quantititavely different results (6.1) and (6.2).Caustic structureAs discussed in section 6.2.1, one of the key quantities that one can calculate is the curvatuve ofthe spacelike curve of caustic points on the \u00E2\u0080\u009Cinseam\u00E2\u0080\u009D of figure 4.5. Similarly, in the case of a head-on collision of black holes in the extreme mass ratio limit, one can use the Rindler approximationdescribed in the previous paragraph to find the curvature of this \u00E2\u0080\u009Cinseam\u00E2\u0080\u009D curve . We have performedsuch a calculation and found the value for the curvature to be:k \u00E2\u0087\u00A1 118M (6.3)Notice that the curvature has dimensions of inverse length, as one would expect. Another way ofdescribing the curvature of the inseam is by using a length scale, which would be given by theinverse of the above curvature. One can also calculate the total proper length of the inseam. Giventhat the inseam extends all the way to t = \u0000\u00E2\u0080\u00A2, it is perhaps surprising that this proper length turnsout to be a finite quantity. This can be understood as arising from the fact that although the inseamis a spacelike curve, it approaches a lightlike curve as one moves away from the merger point. In[14], the authors use a perturbation method to approximate the spacetime of an extreme mass ratiobinary black hole system, and then use this approximate spacetime to analyze the caustic structure1836.2. Applications and future directionsand area of the horizons. One of the quantities they calculate is the total proper length of the inseamcurve.The results (6.2) and (6.3) are part of a publication we are currently preparing. This publication willfocus on using the Rindler approximation to analyze the caustic structure near the merger point ina head on binary black hole merger. This Rindler approximation will compliment the work of [14],and will arrive at some of the same results using a different method. It should be mentioned that theidea of using a Rindler approximation to approximate the horizons of an extreme mass ratio binaryblack hole system has already been partly developed in the thesis [13]. However, our results (6.2)and (6.3) and the work that will follow will go beyond the analysis in that thesis.The Rindler approximation is appealing for at least two reasons. First, the calculations which leadto an understanding of the caustic structure are simpler. This is because the spacetime in the Rindlerapproximation is simply Schwarzschild spacetime, which is known analytically, so that the onlymathematical challenge is the integration of the equations for the null generators. Secondly, whenusing the Rindler approximation we have the possibility of easily generalizing the result to a rotatingblack hole. This is done by considering a Rindler horizon merging with a Kerr black hole. On theother hand, when using perturbation methods, there is no clearly established method for perturbinga large non-rotating black hole by a small rotating black hole, or perturbing a large rotating blackhole by a small non-rotating black hole. In the Rindler approximation, it is easy to incorporate thespin of the smaller black hole. Incorporating the spin of the larger black hole is more difficult, butperhaps possible.On the other hand, there are some advantages to using a perturbation method, as was done in [14].The first is that one can naturally deal with non-radially plunging orbits. By contrast, when usingthe Rindler approximation, it is not as obvious how to deal with such orbits. The second advantageis that one can deal with finite but small ratios of black hole masses, whereas in the Rindler approx-imation one is essentially taking the mass of the larger black hole to be infinite. However, as willbe explained below, it may be possible to introduce small corrections to the Rindler approximationto take into account a large but finite mass for the larger black hole. Lastly, it is easier to answer1846.2. Applications and future directionsquestions about the area of the horizons when using a perturbation method. When using the Rindlerapproximation, the area is infinite at all times, so that without some additional assumptions aboutthe horizons, questions about area increase are ill-posed.We have stated that it is not immediately clear how the Rindler approximation can accomodate eitherthe spin of the larger black hole, or the possibility that the smaller black hole moves on a non-radialtrajectory. However, it should be mentioned that the challenge is technical and not conceptual. Itis clear that the Rindler approximation is exploiting the equivalence principle, which will be validregardless of the spin of the larger black hole, or the type of geodesic trajectory followed by thesmaller black hole. Provided that we ignore the self-force of the smaller black hole, as well as anypossible non-gravitational forces, the smaller black hole is in free fall. We can construct a localinertial frame in the vicinity of this freely falling black hole. In the limit that the mass of the largerblack hole is infinite, tidal distortions of the spacetime curvature due to this larger black hole can beignored. We are then left with Schwarzschild spacetime in the vicinity of the smaller black hole, orKerr spacetime if the smaller black hole is spinning. By focusing on a neighborhood of the smallerblack hole, the spacetime geometry is entirely understood and described analytically, at least tolowest order in the mass ratio of the black holes. Understanding how the event horizon of the largerblack hole merges with the event horizon of the smaller black hole then reduces to understandingthe null generators of the larger black hole, and how these are lensed by the presence of the smallerblack hole. The spin of the larger black hole, or the effect of a non-radial trajectory for the smallerblack hole, can both be taken into account by adjusting the \u00E2\u0080\u009Cinitial conditions\u00E2\u0080\u009D of the null generatorswhich make up the large black hole horizon. That is, they are taken into account by adjusting howthis family of generators behaves far away from the smaller black hole. For a head-on collision inthe case of non-spinning larger black hole, the null generators are a plane of light rays when farfrom the smaller black hole, as one would normally expect for a Rindler horizon. This plane oflight rays is modified when the spin of the larger black hole or the non-radial trajectory of the smallblack hole is taken into account. By using this Rindler approximation in the case where the spins ofthe smaller and larger black holes are taken into account, we could potentially reveal a rich causticstructure near the merger point of the horizons. It would be interesting to investigate the topologyof this structure, as well as its dependence on the mass of the smaller black hole and spins of the1856.2. Applications and future directionstwo black holes.In addition to generalizing the Rindler approximation to deal with spin and non-radial trajectories,it would also be interesting to generalize it to finite mass ratios. One can imagine approximating thespacetime surrounding the small black hole using a series expansion, with the small parameter beingthe ratio of the small black hole mass to the large black hole mass. The Rindler approximation isthen the lowest order approximation, where the mass ratio of the black holes is zero. At this level ofapproximation the spacetime surrounding the small black hole is given by the Schwarzschild or Kerrmetric, and the horizon of the large black hole is the Rindler horizon of an observer acceleratinguniformly away from the black hole. At the next order of approximation, there would be a lowestorder correction to the Schwarzschild or Kerr metric which would incorporate the tidal distortionsto the spacetime surrounding the small black hole, as caused by the larger black hole. There wouldalso be a lowest order correction to the shape of the Rindler horizon due to the large but finite massof the larger black hole. These small corrections to both the spacetime surrounding the small blackhole and the shape of the larger horizon would result in small corrections to the shape of the causticstructure, and to small corrections with associated quantities such as the curvature of the inseam.Unlike the effect of taking into account spin or non-radial trajectories, these small lowest ordercorrections would be unlikely to affect the topology of the structure of the caustic near the mergerpoint, and so would be of interest for their quantitative rather than qualitative effects. Understandingthese quantitative corrections to the caustic structure is a question we are currently working on.Notice that creating a perturbation series starting from the Rindler approximation, as was describedin the previous paragraph, is quite different from the perturbation series that was created in [14].There the authors use the Regge-Wheeler formalism for perturbing Schwarzschild spacetime andapproximate the small black hole as a point mass. The resulting approximate spacetime is not validnear the small point mass. This can be understood as a consequence of the fact that the curvature ofthe spacetime near the small mass diverges as one gets closer to it. This is in contrast to the Rindlerperturbation series discussed in the previous paragraph, which would in fact only be valid nearthe small black hole. At distance scales comparable to the Schwarzschild radius of the larger blackhole, the tidal effects due to the large black hole would no longer be small corrections, and the series1866.2. Applications and future directionswould no longer be valid. These two perturbation approaches, with one valid near the small blackhole and one valid far away, could be joined together into a matched asymptotic approximation,with a transition region between the two and a set of matching conditions in the transition region.Such approximations are well known in the context of black hole perturbation theory, but so farhave not been used to study the event horizons of the black holes (to our knowledge).Area of horizonsRecall the three main results from the chapter on the area of merging cosmological and black holehorizons (chapter 5): (i) The time at which the rate of change of horizon area is at its largest valueis also the merger time of the horizons. (ii) In the limit e ! 0, all the horizon area increase takesplace before merger. (iii) In the limit e ! 0, the area increase has equal contributions from theexpansion of generators always on the horizon, and the joining of generators not previously on thehorizon. These three results regarding horizon area naturally lead to three corresponding hypothesesin the context of binary black holes. The statement of these hypotheses in this new context are nearlyidentical to (i)-(iii) above, with the only difference being that for results (ii) and (iii), the limit e ! 0is replaced with the extreme mass ratio limit of binary black holes.To our knowledge, only hypothesis (iii) has been shown to be true in the context of binary blackholes. In [14], the authors use perturbation theory to approximate the spacetime of an extreme massratio binary black hole system. This approximate spacetime is then used to show that in the limit ofinfinitesimally small black hole mass, the total horizon area increase has equal contributions fromboth existing and new generators, as in (iii) above. The result in this thesis extends this result of[14] to the context of cosmological horizons. It is intriguing that although the spacetime consideredin this thesis and the spacetime of an extreme mass ratio binary black hole are quite different, thehorizons in these spacetimes both satisfy (iii). One might attribute this as being due to a similarityin the horizons near the caustic points, since both the cosmological horizon and the large black holehorizon can be locally approximated as a Rindler horizon in the limit of small black hole mass.1876.2. Applications and future directionsHowever, the increase in horizon area occurs not only near the caustics where new generators enterthe horizon, but also far from the caustic points due to the expansion of existing generators. Thefact that result (iii) is valid for the merging horizons considered in this thesis and for the horizonsin binary black holes is even more surprising when we consider the fact that the horizons in binaryblack holes are observer independent event horizons, whereas the horizons in this thesis are observerdependent causal horizon. Despite this and other differences between the spacetimes, result (iii)holds for both spacetimes. This points to perhaps a deeper principle at work which may be validmore generally. It could be that whenever a small black hole mergers with a much larger horizon, thearea increase can be split into equal contributions from two types of generators, as in (iii). However,we do not at the moment have any reason to believe that this is indeed the case.It would be interesting to investigate the case of a charged and/or rotating black hole merging witha cosmological horizon, and see whether or not (iii) holds in those cases. The analysis wouldbe similar to that performed in this thesis, but with Kerr-Newman-deSitter spacetime replacingSchwarzschild-deSitter spacetime. To investigate statement (iii) for rotating binary black holeswould be more difficult, since there is no established perturbation technique that can be used to ap-proximate the spacetime in those cases. Although the Rindler approximation can deal with rotatingblack holes, it is not well suited to answering questions about area, since the area of a Rindler hori-zon is effectively infinite. One can imagine getting around this difficulty of having an infinite areaby first taking a finite Rindler horizon (for example, by ignoring the generators with large impactparameter), and then taking the limit as the Rindler horizon area goes to infinity. If one is answeringquestions about the rate of change of area, as in (i) above, or the relative area increase, as in (ii),then taking this limit could yield sensible answers. This is a direction that we are currently inves-tigating. We are also currently investigating the possibility of extending the techniques in [14] toprove (i) and (ii) for binary black holes. Proving (i) may require the use of a matched asymptoticapproximation, where an approximate spacetime near the small black hole is joined to an approx-imate spacetime valid far from the small black hole. This may be necessary because proving (i)requires taking into account the horizon area both in vicinity of the merger point and far away fromthe small black hole, so that we are required to have knowledge of the spacetime both close to andfar from the small black hole.188Bibliography[1] Thomas W Baumgarte and Stuart L Shapiro. Numerical relativity: solving Einstein\u00E2\u0080\u0099s equa-tions on the computer. Cambridge University Press, 2010.[2] Jacob D Bekenstein. Black holes and entropy. Physical Review D, 7(8):2333, 1973.[3] George David Birkhoff and Rudolph Ernest Langer. Relativity and Modern Physics. By GDBirkhoff... with the co-operation of Rudolph E. Langer. Harvard University Press, 1927.[4] Scott A Caveny and Richard A Matzner. Adaptive event horizon tracking and critical phenom-ena in binary black hole coalescence. Physical Review D, 68(10):104003, 2003.[5] Tamara M Davis and Charles H Lineweaver. 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