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© 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! • 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 “trouser” 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 Thal f ) for several smallvalues of the paramter eˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 eˆ . . . . . . . . . . . . . . . . . . . . . 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’spositive attitude and affable nature helped me persevere through the more difficult moments ofmy doctoral work. My parents’ 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 Frédéric 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’scosmological 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 , 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  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 “faster than light” description is1For FRW universes accelerated expansion implies a cosmological horizon; for more general cosmological modelsone can find exceptions ( See .)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  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  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’s cosmological horizon (page 129, ). 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  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  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 ), orto use a Rindler horizon approximation for the larger black hole horizon, as in .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, ). 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  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  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 . This solution has been used as a test bed for a numerical event horizonsolver in .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 , therehas been interest in topological transitions of event horizons in higher dimensional spacetimes. Forexample, Emparan and Hassad  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  to Einstein’s newly devel-oped field equations of gravitation . 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 = ✓1 2Mr ◆dt2 ✓1 2Mr ◆ 1 dr2 r2(sin2 qdf 2 +dq 2) , (2.1)where 0 < r < •. 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’s equation is the so-called deSitter space-time, first discovered by Willem deSitter . Mathematically, the deSitter spacetime is the maxi-mally symmetric vacuum solution of Einstein’s 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 Hr)dt2 (1 Hr) 1 dr2 r2(sin2 qdf 2 +dq 2) , (2.2)where H =pL/3 is the Hubble constant and 0 < r < •. 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 “closed” slicing of deSitter spacetime, one can interpretthe full deSitter manifold as a so-called “big bounce” 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 “big bang”and “big crunch”). 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 e2Ht dx2 +dy2 +dz2 , (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’s 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’stheorem . It is possible to generalize this theorem to SdS spacetime and include the cosmologicalconstant (see  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  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  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 “big bounce” 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 “perfect cosmological principle”, which extends the usual “cosmological principle” 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  for similar diagrams constructed for SdS spacetime).As with deSitter spacetime, one can consider only half of SdS spacetime (this is the region labelled“Schwarzschild coordinates region” 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 © , 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 (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 © , page 125, by permission from publisher.142.2. Schwarzschild coordinatesFigure 2.6: Function f (r) = 1 2M/r H2r2.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 ! • 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 f (r) 1dr2 r2(sin2 qdf 2 +dq 2) , (2.4)f (r) = 1 2Mr 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 (+, , , ). 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 3(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)✓ dtdt◆2 f (r) 1✓ drdt◆2 = 1 . (2.9)Substituting (2.8) into the above gives the equation for r(t):E2 ✓ drdt◆2 = f (r) .The above equation can be recast in effective potential form as✓ drdt◆2 +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 = •) = 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 = ±( f (re) f (r))1/2 , (2.12)where the + sign is for trajectories drifting towards larger values of r and the 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  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  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’s trajectory. For-mally, if g(t) is the observer’s 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 (g(R))). The terminology “causal horizon”may not be familiar to some readers, and has not yet gained widespread and popular usage. Forexample, some authors prefer to use the words “cosmological horizon” in the cosmological context,and “Rindler horizon” 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  by using the words “causal horizon”.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 ) 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 , although it is simply called an “event horizon” 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’s 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’s 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’s 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’s 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! • 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! • limit of SdS spacetimeIn section 2.5, we claimed that in the limit that r ! •, SdS spacetime in some sense approachesdeSitter spacetime, and that this could be used to conclude that the late time behavior of a driftingobserver’s 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’s 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! • is the late time limit.The claim that SdS spacetime approaches deSitter spacetime in the limit r ! • 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 ! •, 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 ! • 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 “initial” 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¯ andradial coordinate r¯ as follows: t¯ = Ht,r¯ = Hr .The line element (2.4) now becomesds2 = 1H2 "✓1 2e¯r r¯2◆dt¯2 ✓1 2e¯r r¯2◆ 1 dr¯2 r¯2(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 ⌧ 1 the black hole is much smaller than the cosmological horizon and as e ! e c , 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 . 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 ⌧ 1 by using pertubation methods to find the solution to f (r) = 0. This givesrb = 2M(1+O(e2)) ,rc = 1H 1 e +O(e2) , (2.15)rbrc = e +O(e2) .From the above we see that for e ⌧ 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 ⌧ 1 equivalent to introducing a small black holeinto the spacetime. In this way considering e ⌧ 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 ⌧ 1 introducesa small perturbation to the gt¯t¯ and gr¯r¯ metric components. The limit e ⌧ 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 ⇡ 1012m and L⇡ 10 52m 2, giving e ⇡ 10 14.29Chapter 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 (+, , , ) 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’s self to a local use of GN coordinates. However, according to the Frobenius theorem (see sec-tion 2.3.3 of , or section B.3 of ), 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 e2Ht dr2 +r2(dq 2 + sin2 qdf 2) (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  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 . 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 = • and those going to r = 0. The “energy at infinity” 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 “energy at infinity” 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’s coordinate system for SdS spacetime .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 = •, 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’s position when that observer’s clock value reads some chosen value T = T ⇤.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 = •)=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⇤ > re at a point with coordinates (T,R) =(t,R⇤), where t is the proper time along the curve R = R⇤. This intersection of curves is shown infigure 3.4a. As will be shown in section 3.2, R⇤ 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 ⇤ 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 ⇤ on the metric components in the LP coordinate system amounts tosimply shifting the time variable T , and thus T ⇤ 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⇤ 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⇤ 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 ⇤ (shown ingreen).363.1. Construction of LP coordinatestimelike and spacelike coordinate tangent vectors ∂∂T and ∂∂R , respectively. Expressed in the basisof coordinate tangent vectors in Schwarzschild coordinates, ∂∂ t and ∂∂ r , these directions are∂∂T µ f (re) 12f (r) ∂∂ t + sgn(r re) [ f (re) f (r)]1/2 ∂∂ r , (3.2)∂∂R µ sgn(r re) [ f (re) f (r)] 12 ∂∂ t + f (re)1/2 f (r) ∂∂ 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 ∂∂T , ∂∂Ri = 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⇤) = r0(t = T ;R⇤), where r0(t;R⇤) is a timelike geodesic satisfying r0(t = •) = reand r0(t = T ⇤) = R⇤,(iv), r(T ⇤,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 R=constant = f (re)1/2f (r)( f (re) f (r))1/2 sgn(r 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 ⇤, we have r(R) = Rand the function r(R) is monotonic. It then follows by continuity that for values of T sufficientlynear T ⇤, r(R) is once again monotonic. Therefore, for values of T near T ⇤, 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 T=constant = ( f (re) f (r))1/2f (re)1/2 f (r) sgn(r 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 ⇤ 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 ⇤,R⇤), where it is understood that (T,R) and R⇤ are chosen sothat the restrictions mentioned in the previous paragraph are met. The first path connects the point(T,R) to (T ⇤,R⇤) by first going along R = constant until the point (T ⇤,R), and then going alongthe path T = T ⇤ until the point (T ⇤,R⇤) is reached. The second path connects the point (T,R) to(T ⇤,R⇤) by first going along T = constant until the point (T,R⇤), and then going along the pathR = R⇤ until the point (T ⇤,R⇤) 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 ⇤,R) along the coordinate curve R = constant, and then integrate (3.5) from (T ⇤,R) to(T ⇤,R⇤) along the coordinate curve T = T ⇤. This gives the following equation:r(T ⇤,R)ˆr(T,R) dtdr R=constant dr+ r(T ⇤,R⇤)ˆr(T ⇤,R) dtdr T=constant dr = t(T ⇤,R⇤) t(T,R) . (3.6)Moving along the second path, we integrate (3.5) from (T,R) to (T,R⇤) along the coordinate curveT = constant, and then integrate (3.4) from (T,R⇤) to (T ⇤,R⇤) along the coordinate curve R =constant. This gives the following equation:r(T,R⇤)ˆr(T,R) dtdr T=constant dr+ r(T ⇤,R⇤)ˆr(T,R⇤) dtdr R=constant dr = t(T ⇤,R⇤) 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 ⇤,R)ˆr(T,R) dtdr R=constant dr+ r(T ⇤,R⇤)ˆr(T ⇤,R) dtdr T=constant dr = r(T,R⇤)ˆr(T,R) dtdr T=constant dr+ r(T ⇤,R⇤)ˆr(T,R⇤) dtdr R=constant dr .Manipulating the limits of integration in the above and rearranging terms, we obtain the following:r(T,R)ˆr(T ⇤,R) ✓ dtdr R=constant dtdr T=constant◆dr = r(T,R⇤)ˆr(T ⇤,R⇤) ✓ dtdr R=constant dtdr T=constant◆dr . (3.8)Using (3.4) and (3.5), we havedtdr R=constant dtdr T=constant = 1f (re)1/2 sgn(r re)( f (re) f (r))1/2 .From condition (iii) above, and the geodesic equation (2.12), we have that r(T,R) satisfies thefollowing: ∂ r∂T = sgn(r re)( f (re) f (r))1/2 (3.9)393.1. Construction of LP coordinatesCombining the previous two equations, we havedtdr R=constant dtdr T=constant = 1f (re)1/2 ✓ ∂ r∂T ◆ 1 . (3.10)Substituting the above into (3.8), we obtainr(T,R)ˆr(T ⇤,R) ✓ ∂ r∂T ◆ 1 dr = r(T,R⇤)ˆr(T ⇤,R⇤) ✓ ∂ r∂T ◆ 1 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⇤, definer(T ) = r(T,R⇤). 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⇤)ˆr(T ⇤,R⇤) ✓ ∂ r∂T ◆ 1 dr = TˆT ⇤ dt = T T ⇤ . (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)ˆR dr( f (re) f (r))1/2 = sgn(R re)(T T ⇤) , (3.13)where we have used r(T ⇤,R) = R (condition (iv) above). We have also removed sgn(r re) fromthe integral and replaced it with sgn(R 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 re) = sgn(R 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 188.8.131.523.1. 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 ⇤,R⇤)+ r(T,R⇤)ˆr(T ⇤,R⇤) ✓ dtdr R=constant dtdr T=constant◆dr+ r(T,R)ˆr(T ⇤,R⇤) dtdr T=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⇤)ˆr(T ⇤,R⇤) ✓ dtdr R=constant dtdr T=constant◆dr = 1f (re)1/2 (T T ⇤) .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 ⇤,R⇤)+ 1f (re)1/2 (T T ⇤)+ r(T,R)ˆr(T ⇤,R⇤) ( f (re) f (r))1/2f (re)1/2 f (r) sgn(r 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:∂ r∂T = sgn(r re)( f (re) f (r)) 12 , (3.16)∂ r∂R = f (re) f (r)f (re) f (R) 1/2 . (3.17)Next, taking the partial derivative of (3.15) and using the above where necessary, we get the othertwo components of the Jacobian:∂ t∂T = f (re) 12f (r) , (3.18)∂ t∂R = [ f (re) f (r)]sgn(R re)f (re)1/2 f (r) [ f (re) 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 f (re) f (r)f (re) [ f (re) f (R)]dR2 r2(dq 2 + sin2 qdf 2) , (3.20)where r = r(T,R) is given implicitly by (3.13). The constant R⇤, 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⇤ has no impact on the metric and is essentially arbitrary. The constant T ⇤ (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 ⇤ can be thought of as arbitrary and we will take T ⇤ = 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 184.108.40.206.3 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 H2r2,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 ⇤ = 0. Substituting the above into (3.20) and setting M = 0, we getds2 = dT 2 e2HT dR2 +R2(dq 2 + sin2 qdf 2) , (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 “steady state” 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 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 ✓Rr ◆dR2 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 2Mr dr2 r2dW2 .The above is the line element of Schwarzschild spacetime, with (T,r,q ,f) the well known Lemaitrecoordinates . 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) = ✓32◆2/3(2M)1/3(r T )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 = ✓Rr ◆dR2 r2dW2 . (3.24)Now consider the Jacobian component (3.17) with H = 0:∂ r∂R = ✓Rr ◆1/2 .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 “energy at infinity”. 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 11 ⇣F1⇣ Rre⌘eHT⌘ , R rereF 12 ⇣F2⇣ Rre⌘eHT⌘ , R re (3.25)where F1(r) = ⇣r +1+pr(r +2)⌘" 1 rp3r(r +2)+2r +1# 1p3 ,F2(r) = ⇣r +1+pr(r +2)⌘" r 1p3r(r +2)+2r +1# 1p3 . (3.26)Recall that we have taken T ⇤ = 0 in the above. The functions F1(r) and F2(r) are strictly monotonic,and therefore invertible, for r 1 and r 1, respectively. The ranges of these functions are F1 0and F2 0, so that the domains of the inverses F 11 (y) and F 12 (y) are well defined for all y 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⇤+ 1f (re)1/2 0@T + rˆr⇤ sgn(r re)( f (re) f (r))1/2f (r) dr1A , (3.27)where t⇤ = t(T ⇤,R⇤) and r⇤ = r(T ⇤,R⇤) and we have once again taken T ⇤ = 0. Recall that R⇤ 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⇤, 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⇤ and r⇤ are arbitrary, and we take t⇤ = 0 and r⇤ = (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)) A(r⇤)) , (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 rb)H (ra rb)(rb rc) arctanh0@ rbre + rer + rbrqr2b +2rbrepr2 +2rre1A pr2c +2rcre (re rc)H (ra rc)(rb rc) arctanh rcre + rer + rcrpr2c +2rcrepr2 +2rre! ln✓1+rre + 1repr2 +2rer◆ (3.29) pr2a +2rare (re ra)H (ra rb)(ra rc) ln 2rare + rer + rar +pr2a +2rarepr2 +2rrere (r 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 ! • or t ! •. 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 11 ⇣F1⇣ rre⌘e HT (t,r)⌘ , r rereF 12 ⇣F2⇣ rre⌘e HT (t,r)⌘ , r re (3.30)473.3. Additional features of LP coordinatesT (t,r) = f (re)1/2t +A(r⇤) A(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 220.127.116.11.3.4 Continuity of metric componentsConsider the gRR component of the line element (3.20):gRR = f (re) f (r)f (re) [ f (re) 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 = 1f (re)✓ ∂ r∂R◆2 . (3.33)From the above, we see that we can find the limit of gRR as R! re by calculating ∂ r/∂R 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) ⇠ ⇣2+p3⌘✓1 r6 ◆ 1p3 ,F2(r) ⇠ ⇣2+p3⌘✓r 16 ◆ 1p3 .The inverses F 11 (y) and F 12 (y) therefore have the following asymptotic forms:F 11 (y) ⇠ 6✓ y2+p3◆p3 +1 ,F 12 (y) ⇠ 6✓ y2+p3◆p3 +1 .Substituting the two preceeding equations into 3.25, we obtainr(T,R) = reF 1i ✓Fi✓ Rre◆eHT◆⇠ re +(R re)ep3HT ,where i = 1,2. Notice that the asymptotic form of r(T,R) near R = re is the same for both R reand R re. Also, as one would expect, r(T,R) is linear in R near R = re. From the above we cancompute thatlimR!re ∂ r∂R = ep3HT .Substituting the above in (3.33), we getlimR!re gRR = 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 gRR dR2 r2(dq 2 + sin2 qdf 2) ,with gRR =8><>: f (re) f (r)f (re)[ f (re) f (R)] for R 6= re, 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 • < T < •. 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 18.104.22.168 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) = ∂t 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✓F2✓ Rre◆◆ , (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 •< r < •. 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) f (R)) 1/2 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 ( f (re) f (r)) dr2 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 12 ⇣eH(r+T )⌘ .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 T0will 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 12 ✓F2✓ Rre◆e HT0◆ . (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 ⇤ and R⇤introduced in section 22.214.171.124.6 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 = f (re) f (r)f (re) [ f (re) f (R)]dR2 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 “hybrid” 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 = f (re) f (r)f (re) f (R) 1/2 dr .Substituting the above into (3.20), we getds2 = 1f (re)dr2 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 f (re)1/2 . 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 = 6r2 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 (g(R)) and I (g(R)) be the causal and chronologicalpasts of g(R), respectively, and letH be the horizon, so thatH ⌘ Bd(J (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 , 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 ⌘ Bd(I (g(R))) 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 (g(R))) = Bd(I (g(R))) and g(R)\Bd(I (g(R))) = /0, where the former is discussedat the bottom of page 191 of , and the latter follows from the fact that g(R)⇢ I (g(R)), alongwith the fact that I (g(R)) is an open set (page 190 of ).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 “initial conditions” 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 , 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) ⇢ I (g(R)), we havep2 I (g(R)), and therefore, p2 Int(J (g(R))) (where we have used I (S) = Int(J (S)); page 191of ). Since Int(J (g(R))) is an open set (page 190 of ) and p 2 Int(J (g(R))), there is aneighborhood O of the point p such that O ⇢ Int(J (g(R))). By the definition of future endpoint(page 193 of ), 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 (g(R))) for all l > l0, which contradicts the fact that we must alwayshave C (l ) 2H = Bd(J (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 )). 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 = •) = •, 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 = •) = • becomes r(T = •) = •instead (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 ! •. As discussed in section 2.6, SdS spacetimeapproaches deSitter spacetime in the limit that r ! •. 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 ! •. 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 “trouser” shaped horizon of two merging black holes. The green curves are horizongenerators which enter through caustic points on the “inseam” of the trouser. Notice that thesegenerators cross at the caustic points.closed surface will satisfy r(T =•) =• and f(T =•) = 0. Furthermore, since the null generatorswhich make up the horizon are all future inextendible, they will all reach this final closed surface asT ! •.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 !• and satisfy r(T = •) = • and f(T = •) = 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 ! •. 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 ): (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 “inseam” of the “trouser” 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 126.96.36.199.2 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 = •) = re and r(t) > re. The observer’s 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 = •) < rc,(ii) the trajectory approaches one of the outward radial geodesics as t ! • (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’ 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 = • 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 =•) =•would 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 ! •) 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) ⌘Ve f f (r)sin2a = 1r2 1 2Mr . The green and blue curves are schematic illustrations of trajectories withand without turning points, respectively. The existence of turning points depends on the “initialconditions”, as characterized by the value of the parameter a . The precise conditions are sin2a 27H2M2 (no turning points) and sin2a > 27H2M2 (turning point). The parameter values used toproduce the plot are M = 1, 27H2M2 = 14 , with sin2a ⇡ 3/4 and sin2a ⇡ 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) = ∂∂ t ,x (f) = (0,0,1,0) = ∂∂f .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)✓ dtdl ◆2 f (r) 1✓ drdl ◆2 r2✓dfdl ◆2 = 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 ,✓ drdl ◆2 +✓ lr◆2 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✓ le◆ , (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 “horizon” 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’s 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 ! ⇣l2 + e2H2⌘l and replacing l and e in favor of a , thegeodesic equations become dtdl = H cosaf (r) , (4.2)dfdl = sinar2 , (4.3)✓ drdl ◆2 +Ve f f (r) = H2, (4.4)where Ve f f (r) = sin2ar2 ✓1 2Mr ◆ . (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 = • will have a turningpoint. We have two cases:(I) If sin2a 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 ⇤ and we have drdl < 0 for l < l ⇤and drdl > 0 for l > l ⇤ .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 ⇤ in the second case. This givessin2a 27H2M2 =) drdl =qH2 Ve f f (r) , (4.6)sin2a > 27H2M2 =) drdl =8><>: pH2 Ve f f (r) ifl < l ⇤pH2 Ve f f (r) ifl > l ⇤ (4.7)Alternatively, equation (4.4) can be transformed into a set of coupled first order equations:drdl = w, (4.8)dwdl = 12V 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 “horizons” 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(·) 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 = ∂T∂ r drdl + ∂T∂ 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 ∂T/∂ r and ∂T/∂ t, we getdTdl = H f (re) 12 cosa sgn drdl ( f (re) f (r))1/2 sgn(r re)pH2 Ve 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 = f (r)pH2 Ve f f (r)sgn drdT H f (re) 12 cosa +( f (re) f (r)) 12 sgn(r re)pH2 Ve f f (r) , (4.12)dfdT = ✓sinar2 ◆ f (r)H f (re) 12 cosa + sgn drdT ( f (re) f (r)) 12 sgn(r re)pH2 Ve f f (r) . (4.13)It will be useful to rewrite the above equations more succinctly asdrdT = G±(r), (4.14)dfdT = K±(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∂T/∂ r and ∂T/∂ 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 = H f (re) 12 cosa w( f (re) f (r))1/2 sgn(r 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 = w f (r)H f (re) 12 cosa +w( f (re) f (r))1/2 sgn(r re) , (4.16)dwdT = 12V 0e f f (r) f (r)H f (re) 12 cosa +w( f (re) f (r))1/2 sgn(r re) , (4.17)dfdT = ✓sinar2 ◆ f (r)H f (re) 12 cosa +w( f (re) f (r)) 12 sgn(r 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 188.8.131.524.2. Null geodesic equationsProof that dTdl > 0Let N(r) be the numerator of the right hand side of (4.10):N(r) = H f (re) 12 cosa sgn✓ drdl ◆( f (re) f (r))1/2 sgn(r re)qH2 Ve 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✓ drdl ◆ = 1,sgn(r 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) = H f (re) 12 cosa ( f (re) f (r))1/2qH2 Ve f f (r) . (4.24)In addition, (4.22) implies that the following inequalities hold:( f (re) f (r))1/2 > f (re) 12 , (4.25)rH2 cos2a 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 Ve f f (r) > H | cosa | . (4.27)Applying (4.25) and (4.27) to (4.24), we obtainN(r) <
UBC Theses and Dissertations
Merging black hole and cosmological horizons Prat, Alain 2015
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