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Gravitational geons on the brane Kermode, Daniel John 2011

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Gravitational Geons on the BranebyDaniel John KermodeB. Math, University of Waterloo, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Interdisciplinary Graduate Studies)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)April 2011c⃝ Daniel John Kermode, 2011AbstractGravitational geons, if they exist, represent a candidate for cosmological darkmatter observations. Gravitational Geons that are nonsingular, asymptotically flat,topologically trivial spacetime solutions are not possible in General Relativity. Thepurpose of this thesis is to investigate whether such geons are possible on a 3 di-mensional brane embedded in a 4+1 dimensional spacetime. Our approach is toidentify the characteristics of a candidate spacetime solution on the brane, choosean example spacetime that exhibits those characteristics and determine if it corre-sponds to a gravitational geon.iiPrefaceThe original idea to investigate gravitational geons in brane-world cosmologiesis that of my supervisor Dan Vollick. Together, we collaborated to author an articleentitled Gravitational Geons on the Brane which has been submitted for publica-tion at the journal General Relativity and Gravitation. Chapter 2 of this thesis is aversion of that article.The investigative effort involved in the article required extensive use of bothanalytical and numerical methods. Although we both participated in each effort,Professor Vollick’s contributions were largely analytical whereas mine were largelynumerical.The writing of the article including the sourcing and verification of referenceswas done by myself with extensive advice and support by Professor Vollick.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Differential Geometry and General Relativity . . . . . . . . . . . 21.1.1 The Einstein Field Equations . . . . . . . . . . . . . . . . 21.1.2 The Kretschmann Scalar . . . . . . . . . . . . . . . . . . 41.1.3 Junction Conditions . . . . . . . . . . . . . . . . . . . . 41.2 The Brane-World . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Gravitational Geons . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Static Space-Time Solutions on the Brane . . . . . . . . . . . . . 161.4.1 Brane-World Black Holes . . . . . . . . . . . . . . . . . 161.4.2 Brane-World Wormholes . . . . . . . . . . . . . . . . . . 191.5 Gravitational Geons in Other Theories . . . . . . . . . . . . . . . 202 Gravitational Geons in Brane-World Cosmologies . . . . . . . . . . 232.1 The Candidate Space-Times . . . . . . . . . . . . . . . . . . . . 242.2 Methods Of Investigation . . . . . . . . . . . . . . . . . . . . . . 25iv2.2.1 The Weak Field Approximation . . . . . . . . . . . . . . 252.2.2 Numerical Iteration . . . . . . . . . . . . . . . . . . . . . 262.2.3 Direction Field Plots . . . . . . . . . . . . . . . . . . . . 272.2.4 Potential Singular Points . . . . . . . . . . . . . . . . . . 282.3 Weak Field Behaviour . . . . . . . . . . . . . . . . . . . . . . . 292.4 General Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 Two Critical Points – High Range . . . . . . . . . . . . . 362.4.2 Two Critical Points – Medium Range . . . . . . . . . . . 372.4.3 Two Critical Points – Low Range . . . . . . . . . . . . . 382.4.4 No Critical Points . . . . . . . . . . . . . . . . . . . . . . 393 Discussion And Conclusion . . . . . . . . . . . . . . . . . . . . . . . 433.1 The Weak Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Identifying m and l . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Suggestions for Further Analysis . . . . . . . . . . . . . . . . . . 463.4 Significance of the Result . . . . . . . . . . . . . . . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51vList of FiguresFigure 2.1 The interface of a custom application built to investigate our candidate space-times using a fourth-order Runge-Kutta numerical iteration. . . . . . . . 27Figure 2.2 An example of a direction field plot showing numerical instabilities in A(r)(here plotted for l = 1, m = 1.189 and 1.25 < r < 1.42). . . . . . . . . . 28Figure 2.3 B(r) and A(r) plotted against radius r for l = 1, m = 0.01. A fourth-ordernumerical iteration of (2.5) is labelled A(r) and the analytical weak fieldresult is labelled weak A(r). These two approaches give almost identicalresults for ml << 1. . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 2.4 α1 and β1 plotted against ml for the first zero of B′r +4B. . . . . . . . . 33Figure 2.5 α1 and β1 plotted against ml for the second zero of B′r +4B. . . . . . . . 33Figure 2.6 A(r) iteratively plotted against radius r for l = 1 and m = 1.25 against adirection field plot. Here, both the convergence to A = 0 and A = −α1β1≈−1.84 at the first zero of B′r +4B can be clearly seen. . . . . . . . . . . 36Figure 2.7 Reverse iteration of A(r) plotted against radius r for l = 1 and m = 1.192from the second zero of B′r + 4B to the first, against a direction field plot.The solution is incompatible with the analytical result that A′ < 0 at the firstzero for λ = 0 solutions. . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.8 This direction field plot of A(r) against radius r shows that A(r) is numeri-cally unstable at the zeros of B′r+4B. Here we choose l = 1 and m = 1.189such that the zeros are found at r = 1.281, 1.392. . . . . . . . . . . . . 39Figure 2.9 Forward iteration of A(r) plotted from r = 0 to the first zero of B′r +4B forl = 1 and m = 1.189. A → −α1β1≈ 1.38 as the solution approaches the first zero. 40Figure 2.10 Reverse iteration of A(r) plotted from the second zero of B′r+4B to the firstfor l = 1 and m = 1.189. . . . . . . . . . . . . . . . . . . . . . . 40viFigure 2.11 Forward iteration of A(r) plotted from the second zero of B′r +4B for l = 1and m = 1.189. . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.12 The function A(r) plotted against radius r for l = 1 and m = 1.189. A fourth-order Runge-Kutta numerical iteration was unable to navigate the zeros ofB′r+4B (at r = 1.281, 1.392), so the result is constructed piecewise betweenthese points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.13 Function A(r) plotted against radius r. Here we choose l = 1 and chart avariety of values for m in the specified region. A fourth-order Runge-Kuttanumerical iteration is used to generate each result. . . . . . . . . . . . 42Figure 3.1 B(r) plotted for m = 1 and 1≤ l ≤33.3. Note that all solutions have ml < 3227so these solutions all correspond to gravitational geons. . . . . . . . . . 45viiList of SymbolsThe following list contains primarily symbols whose physical and mathemati-cal meaning remain unchanged throughout this thesis. Cited page numbers refer tothe first occurrence of each symbol.Rµν,(5)RAB Ricci tensor 4D/5D components (contracted from the Riemann tensor) . . . . . . . . . 2R,(5)R Ricci scalar 4D/5D (contracted from the Ricci tensor) . . . . . . . . . . . . . . . . . . . . . . . . 2Tµν,(5)TAB energy momentum tensor 4D/5D components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2gµν,(5)gAB metric tensor 4D/5D components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Λ,Λ5 cosmological constant 4D/5D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2κ,κ5 gravitational coupling constant 4D/5D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2B(r),B gtt metric tensor component of static, spherically symmetric space-time . . . . . . . . 3A(r),A grr metric tensor component of static, spherically symmetric space-time . . . . . . . .3O() Order of notation - terms of specified order and higher are neglected . . . . . . . . . . 4Sµν,(5)SAB surface energy momentum tensor 4D/5D components at a hypersurface . . . . . . . . 5Kµν,(5)KAB extrinsic curvature tensor 4D/5D components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Ln Lie differentiation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Rµνσφ The Riemann Curvature Tensor 4-dimensional components . . . . . . . . . . . . . . . . . . . 9(5)RABCD Riemann curvature tensor 5-dimensional components . . . . . . . . . . . . . . . . . . . . . . . . 9∇µ covariant differentiation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9(5)CABCD Weyl curvature tensor 5-D components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Eµν electric projection of the 5-D Weyl curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . 10Γαµν Christoffel symbol for the Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . . . . . .14√−g square-root of the determinant of the matrix of metric tensor components . . . . . 14∂α partial derivative operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14ηµν Minkowski space-time metric tensor components . . . . . . . . . . . . . . . . . . . . . . . . . . . 16viiib(r),b variation in B(r), defined in the weak field by b(r) = 1−B(r) . . . . . . . . . . . . . . . .25a(r),a variation in A(r), defined in the weak field by a(r) = 1−A(r) . . . . . . . . . . . . . . . .25Bµνσ magnetic projection of the 5-D Weyl curvature tensor . . . . . . . . . . . . . . . . . . . . . . . 47ixAcknowledgmentsI would like to thank my supervisor, Professor Dan Vollick for his support andpatience. It was Professor Vollick’s original idea which led to the paper Grav-itational Geons on the Brane which we collaborated to write and on which thisthesis is based. I would also like to thank my wife Myra as well as my family andfriends who have supported and encouraged me through my degree. I particularlyappreciate the editorial review provided by Stephanie Halldorson.For their excellent academic instruction and unqualified support, my gratitudegoes to the professors of physics and mathematics at UBCO, in particular to Profes-sor Erik Rosolowsky, Professor Murray Neuman and Professor Sylvie Desjardins.This research was supported by the Natural Sciences and Engineering ResearchCouncil of Canada.xChapter 1IntroductionEinstein’s general theory of relativity was published in 1915 and remains un-surpassed as a predictive mathematical model of gravitation. That having beensaid, general relativity also predicts the existence of space-time singularities ingravitational collapse (i.e. in black holes) and in cosmological space-times.In particle physics, quantum field theory shares the predictive success of gen-eral relativity. In quantum field theory, one obstacle to unifying gravitation withthe other forces of nature is the hierarchy problem which is related to the largediscrepancy between the electro-weak scale and the Planck scale. The Randall-Sundrum Brane-world cosmological model [1] was proposed as a new way to solvethe hierarchy problem. In this investigation, we consider Randall-Sundrum typebrane-world cosmologies in which the three spatial dimensions of the observableuniverse are modeled by a 3 dimensional membrane or 3-brane embedded in a 4+1dimensional bulk space-time.Long before brane-world theory was introduced, John Wheeler introduced theconcept of a gravitational-electromagnetic entity or geon [2] with the hope that aconfiguration of gravitational and electromagnetic fields could form a persistentspace-time disturbance that would ultimately provide a self-consistent picture ofwhat a particle of matter is. A gravitational geon is a non-singular configurationof the gravitational field, without horizons that persists for a long period of time.General relativity does not allow for static, topologically trivial and sphericallysymmetric gravitational geon solutions of the Einstein field equations. However,1in the Randall-Sundrum brane-world model, the 4+1 dimensional Einstein fieldequations in the bulk induce field equations on the brane that differ from those ofgeneral relativity.In this thesis, we consider whether there are static, topologically trivial andspherically symmetric solutions to the Einstein field equations on the brane thatcorrespond to gravitational geons. We construct these space-time solutions on thebrane in order to prove that they could exist in Randall-Sundrum type brane-worldcosmologies.Our approach does not yield a general solution for gravitational geon space-times on the brane. However, using approximations and numerical methods, itdoes show that such space-time solutions exist.Although the solutions that we find cannot correspond to conventional particlesof matter, they could correspond to dark matter.1.1 Differential Geometry and General RelativityIt is assumed that the reader is familiar with the mathematics of differentialgeometry and Einstein’s general theory of relativity. In this thesis, we work exten-sively with the Einstein field equations and certain ideas specific to the study ofcurved manifolds. The intention of this section is to state and briefly review thoseresults that form the foundation of this thesis. For further details, see [3–5].We adopt the convention that the gravitational constant G and the speed of lightc are dimensionally adjusted to unity.1.1.1 The Einstein Field EquationsIn general relativity, the Einstein field equations areRµν − 12gµνR =−gµνΛ+κTµν, (1.1)where Rµν is the Ricci tensor (a contraction of the Riemann curvature tensor), R isthe Ricci scalar (a contraction of the Ricci tensor), gµν is the space-time metric, Λis the cosmological constant, κ = 8pi is the gravitational coupling constant and Tµνis the energy-momentum tensor representing the source of the gravitational field.2The left hand side of (1.1) describes the curvature of space whereas, ignoringthe cosmological constant for the moment, the right hand side describes the distri-bution of matter and energy. In Misner, Thorne and Wheeler, [3], this interactionis described as ”Space tells matter how to move” and ”Matter tells space how tocurve”. Einstein added the cosmological constant to this equation to account forthe general belief at the time that the universe is static on a cosmological scale.The space-time metric is of the formds2 = gµνdxµ dxν . (1.2)The metric for a static, spherically symmetric space-time solution can be written inthe form (adopting the - + + + signature convention for this thesis)ds2 =−B(r)dt2 +A(r)dr2 +r2(dθ2 +sin2 θdφ2), (1.3)and the Ricci scalar is given byR = B′′AB −B′2ABuni0028.alt03A′A +B′Buni0029.alt03+ 2Aruni0028.alt03B′B −A′Auni0029.alt03+ 2Ar2 − 2r2 . (1.4)In general relativity, for space-time solutions on scales much smaller than thecosmos (i.e. stars and galaxies), it is reasonable to take Λ = 0 in 1.1. Vacuumsolutions have Tµν = 0 so that the Einstein field equations of general relativityreduce toRµν = 0 (1.5)In this thesis, we are investigating vacuum solutions on the brane. Insofar asthe gravitational field equations on the brane are induced from the Einstein fieldequations in an embedding higher dimensional space-time, the field equations onthe brane do not reduce to (1.5) in general.We will see that there is a correspondence between vacuum solutions on thebrane and solutions to the Einstein field equations in 3+1 dimensional general rel-ativity with a trace-free energy-momentum tensor (i.e. with R = 0 in the absenceof a cosmological constant).31.1.2 The Kretschmann ScalarIn our investigations, we be test candidate gravitational geon space-times. Inparticular, we are interested in static, spherically symmetric space-times that arenon-trivial, non-singular and without horizons. For space-time metrics of the form(1.3), the Kretschmann scalar is derived from the Riemann curvature tensor asRµνρσRµνρσ = 4K21 +8K22 +8K23 +4K24 (1.6)where [6]K1 = 1Auni0028.alt03B′′2B −(B′)24B2 −A′B′4ABuni0029.alt03(1.7)K2 = B′2ABr (1.8)K3 = −A′2A2r (1.9)K4 = A−1Ar2 (1.10)As this is a sum of squares of all components of the Riemann curvature tensor, adivergence in the Kretschmann scalar will correspond to a singularity in the space-time. It is clear from (1.9) and (1.10) that limr→0A(r) = 1 + O(rn) with n ≥ 2 toavoid divergence of the Kretschmann scalar. Similarly, from (1.8) it follows thatlimr→0B′(r)B(r) = O(rm) with m ≥ 1.1.1.3 Junction ConditionsIn the Randall-Sundrum model, the brane is a hypersurface embedded in ahigher dimensional space-time. On the brane, we also encounter embedded hy-persurfaces corresponding to potential singular points in the numerical analysis ofour candidate brane-world space-times. In each situation involving an embeddedhypersurface, the Israel-Darmois junction conditions must be met across the hy-persurface.4Taking y to be a Gaussian normal coordinate orthogonal to the hypersurface(and letting y = 0 at the hypersurface without loss of generality), the first junctioncondition is simply that there be no discontinuity in the metriclimy→0+gµν − limy→0−gµν = 0 (1.11)To get the second, we first observe that, in the embedding space-time around ahypersurface, the energy momentum tensor can be decomposed [7] asTµν = θ(l)T +µν +θ(−l)T−µν +δ(l)Sµν (1.12)where θ(l) is the Heaviside theta function, δ(l) is the Dirac delta function (bothevaluated at the proper distance l from the hypersurface), T±µν is the energy-momentumtensor evaluated on either side of the hypersurface and Sµν is the surface energy-momentum tensor corresponding to the hypersurface.The second Israel-Darmois junction condition states that the surface energy-momentum tensor corresponding to a hypersurface is related to the discontinuityin the extrinsic curvature Kµν across the hypersurface by [3, 8]Sµν = 18pi uni0028.alt01[Kµν]−[K]hµνuni0029.alt01 (1.13)where hµν is the induced metric on the hypersurface, K = Kµµ and the square brack-ets, indicate the discontinuity in a physical quantity crossing the hypersurface (i.e.[K] = K+−K−).In the Randall-Sundrum brane-world cosmology, we will look at solutions tothe gravitational field equations on the brane embedded in a higher dimensionalbulk space-time. The solution on the hypersurface is known, but the solution in theembedding space-time is not. The effective Einstein field equations on the braneinclude terms involving Sµν. (1.13) is used to show that these terms are zero forstatic, vacuum space-time solutions on the brane.We will also examine solutions to the 3+1 dimensional Einstein field equationsthat can only be constructed piece-wise around embedded hypersurfaces where wefind numerical instabilities. The solution on the hypersurface is unknown but thesolution in the embedding space-time is. In this situation, the extrinsic curvature5can be evaluated using [4]Kµν = 12Lngµν (1.14)where Ln is the Lie Derivative with respect to the normal of the hypersurface inquestion. The extrinsic curvature is evaluated in the limit approaching the hy-persurface from either side. Should there be a discontinuity in Kµν across thehypersufrace, Sµν ∕= 0 in general. This is generally taken to mean that there is agravitational source (other than the gravitational field) present at the hypersurface.In this thesis, we are intested in vacuum solutions and so require that gravitationalgeon solutions have continuity of the extrinsic curvature across every such hyper-surface.1.2 The Brane-WorldEinstein’s theory of general relativity is a 4 dimensional theory, the dimen-sions being the 3 spatial dimensions of our universe and time. Space-time modelsthat incorporate more than 4 dimensions date back to the theory of Kaluza whoshowed that 5 dimensional general relativity could incorporate both the 4 dimen-sions of Einstein’s gravitational theory and Maxwell’s theory of electromagnetism[9]. As we only experience a 3+1 dimensional space-time, Kaluza made the as-sumption that the metric would not change with respect to the extra dimension. Ineffect, physics would take place on a 4 dimensional hypersurface in a 5 dimen-sional space-time [11]. Klein showed that this condition would arise naturally ifthe 5th dimension had a circular topology on a small enough scale [10]. That beingthe case, the energies of all modes above the ground state of a Fourier expansion ofthe periodic dependence of any physical field on the 5th dimension, could be madeso high as to be unobservable.The success of this unification of gravitation and electromagnetism led physi-cists to theories with higher dimensions with the hope of unifying gravitation andelectromagnetism with the strong and weak forces. 10 dimensional string theoriesand eventually the 11 dimensional M-theory emerged from these efforts. In stringtheory, the 6 extra dimensions are compact and small as is the extra dimension inthe Kaluza-Klien theory. M-theory extends 10-dimensional string theory with one6extra dimension the size of which depends on the string coupling strength.M-theory introduced the idea that there could be an extra dimension whosescale is large relative to the fundamental scale. This idea forms the basis of Randall-Sundrum models of 5-dimensional gravity in which the three spatial dimensions ofthe empirical universe are a 3 dimensional membrane or 3-brane embedded in a4+1 dimensional space-time (4+1 dimensional referring to four spatial dimensionsand time). Particles and forces of the standard model are confined to the branewhereas the gravitational field can propagate through the bulk.The Randall-Sundrum models were proposed as a new mechanism for solvingthe hierarchy problem. The hierarchy problem is related to the fact that the funda-mental energy scale at which the quantum effects of gravity become strong (knownas the Planck scale) is so much larger than the energy scale at which the quantumeffects of the other forces become strong. In the Randall-Sundrum model, gravita-tion is free to propagate through the bulk whereas the other forces are confined tothe brane. The fundamental energy scale in the bulk can therefore be much smallerthan the Planck scale, which remains the effective fundamental energy scale on thebrane.It is the Randall and Sundrum [1, 12] single brane model which provides themathematical framework for our investigations. The effective 3+1 dimensionalEinstein field equations on the brane were derived by Shiromizu, Maeda, andSasaki [13]. An excellent review of this work is provided by [14] (see also [15–20]).The Einstein field equations in the 4+1 dimensional bulk space-time are(5)RAB − 12(5)gAB(5)R =−(5)gABΛ5 +κ25(5)TAB (1.15)where the terms are the same as for the 3+1 dimensional case (1.1) except (5) pre-ceding a tensor or 5 following a scalar indicates the appropriate dimensionality (theconvention that upper case latin indices indicate bulk tensors with lower case greekindices indicating 3+1 dimensional tensors on the brane is also used).The metric in the bulk will be(5)ds2 = (5)gABdxAdxB (1.16)7If we let y be a Gaussian normal coordinate orthogonal to the brane (letting y = 0at the brane without loss of generality) then(5)ds2 = (4)gµν(xα,y)dxµ dxν +dy2 (1.17)where (4)gµν(xα,y) corresponds to induced metric on the hypersurfaces at y =constant. The 3+1 dimensional metric on the brane is therefore (4)gµν(xα,0)(which we will generally write as gµν ).Our objective in this section is to determine the induced Einstein field equa-tions on the brane. We begin by taking the string theory idea that the fields of thestandard model (i.e. electromagnetic, strong, weak and all matter) are confined tothe brane. Gravitation is the only field that is permitted to extend into the bulk.Thus if Tµν is the 4 dimensional energy-momentum tensor of the particles andfields confined to the brane then the total energy momentum tensor on the brane isT braneµν = Tµν −λgµν (1.18)where λ is the brane tension which tends to prevent the gravitational field fromleaking into the bulk. Insofar as this model represents a departure from standardgeneral relativity, the accuracy to which general relativity predicts gravitation inexperiment puts a lower limit on the brane tension of λ > (1 TeV)4.In (1.17), the use of a 4-dimensional tensor in a 5-dimensional equation isfairly explicit by nature of the coordinate system used. However, we wish to de-compose the the 5-dimensional energy-momentum tensor into a part that resideson the brane and those parts that do not. This requires that we include the tensorT braneµν , which is 4-dimensional in a 5-dimensional equation. We do this by definingthe 4-dimensional tensor in terms of a 5-dimensional tensor asT braneµν = (5)T braneAB gµ Agν B,(5)T braneAB nA = (5)T braneAB nB = 0 (1.19)where nB is normal to the brane. The behaviour of (5)T braneAB off the brane is notimportant insofar as we decompose the bulk energy-momentum as8(5)TAB = (5) ˜TAB +T braneAB δ(y) (1.20)where the delta function expresses the confinement of standard model particles andfields to the brane at y = 0. We can now rewrite (1.15) incorporating this to give(5)RAB− 12(5)gAB(5)R =−(5)gABΛ5 +κ25uni0028.alt02(5) ˜TAB +T braneAB δ(y)uni0029.alt02(1.21)In order to determine the induced Einstein field equations on the brane, inaddition to the induced energy-momentum tensor, we need to know the inducedRiemann curvature tensor on the brane. The bulk Riemann curvature tensor isprojected onto 4 dimensional hypersurfaces y = const. (with extrinsic curvaturecorrections) asRµνσρ = (5)RABCDgµ Agν Bgσ Cgρ D +2Kµ[σ Kρ]ν (1.22)in which the square brackets denote anti-symmetrization. The change in the ex-trinsic curvature Kµν along the y = const. surfaces is projected from the bulk Riccitensor via the Codazzi equation∇νKν µ −∇µK = (5)RABgµ AnB, (1.23)where K = Kν ν.Observing that the 5-dimensional Riemann curvature tensor can be written as(5)RABCD = (5)CACBD + 23uni0028.alt02(5)gA[C(5)RD]B −(5)gB[C(5)RD]Auni0029.alt02− 16(5)gA[C(5)gD]B(5)R,(1.24)where (5)CACBD is the Weyl tensor (which is trace free). Further observing that(5) ˜TAB = 0 (i.e. there are no stresses other than the cosmological constant and thoseon the brane) then combining (1.21) and (1.22) gives an expression for the inducedEinstein field equations on the brane9Rµν − 12gµν R = −12Λ5gµν + 23uni0028.alt03T braneµν − 14T branegµνuni0029.alt03+KKµν− Kµ αKαν + 12(Kαβ Kαβ −K2)gµν −Eµν, (1.25)whereEµν = (5)CACBDnCnDgµ Agν B (1.26)is the electric projection of the 5 dimensional Weyl tensor and, by virtue of Weyltensor symmetries, is trace-free.Now, Randall-Sundrum brane-world cosmologies are Z2-symmetric (have mir-ror symmetry) so that the bulk space-time looks the same approaching the branefrom one side as it does emerging from it on the other, but with the normal re-versed. Consequently, we can say that the extrinsic curvature in the limit as y → 0approaching the brane from either side satisfieslimy→0+KAB =− limy→0−KAB (1.27)The Israel Darmois junction conditions for a 4+1 dimensional bulk space-timeare derived by integrating (1.21) across the brane taking the limit from either sidegivinglimy→0+gAB − limy→0−gAB = 0, (1.28)limy→0+KAB− limy→0−KAB =−κ25uni0028.alt03(5)T braneAB +13TbranegABuni0029.alt03, (1.29)which, applying (1.27) and (1.18) gives the extrinsic curvature on the braneKµν =−12κ25uni0028.alt03Tµν + 13(λ −T)gµνuni0029.alt03, (1.30)So we can write the contributions from the extrinsic curvature in (1.25) in terms ofthe energy momentum tensor. This contribution is written10Sµν = 112T Tµν − 14TµαT α ν + 124gµνuni0028.alt023Tαβ T αβ −T 2uni0029.alt02. (1.31)where, recalling 1.18, Tµν is the 4-dimensional energy-momentum tensor of theparticles and fields confined to the brane.If there are any stresses in the bulk other than the bulk cosmological constant(i.e. if (5) ˜TAB ∕= 0) , they will also affect the induced Einstein field equations on thebrane. In our case we assume there are none so the induced Einstein field equationson the brane becomeRµν − 12gµνR =−gµνΛ+κ2Tµν +6κ2λ Sµν −Eµν, (1.32)where κ2 is an effective coupling constant inherited from the fundamental couplingconstant and the cosmological constant on the braneΛ = 12uni0028.alt01Λ5 +κ2λuni0029.alt01, (1.33)will be zero as we assume that the bulk cosmological constant will be balanced bythe brane tension λ.We will be working only with vacuum solutions on the brane (i.e. Tµν = 0). Asa consequence, equation (1.32) simplifies toRµν − 12gµνR =−Eµν. (1.34)We know that Eµν is trace free (i.e. Eµ µ = 0) so contracting both sides of (1.34)yieldsRµ µ = Eµ µ = 0,Rµν = −Eµν, (1.35)and that, for vacuum solutions,∇µEµν = 0, (1.36)11where ∇µ is the covariant derivative on the brane.In standard 3+1 dimensional general relativity (with no cosmological constant),the Einstein field equations for a solution to R = 0 (so not a vacuum solution ingeneral) will reduce toRµν = 8piTµν, (1.37)with ∇µTµν = 0. We therefore make the correspondence8piTµν =−Eµν. (1.38)The result is that, as stated by Dadhich, Maartens, Papadopoulos and Rezania [21]A stationary general relativity solution with trace-free energy-momentumtensor gives rise to a vacuum brane-world solution in 5-dimensionalgravity.The candidate gravitational geon space-time solutions that we will be investi-gating are stationary vacuum solutions on the brane. In standard general relativity,such solutions would have to satisfy Rµν = 0. However, in the brane-world cos-mologies considered herein, we solve R = 0 to obtain non-vacuum (in general), 3+1dimensional space-time solutions to general relativity and these will correspond to4+1 dimensional, vacuum solutions on the brane.1.3 Gravitational GeonsIn 1955, John Archibald Wheeler coined the term geon as an abbreviation forgravitational-electromagnetic entity [2]. He wrote the following description of ageon:Associated with an electromagnetic disturbance is a mass, the gravita-tional attraction of which under appropriate circumstances is capableof holding the disturbance together for a long time in comparison withthe characteristic periods of the system.Wheeler’s intention was to provide a self-consistent picture of what a body ofmatter is. More than half a century later, we still do not have a consensus on what12that picture should be.In quantum field theory, particles of matter are considered to be field quantabut there is no accepted understanding of how the gravitational field should bequantized. General relativity provides an effective model of gravitation in termsof the geometry of space-time, but in its original form, general relativity does notexplain what matter is or how the other forces of nature come about.In the years immediately following the introduction of the geon, work focusedon geons that included electromagnetic waves. Regge and Wheeler introduced theidea of gravitational geons in 1957 [22]. These were defined as geons that consistedonly of gravitational waves.In general relativity, the gravitational field corresponds to curvature in a 3+1dimensional space-time. Solutions to the field equations of general relativity areconsidered topologically trivial if they are topologically equivalent to a Minkowskispace-time (i.e. the solution can be obtained by continuously deforming a flat 3+1dimensional space-time).In general relativity (and Kaluza’s 5 dimensional generalization [9]), there areno non-singular, topologically trivial solutions representing a gravitational field ofnon-vanishing mass [23, 24] (see also [25, 26] and [27] in particular for historicalcontext).Our investigations are concerned with static, topologically trivial, non-singulargravitational geon solutions for brane-world cosmologies. We know that such so-lutions are not possible in 3+1 dimensional general relativity, but in brane-worldcosmologies, the induced Einstein field equations on the brane differ from those ofgeneral relativity. In particular, in brane-world cosmologies, vacuum solutions onthe brane correspond to solutions to R = 0 in general relativity whereas in vacuumsolutions of general relativity require Rµν = 0.It is useful to briefly consider why gravitational geon solutions cannot satisfythe condition that Rµν = 0. In his 1923 theorem, Birkhoff [28] showed that theSchwarzschild metric is the unique spherically symmetric vacuum solution in gen-eral relativity. As we are interested in non-singular, spherically symmetric vacuumsolutions without horizons of the form (1.3), this result is sufficient for our pur-poses as the Schwarzschild metric has both a horizon and is singular at its radialorigin. However, Einstein and Pauli [23, 24] did not impose spherical symmetry13and worked with Kaluza’s 5 dimensional generalization of general relativity (whichis a foundational concept for string theory and, consequently brane-world cosmolo-gies). Their work is therefore more relevant to the study of gravitational geons inbrane-world cosmologies than Birkhoff’s theorem. For that reason, I present herea simplification to the spherically symmetric case, of Einstein and Pauli’s work .In 3+1 dimensional general relativity, Einstein and Pauli considered the varia-tion of the Ricci tensorδRµν =−∇α(δΓαµν)+∇ν(δΓαµα). (1.39)We can then define the vector density ℵα by∂α ℵα =√−ggµνδRµν, (1.40)so thatℵα =√−g(gµα δΓνµν −gµνδΓαµν). (1.41)We are interested in vacuum solutions, so we only consider variations that preserveRµν = 0. For any such variation, we have that δRµν = 0 and hence∂αℵα = 0. (1.42)In the following treatment we consider only space-time solutions of the form(1.3) thus, gtt and grr are both functions only of r. In our notation, we continue touse lower-case Greek indices (α, β, etc.) to indicate tensors on a 3+1 dimensionalspace-time; we use specific lower-case latin letters (i, j, k) to indicate tensors on3 dimensional space and we use [t, r, θ, φ] as the indices of 3+1 dimensionalspace-time in spherical polar coordinates.To avoid singularities (and for consistency with the candidate space-times thatwe examine in this thesis), we consider only solutions for which gtt and grr areboth non-zero and finite for all r ≥ 0. The variation that we are considering can bewritten as a coordinate transformation˙xα = xα +ξ α. (1.43)14Retaining only terms of the first order in the infinitesimal variation ξ α, we haveδΓαµν =−∂µ∂νξ α +∂σ(ξ α)Γσµν −∂µ(ξ σ)Γασν −∂ν(ξ σ)Γασ µ −ξ σ ∂σ Γαµν. (1.44)We are interested in coordinate transformations that preserve the static natureof the space-time solutions. Einstein and Pauli [24] considered specific transfor-mations that lead to an integral theorem that singles out the regular solutions ofRµν = 0 and are defined byξ t = c1t, ξ r = 0, ξ θ = 0, ξ φ = 0. (1.45)Substituting (1.44) into (1.41), (1.42) therefore becomes∂i(√−ggttΓitt) = 0, (1.46)where i only sums over the 3 spatial dimensions since we are only consideringstatic space-time solutions (i.e. ∂tgµν = 0). This simplifies to∂runi0028.alt01√−ggtt grr∂r(gtt)uni0029.alt01= 0. (1.47)Multiplying (1.47) through by gtt gives∂r(gtt)√−ggttgrr∂r(gtt) = ∂runi0028.alt01gtt√−ggttgrr∂r(gtt)uni0029.alt01, (1.48)which integrating over a 3 dimensional, singularity-free region and applying Gauss’theorem givesuni222B.alt02V√−ggtt grr (∂r(gtt))2 dV +uni222E.alt02S√−ggrr∂r(gtt)nrdS = 0. (1.49)As the surface S approaches infinity, gtt →−1 and grr → 1 so thatuni222E.alt02S√−ggrr∂r(gtt)nrdS → 0. (1.50)Therefore,15uni222B.alt02V√−ggtt grr (∂r(gtt))2 dV → 0, (1.51)as the boundary of the volume V approaches infinity. As gtt < 0, √−g > 0 andgrr > 0 for all r ≥ 0, it must be the case that gtt = constant. For vacuum solutionswe therefore have Ri j = 0, but in three dimensions this implies that the space isEuclidean.The requirement that Rµν = 0 therefore implies that there can be no static,spherically symmetric, topologically trivial and non-singular space-time solutions(i.e. gravitational geons of the type we are investigating) in 3+1 dimensional gen-eral relativity.However, in Randall-Sundrum brane-world cosmologies, vacuum space-timesolutions must satisfy R = Rα α = 0 but not Rµν = 0. Thus, gravitational geonsolutions of brane-world space-times may therefore be possible.1.4 Static Space-Time Solutions on the BraneThis thesis explores the possibility that space-time solutions that correspondto gravitational geons may exist in brane-world cosmologies. Insofar as there hasbeen considerable interest in brane-world cosmological models, there are recentlypublished articles regarding other static space-time solutions on the brane that havesome aspects in common with the investigations of this thesis. The following sec-tions briefly summarize articles that investigate solutions corresponding to blackholes and wormholes on the brane.1.4.1 Brane-World Black HolesThere are a number of different ways that brane-world black holes have beenconsidered. In 2000, Chamblin, Hawking and Reall [29] looked at the metric for ablack string in a 4+1 dimensional anti de Sitter (AdS) space-time. They wrote theAdS space-time metricds2 = e−2yl ηµνdxµdxν +dy2, (1.52)where ηµν is the 4-dimensional Minkowski metric and l is the AdS radius. They16introduced the coordinate z = le yl so that the black string metric could be writtenasds2 = l2z2uni0028.alt01−U(r)dt2 +U(r)−1dr2 +r2(dθ2 +sin2θdφ2)+dz2uni0029.alt01, (1.53)where U(r) = 1− 2Mr . They then pointed out that this metric, with an appropriaterescaling of the coordinates r and t, is in standard Schwarzschild form on the brane(at y = 0 hence z = l). A black string in the bulk thus defined gives rise to aSchwarzschild black hole on the brane.Later in the same year, Dadhich, Maartens, Papadopoulos and Rezania [21]considered the problem a different way. Instead of starting with a solution on thebulk space-time, they started (as do we) by insisting that their solution solve the in-duced Einstein field equations on the brane. They showed that a static, sphericallysymmetric black hole solution of the form (1.3) is given byB(r) = A(r)−1 = 1−uni0028.alt042MM2puni0029.alt041r +uni0028.alt04q˜M2puni0029.alt041r2 , (1.54)where Mp is the Planck mass on the brane, ˜Mp is the Planck mass in the bulk, and qis a dimensionless tidal charge parameter (so named because of its correspondenceto the charge parameter in the Reissner-N¨ordstrom black hole in general relativity).In general relativity, the charge parameter Q in the Reissner-N¨ordstrom blackhole solution is squared and so always acts to weaken the gravitational field of theblack hole (as compared to a Schwarzschild solution which is achieved in the limitas Q → 0). In their discussion of this brane-world black hole solution, the authorspointed out that the tidal charge parameter q is negative corresponding to a negativeeffective energy density on the brane contributed by the free gravitational field inthe bulk. Consequently, the bulk effects tend to strengthen the gravitational field ofthe black hole as compared to a Schwarzschild black hole (achieved in the limit asq → 0).In 2002, Vollick [19] showed that solving R = 0 for space-time metrics of theform (1.3) with A(r) = B(r)−1 has the general solution17B(r) = A(r)−1 = 1+ αr + βr2 , (1.55)where α and β are constants. Thus the Reissner-N¨ordstrom type solution of Dad-hich, Maartens, Papadopoulos and Rezania is the most general of that form. Hethen considered the result if the condition that A(r) = B(r)−1 is removed andB(r) = 1− 2mr (i.e. gtt is Schwarzschild). He obtained the solutionA(r) =uni0028.alt031− 2mruni0029.alt03−1uni005B.alt033m−2rλ −2runi005D.alt03. (1.56)It was suggested that deviations from Schwarzschild geometry (which mustresult from the Weyl term in the induced Einstein field equations on the brane as in(1.26)) could be responsible for observations attributed to dark matter.A third distinct approach to brane-world black holes is that of Shankaranarayananand Dadhich in 2004 [30]. They considered the possibility of non-singular blackholes on the brane. They obtained a non-singular Reissner-N¨ordstrom type blackhole solution in a D-dimensional, spherically symmetric space-time (where D ≥ 5)with a metric of the formds2 =−B(r)dt2 +A(r)dr2 +r2dΩ2, (1.57)where dΩ2 is the (D−2)-dimensional angular line-element. The solution that theyobtained hadB(r) = A(r)−1 = 1− 4Rg(r)(D−2)rD−3 , (1.58)whereRg(r) = 4piuni222B.alt02 r0ρ(x)xD−2dx . (1.59)Having found the form of their solution, the authors proceeded to choose massand charge distributions in order to set ρ(x) in (1.59). This led to them arriving(with some simplifying assumptions) at the expression18B(r) = A(r)−1 = 1− 16pi(D−2)(D−1) 1rD−3 c1c2uni0028.alt011−exp(−c2rD−1)uni0029.alt01+ 16pic3(D−2)(D−3) 1r2(D−3)uni0028.alt021−exp(−c4r2(D−2))uni0029.alt02(1.60)where the ci’s are constants.The induced black hole solution on the brane is non-singular and admits onehorizon. In order to achieve this result, they had to relax the condition that R = 0. Inour investigation, we insist on a vanishing Ricci scalar in the induced Einstein fieldequations on the brane (1.34). In the brane-world black hole of Shankaranarayananand Dadhich contributions to R cannot come from the bulk Weyl tensor via Eµνas this tensor is trace-free. The non-vanishing Ricci scalar indicates a varyingcosmological constant-like term, thus the gravitational geons that we investigate inthis thesis differ fundamentally from their result.1.4.2 Brane-World WormholesAs with black hole solutions on the brane, when investigating wormhole solu-tions on the brane there are two fundamental approaches. One can either create awormhole solution in the bulk space-time and examine the induced solution on thebrane (see for example [31]), or one can create a wormhole solution that satisfiesthe induced Einstein field equations on the brane.In their 2002 paper, Bronnikov and Kim [6] take the latter approach. Theybegin with a generic, static, spherically symmetric space-time metric on the brane,as do we (1.3). Our method is comparable to theirs insofar as they also solvefor R = 0 and test for finiteness of the Kretschmann scalar (1.1.2) as a regularitycriterion for the resulting space-time solutions.The brane-world wormhole solutions considered were, like our gravitationalgeon solutions, static, spherically symmetric, non-singular and without horizons.However, for a wormhole, the solution need only be non-singular for r > r0 forsome r0 > 0. Two space-time solutions meeting this condition can be sewn togetherat the spherical hypersurface at r0 (which is identified as the throat). The result-ing piece-wise constructed space-time forms the wormhole provided the Israel-19Darmois junction conditions (1.1.3) are met at the throat.The authors considered both symmetric and asymmetric brane-world worm-hole metrics. One example given of a symmetric wormhole solution used the met-ric [32]ds2 =−dt2 +uni0028.alt021− r0runi0029.alt02−1dr2 +r2(dθ2 +sin2 θdφ2). (1.61)It is clear that grr → −∞ as r+ → r0. Bronnikov and Kim assert that the cre-ation of wormholes involves the violation of the null energy condition at least ina neighbourhood of the throat. In brane-world cosmologies, a negative energydensity that would provide for this violation is possible (see [33]). The surfaceenergy-momentum tensor (1.12) associated with the throat of the wormhole mayhave contributions from the Weyl term in the induced field equations on the brane(1.26).The authors do not attempt to evolve the brane-world metric into the bulk inorder to discover what those contributions are for any particular solution.1.5 Gravitational Geons in Other TheoriesAs we have seen, general relativity does not admit static, non-singular andtopologically trivial gravitational geon solutions. As a consequence, most workon gravitational geons has considered concentrations of gravitational waves (seefor example [34] and, for an argument that this type of solution is inadmissible asa gravitational geon in general relativity, [35, 36]) or topological geons (particlesmade from non-trivial spatial topology [37]).This thesis investigates whether topologically trivial gravitational geons arepossible in brane-world cosmologies based on the observation that the effectiveEinstein field equations on the brane are different from those of 3+1 dimensionalgeneral relativity. However, brane-world space-times are not the only theories thatmay potentially allow the existence of topologically trivial gravitational geons.One possibility, recently investigated by Dymnikova and Galaktionov, is to assumethat vacuum solutions allow a varying Ricci scalar in 3+1 dimensional general rel-ativity.They investigated geon-like particles that they described as vacuum non-singular20black holes [38] in a spherically symmetric space-time with a de-Sitter centre. Theyexamined a metric of the form (1.3) defined byB(r) = A(r)−1 = 1− 2mruni0028.alt031−exp( −r32mr20 )uni0029.alt03(1.62)where r20 = 3Λ, Λ being a cosmological constant that appears at the origin (there be-ing no cosmological constant in the solution asymptotically). The resulting space-time is de-Sitter at its origin (i.e. having a positive cosmological constant) butSchwarzschild asymptotically (i.e. having no cosmological constant as r → ∞).The authors showed that the solution has horizons for m greater than a criticalvalue mcr (mcr being dependent on Λ). Solutions for m < mcr are static, spheri-cally symmetric and topologically trivial gravitational geons. However, owing tothe radially dependent cosmological constant, R ∕= 0 in general for this solution.Consequently, these solutions are not vacuum brane-world solutions of the type weare concerned with in this thesis.Another interesting example is Vollick’s 2008 paper [39] in which he looksat the possibility of stationary, asymptotically flat, non-singular and topologicallytrivial gravitational geons in 1+1 dimensions. He looks at two different 1+1 di-mensional theories of gravity, the first being a modified Jackiw-Teitelboim theory[40, 41] of the formR+αR2 +β∇µ∇µR−Λ = 8piT (1.63)where Λ is a cosmological constant (in this case Λ = 0). The second theory is basedon the actionL =√−guni0028.alt031φ R+V(φ)uni0029.alt03. (1.64)In each case, working in 1+1 dimensions and treating r as a radial-like coordinatewith r ≥ 0, exact solutions that correspond to gravitational geons are found.In a published addendum [42], Vollick pointed out that each of the solutionshas a jump discontinuity in d2tdτ2 if one imposes a reflecting boundary condition atr = 0 (which is necessary if there is to be a correspondence between the derivedsolutions and spherically symmetrical space-time solutions in 3+1 dimensions).21The author removes these discontinuities for (1.63) by allowing a non-zerocosmological constant.In the original paper, it was shown that the field equations resulting from (1.64)would be solved by potentials of the formV(φ) = A f′ and dV(φ)dφ =−A2r2 f′′. (1.65)The choice of f(r) allows one to solve for the potential. In the addendum, thefunctionf(r) = 1− 2mr2r3 +2ml2 (1.66)is shown to remove the jump discontinuity and has Schwarzschild behaviour atlarge r.The existence of gravitational geons similar to what we investigate herein havetherefore been explored, but not previously in brane-world cosmologies.22Chapter 2Gravitational Geons inBrane-World CosmologiesIn this thesis, we investigate whether the free gravitational field in the bulkmodifies the gravitational field on the brane so that gravitational geons are possibleon the brane. Our approach is to look at static, spherically symmetric solutions ofthe formds2 =−B(r)dt2 +A(r)dr2 +r2(dθ2 +sin2 θdφ2). (2.1)The Ricci scalar for metrics of this form isR = B′′AB −B′2ABuni0028.alt03A′A +B′Buni0029.alt03+ 2Aruni0028.alt03B′B −A′Auni0029.alt03+ 2Ar2 − 2r2 . (2.2)As explained in the introduction, a solution to R = 0 substituted into the Einsteinfield equations of 3+1 dimensional general relativity, will give rise to a vacuumbrane-world solution in 4+1 dimensions via a correspondence between the energy-momentum tensor in the 3+1 dimensional solution and the projection of the bulkWeyl tensor in the 4+1 dimensional solution.Our approach is to solve R = 0 in (2.2) for A(r). We do this generally in theweak field, but for strong field results, we choose a specific function B(r).232.1 The Candidate Space-TimesWe were unable to find a general solution to R = 0 for (2.2). For our strong fieldanalysis, each potential choice of B(r) represents a candidate space-time solutionthat may or may not correspond to gravitational geons on the brane.We are not interested in space-time solutions with singularities so we exam-ined the Kretschmann scalar for divergent behaviour in order to narrow down ouravailable choices. From (1.9) and (1.10) it is clear that we must have limr→0A(r) =1 + O(rn) with n ≥ 2 to avoid divergence of the Kretschmann scalar. Similarly,from (1.8) it follows that limr→0 B′(r)B(r) = O(rm) with m ≥ 1.In this thesis, we takeB(r) = 1− 2mr2r3 +2ml2 , (2.3)for which B′(0) = 0, B(0) = 1, both B(r) and B′(r) are continuous for all r ≥ 0 andlimr→∞ B(r) = 1− 2mr is Schwarzschild.In the limit as r → ∞ the gravitational field is weak and static. If we make thefurther assumption that any test particles are travelling at sub-relativistic speeds,we find that gtt =−(1+2Φ) where Φ corresponds to the Newtonian gravitationalpotential. Thus knowing B(r) is sufficient for us to identify the parameter m withthe gravitational mass of the space-time solution.For (2.3) we have that B ≈ 1−uni0028.alt01rluni0029.alt012 for r <<uni0028.alt01ml2uni0029.alt0113 . Thus l has the effect ofcontrolling the strength of a positive cosmological constant like term (with Λ = 3l2 )that appears in the limit as r → 0 (see 3.2).To be certain that the resulting space-times are non-singular and without hori-zons, we only consider solutions A(r) and B(r) greater than zero and finite for allr ≥ 0.Considering that our choice of B(r) is not unique, having chosen (2.3) and beenunsuccessful at finding a general solution, we considered other candidate solutionssuch as those for whichB(r) = 1− 2mr2(r +l)3 . (2.4)24None of the alternate expressions for B(r) that we looked at yielded a general solu-tion to R = 0 in (1.4). We decided to focus on (2.3) and use a variety of approximateand numerical methods to determine the nature of the resulting solution.2.2 Methods Of InvestigationOur investigation concerns solutions to 3+1 dimensional general relativity forwhich the Ricci Scalar vanishes everywhere. As a consequence, (2.2) can be writ-ten asA′ = (2BB′′r2 −(B′)2r2 +4BB′r +4B2)A−4B2A2Br(B′r +4B) . (2.5)We did not find a general solution to (2.5) for our choice of B(r) (2.3). In theabsence of a general solution, we used a variety of analytical and numerical meth-ods to investigate the behaviour of the candidate space-times. Using a combinationof the methods described in the following sections, we were able to determine forwhich choices of the parameters m and l in (2.3) the solutions correspond to gravi-tational geons.2.2.1 The Weak Field ApproximationIn the absence of a cosmological constant on the brane, for B(r)= A(r)= 1, ourgeneral, spherically symmetric metric (2.1) corresponds to flat space. We thereforetake the weak field limit to be B(r)= 1+b(r) and A(r)= 1+a(r) with ∣b(r)∣<< 1and ∣a(r)∣ << 1.In the weak field, the differential equation that we must solve (2.5) thereforesimplifies (by removing terms of second order in a, a′, b or b′) toa′ = b′′r2 +2b′r−2a2r . (2.6)With an analytical solution to this, we can determine whether there is a class offunctions B(r) that correspond to gravitational geons in the weak field.For the specific choice of B(r) that we have made (2.3), we will use our analyt-ical solution to the weak field to verify the results that we obtain using numericalmethods.252.2.2 Numerical IterationIn the absence of a general solution to (2.5), numerical methods proved veryhelpful in the investigation of the behaviour of the candidate space-times.We chose to use a fourth-order Runge-Kutta method to explore (2.5) iteratively.This is expressed given an initial value problem, which in our case isA′ = f(r,A), A(r0) = A0 . (2.7)The Runge-Kutta method is then given byAn+1 = An + K16 + K23 + K33 + K46rn+1 = rn +∆r (2.8)where ∆r is the iteration length, An+1 ≈ A(rn+1) andK1 = ∆r f(rn,An)K2 = ∆r funi0028.alt03rn + ∆r2 ,An + K12uni0029.alt03K3 = ∆r funi0028.alt03rn + ∆r2 ,An + K22uni0029.alt03K4 = ∆r f (rn +∆r,An +K3) . (2.9)We found that working with 3rd party software (Maple) was limiting insofar asit was slow when choosing a very small iteration length, and it proved an inflexibleenvironment for investigating the behaviour of several variables at once. We builta custom application to address these concerns. It allowed us to choose a startingpoint for iteration, an iteration length, the parameters m and l as well as the rangeof r and the granularity for the displayed values. This produced results for certainvariables of interest (see fig. 2.1). These results were then plotted using MicrosoftExcel.Knowing that there are potential singular points at B = 0 and B′r+4B = 0 (2.5),using this custom application allowed us to iteratively examine the behaviour of A26Figure 2.1: The interface of a custom application built to investigate our candidate space-times using a fourth-order Runge-Kutta numerical iteration.alongside various expressions including B, B′r + 4B and the Kretschmann Scalar(Rµνρσ Rµνρσ ) for a variety of choices of the parameters m and l. As we continuedto analyse our results, we added additional expressions to the software to providefurther insight into the behaviour of the solution.It should be noted that Maple provided additional choices for methods of nu-merical iteration. We compared our results to those provided Maple such as theRunge-Kutta-Fehlberg adaptive numeric procedure. We found that the Maple basedmethods were no more successful at navigating the potential singular points in oursolution. However, this comparison was invaluable for finding coding errors inour custom developed software solution as well as validating the results that weobtained.2.2.3 Direction Field PlotsNumerical iteration alone was not sufficient to obtain a complete and reliablepicture of the behaviour of A. In particular, we found regions of instability wherethe numerical solution diverged.We used the Maple mathematics and modelling software package to producedirection field plots (for example see fig. 2.2)Whereas the iterative approach provides results for our specific initial value27Figure 2.2: An example of a direction field plot showing numerical instabilities in A(r)(here plotted for l = 1, m = 1.189 and 1.25 < r < 1.42).problem, the direction field plots indicate where there are critical points and howthe behaviour of the solution depends on changes in initial conditions.2.2.4 Potential Singular PointsThere are potential singular points in (2.5) at r = 0, B = 0 and B′r + 4B = 0.We found solutions near these points using series approximation. Specifically, wewrite (2.5) in the formA′(r) = f(r)A(r)+g(r)A2(r) . (2.10)Letting F′(r) = f(r) then (2.10) has the solution [6]A(r) = −1e−F(r)uni222B.alt01 g(r)eF(r)dr . (2.11)We obtain series approximations of the form28f(x) =∞∑n=0αnxn+sg(x) =∞∑m=0βmxm+t (2.12)where x = r−r0 for some potential singular point at r = r0, the αn and βm are realconstants and s and t are integer constants. We can then solve (2.11) for limx→0A(x).To ensure that the resulting space-times be non-singular and without horizons,we rule out solutions for which we find points where A = 0 or A → ∞. Fromthe Kretschmann Scalar, (1.9) in particular, we know that solutions with A′ → ∞must also be ruled out. To that end, we differentiate our approximate solution A(x)around x = 0 to obtain limx→0A′(x).With an understanding of the behaviour of A and A′ in the vicinity of a potentialsingular point, we can characterize the behaviour of the space-time on the sphericalhypersurface corresponding to that point.2.3 Weak Field BehaviourIn the weak field limit, (2.5) simplifies to (2.6) which has the solutionA = 1+ b′r2 +cr , (2.13)where c is a constant of integration. Noting that c = 0 to ensure that A be finite atr = 0, any choice of B(r) = 1+b(r) with ∣b(r)∣<< 1 for which b′r is small for allr ≥ 0 would give suitable geon solutions in the weak field.In order to analyse candidate geons in the strong field, we have chosen B(r) =1− 2mr2r3+2ml2 . It is useful to solve this for A(r) in the weak field for which we obtainA = 1+ mr5 −4m2l2r2(r3 +2ml2)2 . (2.14)For r <<uni0028.alt01ml2uni0029.alt0113 we have B ≈ 1−uni0028.alt01rluni0029.alt012 and A ≈ 1−uni0028.alt01rluni0029.alt012. A(r) therefore satisfiesthe requirements that A(0) = 1 and that limr→0A(r) = 1+O(rn) with n ≥ 2. Further-more, A(r) is non-zero and continuous for all r ≥ 0.29It is worth noting that for large r, A differs from Schwarzschild with A≈1+ mr .Also, the conditions for the weak field from section (2.2.1)∣b(r)∣ = 2mr2r3 +2ml2 << 1 and (2.15)∣a(r)∣ = mr5 −4m2l2r2(r3 +2ml2)2 << 1 (2.16)are both met for rl << 1 and for ml << 1.For our specific choice of B(r) (see fig. 2.3), the solutions in the weak fieldlimit correspond to gravitational geons.We also examined the behaviour of our candidate space-times for weak fieldsusing numerical iteration (shown in fig. 2.3 for comparison). We found that thenumerical results were almost identical to the analytical results for the weak fieldlimit.Figure 2.3: B(r) and A(r) plotted against radius r for l = 1, m = 0.01. A fourth-order nu-merical iteration of (2.5) is labelled A(r) and the analytical weak field result is labelledweak A(r). These two approaches give almost identical results for ml << 1.302.4 General BehaviourIn the absence of a general solution to (2.5) for strong gravitational fields, weexamined the behaviour of A(r) using a numerical iteration. Using this method,we were unable to begin at exactly A(0) = 1. However, A ≈ 1−uni0028.alt01rluni0029.alt012 for small r.Thus, if we choose an iteration length ∆r so that ∆rl = 10−5 then A(∆r)≈1 to within10−10 (independent of m). Also, having made the observation from direction fieldplots that the solution is insensitive to small changes in the initial value of A, wechose ∆r appropriately and used A(∆r) = 1 as the starting point of the iteration.We are interested in solutions for which both A(r) and B(r) are greater thanzero and finite for all r ≥ 0. The zeros of B can be determined by noting thatB(0) = 1, limr→∞ B(r) = 1 and that B′ = 0 at r = (4ml2)12 where B has its minimumvalue of 1− 13(4ml )23 . ThusB has⎧⎨⎩no zeros if ml <√274one zero if ml =√274two zeros if ml >√274 .(2.17)We proceeded to examine the numerical iteration of A(r). We found that itworks well provided we choose m and l for which B′r + 4B ∕= 0 for all r > 0 butthat it is unable to navigate instabilities in the solution otherwise. The zeros ofB′r +4B are found by observing that B′r +4Bvextendsinglevextendsingler=0 = 4, limr→∞B′r +4B = 4 and thatddr(B′r + 4B) = 0 at r = (2ml2)13 where B′r + 4B has its minimum value of 4−(27m4l )23 . ThusB′r +4B has⎧⎨⎩no zeros if ml < 3227one zero if ml = 3227two zeros if ml > 3227 .(2.18)Noting that there will be choices of m and l for which there will be zeros of B′r +4B but no zeros of B, we investigated the behaviour of A(r) around the zeros ofB′r +4B.In general, from (2.10), we have31f(r) = 2BB′′r2 −(B′)2r2 +4BB′r +4B2Br(B′r +4B) and (2.19)g(r) = −4Br(B′r +4B) . (2.20)Taking x = r−r0 where B′r +4Bvextendsinglevextendsingler=r0= 0 we get series approximations for (2.19)and (2.20)f(x) = α1x +α2 +α3x+O(x2) and (2.21)g(x) = β1x +β2 +β3x+O(x2) . (2.22)In this case, for α1 /∈{0,−1,−2,...}, solutions to (2.11) are of the formA(x) = −1β1α1 −β1α2−β2α1α1(α1+1) x+λ∣x∣−α1 +O(x2,λx∣x∣−α1)(λ - const. of integration).(2.23)For α1 ∈ {0,−1,−2,...} we get a different solution for each choice of α1. Forexample, choosing α1 =−1 givesA(x)= −1−β1 +(β2 +β1α2)xln∣x∣+β1α2x+λx+O(x2ln∣x∣)(λ - const. of integration).(2.24)However, for all choices of α1 the dominant term in the denominator as x → 0results in the same outcome thatlimx→0A(x) =uni007B.alt03−α1β1 if α1 ≤ 0 or λ = 00 if α1 ≥ 0 and λ ∕= 0 . (2.25)We calculated values for α1 and β1 for each zero of B′r+4B given a variety ofvalues of m and l (see figs. 2.4 and 2.5).Having verified our results for a range of choices 0.02 < l < 750, we found that32Figure 2.4: α1 and β1 plotted against ml for the first zero of B′r +4B.Figure 2.5: α1 and β1 plotted against ml for the second zero of B′r +4B.at each zero the values of α1 and β1 depend on ml but not on m and l individually.Knowing that zeros of A(r) correspond to singularities in the space-time, we soughtto determine where α1 = 0 in terms of ml .With x = r−r0, we have B′r +4B ≈ x(B′′r +5B′)vextendsinglevextendsingler=r0in the vicinity of eachzero of B′r +4B, soα1 =2B′′r− (B′)2rB +4B′+ 4BrB′′r +5B′vextendsinglevextendsinglevextendsinglevextendsinglevextendsingler=r033= 2B′′r +7B′B′′r +5B′vextendsinglevextendsinglevextendsinglevextendsingler=r0. (2.26)Substituting (2.26) into (2.3) for α1 = 0, we get the following quadratic in r3r6 +14ml2r3 −24m2l4 = 0 (2.27)with one positive real root corresponding to r ≈(1.544ml2)13 . Substituting this intoB′r +4B = 0 yields ml ≈ 1.202.From (2.18), we know that there will be two zeros of B′r +4B for ml ≈ 1.202.Figure 2.5 shows that α1 > 0 for all ml at the second zero, thus at the first zeroα1 > 0 when ml > 1.202 and α1 < 0 when ml < 1.202.The preceeding analysis shows how the value of α1 will help us to categorizethe behaviour of A around the zeros of B′r+4B given ml . As we know that A′→±∞corresponds to singular behaviour in the space-time, we perform a similar analysisof the behaviour of A′. Differentiating (2.23) givesA′(x) =−β1α2 −β2α1α1(α1 +1)− α1λ∣x∣−α1x +...uni0028.alt03β1α1 −β1α2 −β2α1α1(α1 +1) x+λ∣x∣−α1 +...uni0029.alt032 (2.28)limx→0 A′(x) =⎧⎨⎩α1(β2α1−β1α2)β 21 (α1+1) if α1 <−1 or λ = 0−sgn(α1λx)∞ if −1 < α1 < 1 and λ ∕= 00 if α1 > 1 and λ ∕= 0 .(2.29)We know from (2.25) that we need not consider α1 ≥ 0 (unless λ = 0), but thebehaviour of A′ does depend on whether α1 is greater or less than -1 at the zerosof B′r +4B. We therefore sought to determine where α1 = −1 in terms of ml andnumerically found this corresponds to ml ≈ 1.191.From (2.18), we know that there will be two zeros of B′r +4B for ml ≈ 1.191.Figure 2.5 shows that α1 > −1 for all ml at the second zero, thus at the first zeroα1 >−1 when ml > 1.191 and α1 <−1 when ml < 1.191.At each zero, the values of α1 and β1 depend on ml but not on m and l individ-ually. That is not true of the values of α2 and β2 and (2.29) tells us that we may34need α2 and β2 in order to categorize the behaviour of A around the first zero ofB′r + 4B for 3227 < ml < 1.191. However, we did determine (again verifying ourresults for a range of choices 0.02 < l < 750) that at each zero sgn(α2), sgn(β2)and the ratio α2β2 depend on ml but not on m and l individually. We can therefore saythat sgnuni0028.alt02α1(β2α1−β1α2)β 21 (α1+1)uni0029.alt02= sgnuni0028.alt04α1β2uni0028.alt02α1−β1 α2β2uni0029.alt02β 21 (α1+1)uni0029.alt04depends on ml but not on m and lindividually.The important thing to take away from the preceeding analysis is that the be-haviour of A around the zeros of B′r+4B can be effectively characterized in termsof ranges of values of ml . We are not concerned with solutions for which thereexist points where B = 0 (i.e. ml ≥√274 ), so this leaves five ranges of values toconsider with respect to the behaviour of A. Three of these correspond to solutionsin which there are two zeros of B′r+4B and are distinguished by values taken by aspecific coefficient α1 in the series approximation (2.21). We refer to these rangesas low, medium and high referring to the relative strength of the correspondinggravitational field.∙ Two Critical Points – High Range: There are two zeros of B′r + 4B for1.202 < ml <√274 . Both zeros have α1 > 0.∙ Two Critical Points – Medium Range: There are two zeros of B′r+4B for1.191 < ml < 1.202. The first zero has −1 < α1 < 0 and the second zero hasα1 > 0.∙ Two Critical Points – Low Range: There are two zeros of B′r + 4B for3227 <ml < 1.191. The first zero has α1 <−1 and the second zero has α1 > 0.∙ No Critical Points: There are no zeros of B′r +4B for ml < 3227∙ The Weak Field: See section (2.3)With the exception of the weak field analysis (which has already been covered),we look at each of the above ranges in the following sections.352.4.1 Two Critical Points – High RangeFor 1.202 < ml <√274 , in (2.23) α1 > 0 at each zero of B′r +4B. We thereforeconsidered the following two situations.λ ∕∕∕= 0 at either zero of B′r + 4B: A = 0 at either zero of B′r +4B when λ ∕= 0.Our analysis of the Kretschmann scalar (1.10) indicates singular behaviour of thespace-time at the spherical hypersuface corresponding to either zero.λ = 0 at the first zero of B′r + 4B: Noting that α1 > 0 and β1 > 0 (see fig. 2.4),at the first zero of B′r + 4B we have A = −α1β1 < 0 (see fig. 2.6). Since A(0) = 1then A(r0) = 0 for some r0 > 0 but inside the spherical hypersurface correspondingto the first zero of B′r + 4B. Referring again to our analysis of the Kretschmannscalar, this indicates singular behaviour in the space-time at r0. Therefore, therecan be no solutions that correspond to gravitational geons for these values of ml .Figure 2.6: A(r) iteratively plotted against radius r for l = 1 and m = 1.25 against a directionfield plot. Here, both the convergence to A = 0 and A = −α1β1≈−1.84 at the first zeroof B′r +4B can be clearly seen.362.4.2 Two Critical Points – Medium RangeFor 1.191 < ml < 1.202, in (2.23) −1 < α1 < 0 at the first zero of B′r + 4B,whereas α1 > 0 at the second zero. We therefore considered the following situa-tions.λ ∕∕∕= 0 at the first zero of B′r + 4B: A = −α1β1 and A′ →−sgn(α1λx)∞ at the zerowhich, from (1.9), indicates singular behaviour of the space-time.λ ∕∕∕= 0 at the second zero of B′r +4B: A = 0 at the second zero indicating singularbehaviour in the space-time.λ = 0 at both zeros of B′r + 4B: Noting that α1 < 0, β1 > 0, α2 < 0 and β2 > 0at the first zero we have that A = −α1β1 > 0 and A′ = α1(β2α1−β1α2)β 21 (α1+1)< 0. At thesecond zero, we have α1 > 0, β1 < 0, α2 < 0 and β2 > 0 so that A = −α1β1 > 0 andA′ = α1(β2α1−β1α2)β 21 (α1+1)> 0. There is therefore no singular behaviour and no horizonindicated at either zero.However, direction field plots indicate that solutions with A(0) = 1 have A′ > 0as the first zero of B′r+4B is approached. Iterating backward from A = −α1β1 at thesecond zero results in A′ > 0 as A approaches the first zero (see fig.2.7).Figure 2.7: Reverse iteration of A(r) plotted against radius r for l = 1 and m = 1.192 fromthe second zero of B′r + 4B to the first, against a direction field plot. The solution isincompatible with the analytical result that A′ < 0 at the first zero for λ = 0 solutions.37It is apparent that λ = 0 solutions are inconsistent with our boundary condi-tions. We must therefore conclude that there are no solutions that correspond togravitational geons for these values of ml .2.4.3 Two Critical Points – Low RangeFor 3227 < ml < 1.191, in (2.23) α1 < −1 at the first zero of B′r +4B, whereasα1 > 0 at the second zero. We therefore examined the behaviour of A at each zerodepending on the value of the constant of integration λ.Any choice of λ at the first zero of B′r+4B: A = −α1β1 > 0 and A′ = α1(β2α1−β1α2)β 21 (α1+1)>0 at the first zero of B′r+4B. There is therefore no singular behaviour and no hori-zon indicated.λ ∕∕∕= 0 at the second zero of B′r +4B: A = 0 at the second zero indicating singularbehaviour in the space-time.λ = 0 at the second zero of B′r + 4B: A = −α1β1 > 0 and A′ = α1(β2α1−β1α2)β 21 (α1+1)> 0 atthe second zero. No singular behaviour or horizon is indicated.Our analysis shows the possibility of solutions corresponding to gravitationalgeons with λ = 0 at the second zero, but instability at each zero prevents us fromconfirming any particular solution using iterative methods (see fig.2.8).However, we have been able to confirm that limx→0 A = −α1β1and limx→0 A′ = α1(β2α1 −β1α2)β 21 (α1 +1)around each zero for each section constructed numerically (see figs. 2.9, 2.10 and2.11).We piecewise constructed A around each zero of B′r+4B using the numericallyconstructed sections (see fig.2.12). The resulting solution is sewn together at thespherical hypersurfaces corresponding to each zero. The sewing together of twomanifolds may induce a surface energy-momentum tensor (1.13) on the surfacewhere they are joined.The extrinsic curvature Kµν depends on the metric and its first derivatives [4].Given that A, B and B′ are all continuous in our piecewise constructed solution,there will be a non-vanishing Sµν only if A′ is discontinuous across either hypersu-face corresponding to a zero of B′r+4B. Our analysis shows that A′ is continuous,thus implying that Sµν = 0.It should be noted that there could be contributions to Sµν from the Weyl term38Figure 2.8: This direction field plot of A(r) against radius r shows that A(r) is numericallyunstable at the zeros of B′r + 4B. Here we choose l = 1 and m = 1.189 such that thezeros are found at r = 1.281, 1.392.(1.26). If that were the case, Sµν = 0 in our piecewise constructed solution wouldimply that there is a localized gravitational source (such as matter) at the hyper-surface and the stresses contributed by each cancel one another exactly. As thespace-time off the brane is not known, it is not possible to check to see if suchcontributions exist. If there are no contributions to Sµν from the Weyl tensor, thepiecewise constructed solutions correspond to gravitational geons.2.4.4 No Critical PointsFor these values of ml , all solutions with A(0) = 1 correspond to gravitationalgeons. Choosing l = 1 we confirmed the behaviour of A using iterative numericalanalysis (see fig.2.13). Also evident is the progression from the weak field solutionfor ml << 1 (fig.2.3) to the piecewise constructed solution for 3227 < ml < 1.191(fig.2.12).39Figure 2.9: Forward iteration of A(r) plotted from r = 0 to the first zero of B′r + 4B forl = 1 and m = 1.189. A → −α1β1≈ 1.38 as the solution approaches the first zero.Figure 2.10: Reverse iteration of A(r) plotted from the second zero of B′r +4B to the firstfor l = 1 and m = 1.189.40Figure 2.11: Forward iteration of A(r) plotted from the second zero of B′r + 4B for l = 1and m = 1.189.Figure 2.12: The function A(r) plotted against radius r for l = 1 and m = 1.189. A fourth-order Runge-Kutta numerical iteration was unable to navigate the zeros of B′r + 4B(at r = 1.281, 1.392), so the result is constructed piecewise between these points.41Figure 2.13: Function A(r) plotted against radius r. Here we choose l = 1 and chart avariety of values for m in the specified region. A fourth-order Runge-Kutta numericaliteration is used to generate each result.42Chapter 3Discussion And ConclusionIn this paper, we examined solutions to R = 0 in 3+1 dimensional general rel-ativity that correspond to static, spherically symmetric vacuum solutions on thebrane in 5 dimensional Randall-Sundrum brane-world cosmologies. We investi-gated these solutions for behaviour consistent with that of gravitational geons. Inthe absence of a general solution, we investigated the behaviour of the space-timein the weak field approximation and used a variety of numerical and analyticalmethods to discover the behaviour for strong field solutions. We were able to as-certain that, for a particular range of parameter choices, gravitational geons arepossible.3.1 The Weak FieldIn section (2.3), we found that, for metrics of the form B(r) = 1 + b(r) andA(r) = 1 + a(r) in (2.1) with weak field approximations with ∣b(r)∣ << 1 and∣a(r)∣<< 1 for all r ≥ 0, gravitational geons will exist as long as ∣B′r∣<< 1 for allr ≥ 0.The main result of this thesis, that gravitational geons are possible in Randall-Sundrum brane-worlds, is achieved in the weak field, independent of the choiceof B(r). However, in order to quantitatively describe the potentially measurablepresence of a gravitational geon, a specific choice of B(r) must be made.We chose a specific B(r) and examined the corresponding space-time solution43numerically for both weak and strong fields. Our numerical analysis was largelydone using a custom developed software application, the implementation of whichwas validated in part by the very close agreement between numerical results andanalytical results for weak fields (see fig. 2.3).Further analysis of the weak field solution could potentially show that somechoices of B(r) are viable whereas others are not. This analysis would involveevolving the solution into the bulk (to be discussed in section (3.3)), and havingdone that, determining the stability of the solution with respect to perturbations.Without making weak field approximations, this work must be done numericallyfor each specifically chosen B(r). For the analytical weak field solution, it is pos-sible that the evolution into the bulk could be performed without resorting to nu-merical methods, but the degree of inaccuracy that would be introduced due toapproximation is unknown.3.2 Identifying m and lThis thesis shows that in general for brane-world space-times, static, spheri-cally symmetric and topologically trivial gravitational geons are possible. How-ever, for the specific B(r) that we chose for our strong field analysis (2.3), gravita-tional geons are only viable for particular choices of the parameters m and l. Thechoice of B(r) is by no means unique, but it is worth looking at what these parame-ters represent physically as an example of how a gravitational geon solution wouldmanifest itself as a physical entity.B(r) is Schwarzschild in the limit as r → ∞ (i.e. limr→∞B(r) = 1− 2mr ). In thelimit as r → ∞ the gravitational field is weak and static. If we make the furtherassumption that any test particles are travelling at sub-relativistic speeds, we findthat gtt = −(1+2Φ) where Φ corresponds to the Newtonian gravitational poten-tial. Thus knowing B(r) is sufficient for us to identify the parameter m with thegravitational mass of the space-time solution.To get a sense of the physical significance of the parameter l it is useful toconsider the metric of de Sitter space in static coordinates,44ds2 =−uni0028.alt031− Λ3 r2uni0029.alt03dt2 +uni0028.alt031− Λ3 r2uni0029.alt03−1dr2 +r2(dθ2 +sin2 θdφ2), (3.1)where Λ is a cosmological constant. For our choice of B(r) we have that B ≈1−uni0028.alt01rluni0029.alt012 for r <<uni0028.alt01ml2uni0029.alt0113 . Thus, l has the effect of controlling the strength of apositive cosmological constant like term (with Λ = 3l2 ) that appears in the limit asr → 0.Figure 3.1: B(r) plotted for m = 1 and 1 ≤ l ≤ 33.3. Note that all solutions have ml < 3227 sothese solutions all correspond to gravitational geons.To see how l affects the space-time away from the radial center, see figure 3.1in which B is shown for m = 1 and varying l. For a particular choice of m there willbe solutions (via appropriate choice of l) that correspond to gravitational geons thatare radially concentrated (as ml → 3227) or as spread out as desired (as ml → 0).If we choose B′(r) = 0 as a radial marker with which to assign a radial scalefor our gravitational geon solutions, then observing thatB′(r) = 2mr4 −8m2l2r(r3 +2ml2)2 (3.2)we have B′(r) = 0 at45r = (4ml2)13 . (3.3)We have shown that there are no solutions for the candidate space-times corre-sponding to gravitational geons for ml > 1.191. Therefore, the lower limit on whereB′(r) = 0 will occur isr = 1.78Gmc2 , (3.4)which means that in the lower limit, the radial scale of the candidate space-timesolutions is of the same order as that of the Schwarzschild solution.3.3 Suggestions for Further AnalysisThe conclusion that static, spherically symmetric and topologically trivial grav-itational geons are possible in brane-world cosmologies has been achieved in theideal circumstance that the 3+1 dimensional space-time on the brane is an un-perturbed vacuum. We have shown that such solutions (unlike the case for 3+1dimensional general relativity) are possible. However, further analysis is necessaryto determine if the candidate space-times are likely to actually occur should ouruniverse have a Randall-Sundrum brane-world type cosmological structure.In order to test the dynamical behaviour of our candidate space-times, a logicalnext analytical step would be to test for stability against small perturbations.In this thesis, we have looked in detail at the 3+1 dimensional space-time onthe brane. We are confident that solving R = 0 for our candidate space-times willgive rise to a vacuum brane-world solution in the 4+1 dimensional bulk [21], butthe effective field equations on the brane are not a closed system. In order to testthe stability of the candidate space-times, our solution must be evolved into thebulk.The projection of the bulk Weyl tensor, which factors into the induced 3+1dimensional field equations on the brane (1.34), is governed by a coupled systemof 4+1 dimensional equations obtained from the bulk Bianchi identities [15] (roundbrackets around indices denote symmetrization whereas square brackets denoteanti-symmetrization)46LnEAB = ∇CBC(AB) + 16κ25 Λ(KAB −gABK)+KCD(4)RCADB+ 3KC(AEB)C −KEAB +(KACKDB −KABKCD)KCD,LnBABC = −2∇[AEB]C +KCDBABD−2BCD[AKB]D,Ln(4)RABCD = −2RABE[CKD]E −∇ABCDB +∇ABDCB, (3.5)in which EAB is the electric part of the bulk Weyl tensor (1.26) whose projectioncomes into the induced Einstein field equations on the brane 1.32, KAB is the ex-trinsic curvature (1.14) (K = KAA) andBµνσ = (5)CABCDnDgµ Agν Bgσ C (3.6)is the magnetic part of the bulk Weyl tensor. In (3.6), Bµνσ is expressed as a 3+1dimensional tensor projected onto the brane from the 4+1 dimensional Weyl tensor,whereas in (3.5) BABC is expressed as a 4+1 dimensional tensor. When evolvinga solution from the brane into the bulk, the 3+1 dimensional tensors B and E areequated with their 4+1 dimensional counterparts by evaluating the bulk tensors onhypersurfaces in the limit approaching the brane. We therefore define boundaryconditions for B and E at the brane in order to solve (3.5) as∇AEAB = κ45 ∇ASAB,BABC = 2∇[AKB]C = κ25 ∇[Auni0028.alt03TB]C − 13gB]CTuni0029.alt03. (3.7)We have R = 0, so for our candidate space-times, the induced field equationson the brane (1.34) becomeEµν =−Rµν. (3.8)For a space-time metric of the form (2.1), the components of the Ricci tensorare [5]47Rtt = −B′′2A +B′4Auni0028.alt03A′A +B′Buni0029.alt03− B′rARrr = B′′2B −B′4Buni0028.alt03A′A +B′Buni0029.alt03− A′rARθ θ = −1+ r2Auni0028.alt03B′B −A′Auni0029.alt03− 1ARφφ = Rθ θ sin2θ, (3.9)which for the weak field limit of our candidate space-times becomeRtt = −24m2l2(−r3 +ml2)(r3 +2ml2)3Rrr = 3m(r6 −18r3ml2 +8m2l4)(r3 +2ml2)3Rθ θ = 3mr2(−r6 −6r3ml2 +16m2l4)2(r3 +2ml2)3Rφφ = Rθ θ sin2θ. (3.10)It is worth noting that, for strong fields, we would obtain these components nu-merically. In either case, the components of the Ricci tensor (and therefore E) arenon-zero in general. When a perturbation is introduced, discovering the dynamicalbehaviour of E (and hence the space-time both on and off the brane) depends onsolving the system of equations (3.5).The intention of performing this analysis would be to determine if there areany specific choices of the parameters m and l for which the candidate space-timesare dynamically stable. Of course, in the event that none of the gravitational geonsolutions that we have investigated here are stable, it is understood that the form ofB(r) that we have investigated is by no means a unique choice.Although there has been much research published regarding brane-world cos-mologies in recent years, there are other modifications to the 3+1 dimensional Ein-stein field equations that may more accurately represent the universe in which welive. One such class of theories are known as f(R) theories (see [43] for a review).These theories are obtained when one replaces the Ricci scalar R in the Lagrangian48formulation of general relativity with a function of the Ricci scalar f(R). TheEinstein-Hilbert actionSEH = 116piGuni222B.alt02d4x√−gR (3.11)becomesS = 116piGuni222B.alt02d4x√−g f(R). (3.12)Depending on the definition of f(R), the modified Einstein field equations mayor may not admit space-time solutions corresponding to gravitational geons similarto those we have investigated here.3.4 Significance of the ResultThere is currently a great deal of interest in the Randall-Sundrum brane-world,which models the 3 spatial dimensions of our universe as a brane embedded ina 4+1 dimensional space-time. In 3+1 dimensional general relativity, there areno static, spherically symmetric, topologically trivial vacuum space-time solutionsother than flat, empty space. In this thesis, we have shown that the same is nottrue for Randall-Sundrum brane world models. On the brane, there do exist static,spherically symmetric, topologically trivial vacuum space-time solutions that havemass. The source of mass for these gravitational geons derives from the bulk Weyltensor via its contribution to the effective Einstein field equations on the brane.Further analysis is required to show whether the candidate space-times pre-sented here (or other candidate space-times) are likely to occur and persist in abrane-world teeming with matter and energy. Insofar as gravitational geons onlyinteract gravitationally, they would only be observable indirectly. The possibilitythat gravitational geons may represent a form of dark matter has been raised before(see for example [39, 44]).The significance of the result of this thesis is therefore that, should our uni-verse be a Randall-Sundrum brane-world, gravitational geons represent a viableexplanation for dark matter observations.This result is also significant insofar as it adds to a body of work that shows49how space-time solutions differ between standard general relativity and Randall-Sundrum brane-worlds. It remains to be seen whether this body of work will ul-timately provide evidence for or against the Randall-Sundrum brane-world modelas an effective model of our universe.50Bibliography[1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221]. → pages 1, 7[2] J. A. Wheeler, Phys. Rev. 97, 511 (1955). → pages 1, 12[3] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, (Freeman, S. Fran-cisco, 1973), pp. 551–555 → pages 2, 3, 5[4] S. M. 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