Cosmological Milestones and Gravatars - Topics in General Relativity [thesis]

171 Pages·2006·0.972 MB·English

by  C. Cattoen

Cosmological milestones and
Gravastars — topics in General
Relativity
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C´eline CATTOEN
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0 Supervisor: Dr. Matt Visser
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A thesis submitted to the Victoria University of Wellington in fulfilment of the
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Victoria University of Wellington
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New Zealand

Abstract
In this thesis, we consider two different problems relevant to general relativity. Over
the last few years, opinions on physically relevant singularities occurring in FRW
cosmologies have considerably changed. We present an extensive catalogue of such
cosmological milestones using generalized power series both at the kinematical and
dynamical level. We define the notion of “scale factor singularity” and explore its re-
lationtopolynomialanddifferentialcurvaturesingularities. Wealsoextractdynamical
information using the Friedmann equations and derive necessary and sufficient condi-
tions for the existence of cosmological milestones such as big bangs, big crunches, big
rips, sudden singularities and extremality events. Specifically, we provide a complete
characterization of cosmological milestones for which the dominant energy condition
is satisfied. The second problem looks at one of the very small number of serious
alternatives to the usual concept of an astrophysical black hole, that is, the gravastar
model developed by Mazur and Mottola. By considering a generalized class of similar
models with continuous pressure (no infinitesimally thin shells) and negative central
pressure, we demonstrate that gravastars cannot be perfect fluid spheres: anisotropc
pressures are unavoidable. We provide bounds on the necessary anisotropic pressure
and show that these transverse stresses that support a gravastar permit a higher com-
pactness than is given by the Buchdahl–Bondi bound for perfect fluid stars. We also
comment on the qualitative features of the equation of state that such gravastar-like
objects without any horizon must have.
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Acknowledgment
I would like to thank my supervisor, Dr. Matt Visser for his assistance, guidance and
support. I would also like to give some special thanks to Petarpa Boonserm, Tristan
Faber and Silke Weinfurtner for numerous interesting discussions relating to topics
discussed in this thesis.
I wish to acknowledge the School of Mathematics, Statistics, and Computer Sci-
ence for providing me with office space and all the facilities.
I am also very grateful to the Marsden fund that has indirectly supported me fi-
nancially during my work at Victoria University of Wellington.
Finally, I would like to thank my family and Ian for being there for me.
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Contents
Abstract i
Acknowledgment iii
1 Introduction 1
2 Some essential notions in General Relativity 5
2.1 Notions of metric, geodesics, affine connexion, Killing vector field . . . 6
2.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Ricci tensor, Ricci scalar, Weyl tensor . . . . . . . . . . . . . . . 9
2.2.3 Einstein tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.4 Energy-momentum tensor . . . . . . . . . . . . . . . . . . . . . 10
2.3 Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Weak field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Strong field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.1 Black holes, event horizon . . . . . . . . . . . . . . . . . . . . . 13
2.5.2 Schwarzschild’s solution to the Einstein equations . . . . . . . . 13
2.5.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Energy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.1 Null Energy condition (NEC) . . . . . . . . . . . . . . . . . . . 18
2.6.2 Weak Energy condition (WEC) . . . . . . . . . . . . . . . . . . 18
2.6.3 Strong Energy Condition (SEC) . . . . . . . . . . . . . . . . . . 19
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2.6.4 Dominant Energy Condition (DEC) . . . . . . . . . . . . . . . . 19
2.6.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7.2 “Cosmography” . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7.3 “Cosmodynamics” . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7.4 Cosmological parameters . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Necessary and sufficient conditions for big bangs, bounces, rips... 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Cosmological milestones and kinematics. . . . . . . . . . . . . . . . . . 33
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.2 Spacetime curvature . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Cosmological milestones and dynamics. . . . . . . . . . . . . . . . . . . 56
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.2 NEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.3 WEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.4 SEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.5 DEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Total age of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Age of the universe τ . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.2 Remaining lifetime of the universe T . . . . . . . . . . . . . . . 80
3.4.3 Total age of the universe T . . . . . . . . . . . . . . . . . . . 83
total
3.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 Gravastars 91
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.1 GRAvitational VAcuum STARS . . . . . . . . . . . . . . . . . . 91
4.1.2 Common models . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.3 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
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4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.1 A static spherically symmetric geometry . . . . . . . . . . . . . 93
4.2.2 A TOV equation . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.3 Specific key features . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 The problem of the negative-pressure perfect fluid sphere gravastar. . . 104
4.3.1 Four inconsistencies . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3.2 Case study: Schwarzschild interior and Buchdahl-Bondi bound . 107
4.3.3 Case study: the fate of a negative pressure perfect fluid sphere . 116
4.4 Gravastars with anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4.1 Bounds on the anisotropy pressure . . . . . . . . . . . . . . . . 121
4.4.2 Minimizing the anisotropy . . . . . . . . . . . . . . . . . . . . . 124
4.4.3 Features of the anisotropic equation of state . . . . . . . . . . . 127
4.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Conclusion 135
A Main spacetime metrics considered 139
A.1 Static spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2 Schwarzschild exterior solution . . . . . . . . . . . . . . . . . . . . . . . 139
A.3 Friedmann-Robertson-Walker geometry . . . . . . . . . . . . . . . . . . 140
A.4 Static spherically symmetric spacetime . . . . . . . . . . . . . . . . . . 141
A.5 Interior of a gravastar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.6 General Schwarzschild interior solution . . . . . . . . . . . . . . . . . . 141
A.7 Schwarzschild interior solution as a function of the mass and radius . . 142
B Necessary and sufficient conditions for big bangs, bounces, rips... 143
C Gravastars must have anisotropic pressures 145
D Effective refractive index tensor for weak-field gravity 147
Bibliography 149
Index 157
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