Black Holes, Geons, and Singularities in Metric-Affine Gravity

196 Pages·2017·3.54 MB·English

Black Holes, Geons, and
Singularities in Metric-Affine
Gravity
Phd Thesis by
Antonio S´anchez Puente
Under the supervision of
Gonzalo Olmo Alba
Programa de Doctorado en F´ısica
Diciembre de 2016

A mis padres
y hermanos

i
List of Publications
This PhD thesis is based on the following publications:
• Classical resolution of black hole singularities via wormholes. [1]
Gonzalo J. Olmo, D. Rubiera-Garcia, and A. Sanchez-Puente
Eur. Phys. J., C76(3):143, 2016
• Classical resolution of black hole singularities in arbitrary dimension. [2]
D.Bazeia,L.Losano,GonzaloJ.Olmo,D.Rubiera-Garcia,andA.Sanchez-
Puente
Phys. Rev., D92(4):044018, 2015
• Geodesic completeness in a wormhole spacetime with horizons. [3]
Gonzalo J. Olmo, D. Rubiera-Garcia, and A. Sanchez-Puente
Phys. Rev.,D92(4):044047, 2015
• Impactofcurvaturedivergencesonphysicalobserversinawormholespace–time
with horizons.[4]
Gonzalo J. Olmo, D. Rubiera-Garcia, and A. Sanchez-Puente
Class. Quant. Grav., 33(11):115007, 2016
ii
Contents iii
Contents
Notation vii
Resumen en Espan˜ol ix
1 Introduction: General Relativity and the Schwarzschild Geom-
etry 1
1.1 Einstein Equivalence Principle . . . . . . . . . . . . . . . . . . . 6
1.1.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Description of Physical Observers . . . . . . . . . . . . . . 10
1.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 The Schwarzschild Black Hole . . . . . . . . . . . . . . . . . . . . 12
1.3.1 GeodesicsofaSphericallySymmetricandStaticSpace-time 15
1.3.2 TrajectoryofInfallingRadialLightRaysintheSchwarzschild
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Geodesic Congruences In The Schwarzschild Geometry . . . . . . 20
1.4.1 Evolution of a Geodesic Congruence . . . . . . . . . . . . 20
1.4.2 Congruence Around A Time-like Radial Geodesic For A
Spherically Symmetric And Static Space-time . . . . . . . 23
1.4.3 EvolutionofthecongruencenearthesingularityofaSchwarzschild
black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Charged Black Holes in GR . . . . . . . . . . . . . . . . . . . . . 26
1.5.1 Spherically Symmetric Electrovacuum Field . . . . . . . . 26
1.5.2 The Reissner-Nordstr¨om Metric and its Geometry . . . . 27
2 Introduction: Singularities and Quadratic Gravity 31
2.1 Defining a Singular Space-time . . . . . . . . . . . . . . . . . . . 31
2.2 Extension of Geodesics . . . . . . . . . . . . . . . . . . . . . . . . 34
iv Contents
2.2.1 Conjugated Points . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 Singularity Theorems . . . . . . . . . . . . . . . . . . . . 39
2.3 Extension of Geodesics for Discontinuous Metrics . . . . . . . . . 42
2.3.1 Two Dimensional Study . . . . . . . . . . . . . . . . . . . 44
2.4 Quadratic Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.1 Linearised GR . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.2 Fourth Order Derivatives and Ghosts in Quadratic Gravity 53
3 Metric-Affine Gravity 57
3.1 Connections and Curvature . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . 59
3.1.2 Curvature Tensors . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Metric-Affine Formalism . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 General Lagrangian . . . . . . . . . . . . . . . . . . . . . 68
3.2.2 Role of Torsion in Metric-Affine Formalism . . . . . . . . 73
3.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.1 Analogy with Bravais Crystals . . . . . . . . . . . . . . . 74
4 Geonic Wormhole 77
4.1 General Method for Solving a Space-time with a f(R,Q) Action
and Spherical Symmetry . . . . . . . . . . . . . . . . . . . . . . . 78
4.1.1 Spherically Symmetric Electrovacuum Field . . . . . . . . 80
4.1.2 Charged Black Hole for a Generic L =f(R,Q) . . . . . 81
G
4.1.3 Solutions for Quadratic Gravity . . . . . . . . . . . . . . . 83
4.2 Geometry of Solutions for Quadratic Gravity . . . . . . . . . . . 87
4.2.1 Large r limit . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.2 r →r limit . . . . . . . . . . . . . . . . . . . . . . . . . . 89
c
4.2.3 Coordinate Choices. . . . . . . . . . . . . . . . . . . . . . 91
4.2.4 Beyond r , Geonic Wormhole . . . . . . . . . . . . . . . . 93
c
4.2.5 Horizons and Conformal Diagrams of the Geonic Wormhole 95
4.2.6 Euclidean Embeddings . . . . . . . . . . . . . . . . . . . . 99
5 Geodesics 103
5.1 Geodesics of the Geonic Wormhole . . . . . . . . . . . . . . . . . 103
5.1.1 Radial Null Geodesics . . . . . . . . . . . . . . . . . . . . 106
5.1.2 Null Geodesics with L(cid:54)=0 . . . . . . . . . . . . . . . . . . 107
5.1.3 Radial Time-like Geodesics . . . . . . . . . . . . . . . . . 112
5.1.4 Time-like Geodesics with L(cid:54)=0 . . . . . . . . . . . . . . . 113
5.2 Extension of Geodesics . . . . . . . . . . . . . . . . . . . . . . . . 116
Contents v
5.3 Congruences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Waves 123
6.1 Scalar Waves and Regularity . . . . . . . . . . . . . . . . . . . . 123
6.2 Transmission Through The Wormhole Throat . . . . . . . . . . . 128
7 Wormholes in d-Dimensions 137
7.1 Born-Infeld Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1.1 Born-Infeld Model for Electromagnetism . . . . . . . . . . 137
7.1.2 Born-Infeld inspired Gravity . . . . . . . . . . . . . . . . 138
7.2 Charged Black Holes in an Arbitrary Number of Dimensions. . . 139
7.2.1 Electrovacuum Stress-Energy Tensor in d-Dimensions . . 140
7.2.2 SolutionforSphericallySymmetricandStaticElectrovac-
uum Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 Conclusions 153
Agradecimientos 159
vi