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Thesis: Author (Year of Submission) "Full thesis title", University of Southampton, name of the
University Faculty or School or Department, PhD Thesis, pagination.
Data: Author (Year) Title. URI [dataset]
University of Southampton
Faculty of Social, Human and Mathematical
Sciences
AN INQUIRY INTO THE NATURE OF
GRAVITATIONAL SINGULARITIES
By: Yafet Erasmo Sanchez Sanchez
Thesis for the degree of Doctor of Philosophy
August 2016

University of Southampton
ABSTRACT
Faculty of Social, Human and Mathematical Sciences
Doctor of Philosophy
AN INQUIRY INTO THE NATURE OF GRAVITATIONAL SINGULARITIES
by Yafet Erasmo Sanchez Sanchez
We provide different mathematical frameworks to describe singularities in General
Relativity. The main idea is to regard singularities as obstructions to the dynamics of
different matter models.
The first part of the thesis initiates our examination of spacetime by probing space-
time with matter that can be modelled as a point-particle. In particular, we dis-
cuss the case of two-dimensional Lorentzian metrics. We give concrete applications
of the framework in the case of the Minkowski spacetime (which is regular) and the
Friedmann-Robertson-Walker spacetime (which is geodesically incomplete).
In the second part of the thesis, we probe the geometry of spacetime with classical
scalar fields. The general motivation is to redefine a singularity in spacetime not
as an obstruction to geodesics or curves but as an obstruction to the dynamics of
test fields. We discuss curve-integrable spacetimes, spacetimes with surface layers,
impulsive gravitational waves and brane-world scenarios, and spacetimes that contain
string-like singularities.
In the third part of the thesis, we present the outline of a framework to analyse the
geometry of the spacetime by probing it with quantum scalar fields. The main focus of
this chapter is to describe what is meant by a quantisation in a spacetime with finite
differentiability.
The fourth part of the thesis presents future outlooks and some open problems.

Contents
Abstract iii
Declaration of Authorship ix
1 Introduction 1
I The point-particle probe 5
2 Introduction 7
2.1 Singularity Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Proof of the Singularity theorems. . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Outline of the proof. . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 The b-boundary 11
3.1 The boundary of spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 The Schmidt metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 The classification of singularities . . . . . . . . . . . . . . . . . . . . . . 13
3.4 The b-boundary for 1+1 spacetimes . . . . . . . . . . . . . . . . . . . . 14
3.4.1 The Schmidt metric for 1+1 conformal spacetimes . . . . . . . . 14
3.4.2 The Schmidt metric of Minkowski spacetime . . . . . . . . . . . 16
3.4.3 The Schmidt metric of a FRW spacetime. . . . . . . . . . . . . . 18
3.5 The topology of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II The classical test-field probe 21
4 Introduction 23
4.1 Test-field Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Well-posedness and energy estimates . . . . . . . . . . . . . . . . . . . . 25
5 Curve Integrable Spacetimes 31
5.1 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1.1 The general setting . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1.2 Weak solutions and the main theorem . . . . . . . . . . . . . . . 33
5.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2.1 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2.2 Existenceofatimelikevectorfieldwithboundedcovariantderiva-
tive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
v
vi CONTENTS
5.2.3 Energy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.4 Self-adjointness and existence . . . . . . . . . . . . . . . . . . . . 46
5.2.5 Uniqueness and continuity with respect to the source function. . 48
5.2.6 H1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 C0,1 Spacetimes 55
6.1 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.1 The general setting . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.2 Weak solutions and the main theorems . . . . . . . . . . . . . . . 57
6.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2.1 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2.2 Approximate solutions and energy estimate . . . . . . . . . . . . 59
6.2.3 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2.4 Existence, Uniqueness and Stability . . . . . . . . . . . . . . . . 62
6.2.5 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 Spacetimes with string-like singularities 69
7.1 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1.1 The general setting . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1.2 Weak solution and the main theorem . . . . . . . . . . . . . . . . 70
7.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2.1 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2.3 Uniqueness and stability with respect to the initial data . . . . . 77
7.2.4 Integrability of the energy momentum tensor . . . . . . . . . . . 78
7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
III The quantum test-field probe. 83
8 Introduction 85
8.1 Quantum fields on spacetimes with rough metrics . . . . . . . . . . . . . 85
9 Quantisation 87
9.1 Rough metrics and weak solutions . . . . . . . . . . . . . . . . . . . . . 87
9.1.1 The general setting . . . . . . . . . . . . . . . . . . . . . . . . . 87
9.1.2 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
9.2 Classical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.2.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
9.2.2 Space of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.3 Quantum observables and states . . . . . . . . . . . . . . . . . . . . . . 98
9.4 Unitary equivalence and quantum evolution . . . . . . . . . . . . . . . . 102
9.4.1 Unitary equivalence . . . . . . . . . . . . . . . . . . . . . . . . . 102
9.4.2 Quantum evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.5.1 Static Ck(k (cid:62) 2) spacetimes . . . . . . . . . . . . . . . . . . . . . 104
CONTENTS vii
IV Future outlook and open problems 109
V Appendix 117
10 Differential Geometry 119
10.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.2 Vectors, One forms, and Tensors . . . . . . . . . . . . . . . . . . . . . . 120
10.3 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.4 Hypersurfaces and volume forms . . . . . . . . . . . . . . . . . . . . . . 122
10.5 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.6 Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.7 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.8 G-Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
10.9 Canonical one forms and Connections . . . . . . . . . . . . . . . . . . . 125
10.10Conjugate points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
11 Causality theory 129
12 Banach and Hilbert spaces 133
13 Measure Theory 135
13.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
13.2 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
13.3 Banach space-valued functions . . . . . . . . . . . . . . . . . . . . . . . 136
14 Functional Analysis 139
14.1 Regularity theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
14.2 Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
15 Elliptic Theory 143
16 Parabolic Theory 145
17 C∗-algebra 151
18 Gelfand-Naimark-Segal (GNS) representation theorem 155
Bibliography 160

Declaration of Authorship
I, Yafet Erasmo Sanchez Sanchez, declare that the thesis entitled An Inquiry into the
Nature of Gravitational Singularities” and the work presented in the thesis are both
my own, and have been generated by me as the result of my own original research. I
confirm that:
• this work was done wholly or mainly while in candidature for a research degree
at this University;
• where any part of this thesis has previously been submitted for a degree or any
otherqualificationatthisUniversityoranyotherinstitution,thishasbeenclearly
stated;
• where I have consulted the published work of others, this is always clearly at-
tributed;
• where I have quoted from the work of others, the source is always given. With
the exception of such quotations, this thesis is entirely my own work;
• I have acknowledged all main sources of help;
• wherethethesisisbasedonworkdonebymyselfjointlywithothers, Ihavemade
clear exactly what was done by others and what I have contributed myself;
• parts of this work have been published as:
– Y. Sanchez Sanchez:
Regularity in curve integrable spacetimes
General Relativity and Gravitation 47 80 (2015)
ArXiv:1502.06458
– Y. Sanchez Sanchez, J.A Vickers
Generalized Hyperbolicity in Spacetimes with string-like singularities
Classical and Quantum Gravity 33 20 (2016) ArXiv:1602.03584
– Y. Sanchez Sanchez, J.A Vickers
Generalized Hyperbolicity in Spacetimes with Lipschitz regularity
ArXiv:1507.06463
Signed:.......................................................................................................................
Date:..........................................................................................................................