Imperial College London

Numerical General Relativity in

Exotic Settings

Alexander Adam

December 19, 2013

Submitted in part fulfillment of the requirements for the degree of

Doctor of Philosophy in Theoretical Physics of Imperial College London

and the Diploma of Imperial College London

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Abstract

In this thesis, we discuss applications of numerical relativity in a variety of complex

settings. After introducing aspects of black hole physics, extra dimensions, holog-

raphy, and Einstein-Aether theory we discuss how one can frame the problem of

solving the static Einstein equations as an elliptic boundary value problem by inclu-

sion of a DeTurck gauge fixing term. Having setup this background, we turn to our

simplest application of numerical relativity, namely fractionalisation in holographic

condensed matter. We explain how one may describe this phenomenon by studying

particular classes of hairy black holes and analysing whether bulk flux is sourced by

a horizon or charged matter. This problem is our simplest application of numerical

relativity as the Einstein equations reduce to ODEs and the problem may be solved

by shooting methods. We next turn to a discussion of stationary numerical relativ-

ity and explain how one can also view the problem of finding stationary black hole

solutions as an elliptic problem, generalising the static results discussed earlier. Er-

goregions and horizons are naively a threat to ellipticity, but by considering a class

of spacetimes describing a fibration of the stationary and axial Killing directions

over a Riemannian base space manifold, we show how the problem can neverthe-

less still be phrased in this manner. Finally we close with a discussion of black

holes in Einstein-Aether theory. These unusual objects have multiple horizons as a

consequence of broken Lorentz symmetry, and in order to construct such solutions

we explain how to generalise the PDE methods of previous sections to construct

solutions interior to a metric horizon where the Harmonic Einstein equations cease

to be elliptic. Using this new machinery we rediscover the spherically symmetric

static black holes that have been found in the literature and moreover present the

first known rotating solutions of the theory.

2

In loving memory of my grandmother, Ida Ciancabilla

Benocci.

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Preface

Declaration of Originality

I declare that this thesis has been written by myself and constitutes a survey of

my own research, except in cases where references are explicitly made to the work

of others or to work that was done as part of a collaboration. In detail, the work

discussed in chapters 2 and 3 of this thesis is based on material taken from the

following publications:

A. Adam, S. Kitchen, and T. Wiseman, A numerical approach to finding

general stationary vacuum black holes, Class.Quant.Grav. 29 (2012) 165002,

[arXiv:1105.6347]

A. Adam, B. Crampton, J. Sonner, and B. Withers, Bosonic Fractionalisation

Transitions, JHEP 1301 (2013) 127, [arXiv:1208.3199]

All calculations presented in chapter 3 are taken from the former and were per-

formed by the present author. Chapter 2 is based on the second reference above

and all analytic calculations, as well as the T = 0 numerics were carried out by the

present author in collaboration with Crampton. Chapter 4 is based on work done

in collaboration with Wiseman and Pau Figueras that is soon to be submitted for

publication. The static calculations in this chapter (section 4.4) were also done in

collaboration with Yosuke Misonoh.

All research discussed in this thesis was carried out during the time I was registered

at Imperial College London, between October 2009 and October 2013. No part of

this work has been submitted for any other degree at this or another institution.

Acknowledgments

I wold like to thank my supervisor, Dr Toby Wiseman for his advice during my

time at Imperial College London and in particular for encouraging me to develop

my numerical skills despite much initial opposition! I would also like to thank

Professor Fay Dowker for many years of guidance and support throughout both

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http://xxx.lanl.gov/abs/1105.6347http://xxx.lanl.gov/abs/1208.3199

my undergraduate and postgraduate years at Imperial. Thanks are also due to

Benedict Crampton, Julian Sonner, Benjamin Withers, Sam Kitchen, Pau Figueras

and Yosuke Misonoh for enjoyable collaborations.

More thanks than I can say here are owed to my colleagues in H606 for many

enlightening and entertaining discussions over the years and for ensuring that no

day was without excitement!

Id also like to thank particularly my parents and close friends who have provided

me with invaluable moral support whilst Ive been carrying out my research without

which none of this would have been possible. The author is also grateful for the

financial support that has been provided under a doctoral training grant by the

Science and Technologies Facilities Council (STFC).

Copyright Declaration

The copyright of this thesis rests with the author and is made available under a Cre-

ative Commons Attribution Non-Commercial No Derivatives license. Researchers

are free to copy, distribute or transmit the thesis on the condition that they at-

tribute it, that they do not use it for commercial purposes and that they do not

alter, transform or build upon it. For any reuse or redistribution, researchers must

make clear to others the license terms of this work

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Contents

1. Gravitational Theory in Exotic Settings 11

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2. Higher Dimensional Black Holes, AdS/CFT and Holography . . . . . 13

1.2.1. Killing Fields, Static and Stationary Spacetimes . . . . . . . . 15

1.2.2. No Hair Theorems and Black Hole Uniqueness . . . . . . . . . 17

1.2.3. Black Hole Mechanics and Thermodynamics . . . . . . . . . . 20

1.2.4. Examples of Asymptotically Flat Black Holes . . . . . . . . . 26

1.2.5. Holography and the AdS/CFT Correspondence . . . . . . . . 32

1.3. Static Elliptic Numerical Relativity . . . . . . . . . . . . . . . . . . . 37

1.3.1. Ansatz for Static Black Holes . . . . . . . . . . . . . . . . . . 38

1.3.2. Hyperbolicity, Ellipticity and the Harmonic Einstein Equation 39

1.3.3. Ricci Flatness, Solitons, and Maximum Principles . . . . . . . 47

1.3.4. Numerical Implementations . . . . . . . . . . . . . . . . . . . 50

1.4. Einstein-Aether Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.4.1. Action and Field Equations . . . . . . . . . . . . . . . . . . . 55

1.4.2. Wave Modes and Physical Degrees of Freedom . . . . . . . . . 60

1.4.3. Constraints on the Parameter Space . . . . . . . . . . . . . . . 65

2. Bosonic Fractionalisation in AdS/CFT 69

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.2. General Features, Action and Field Content . . . . . . . . . . . . . . 72

2.3. Ultraviolet Expansions and Asymptotic Charges . . . . . . . . . . . . 77

2.4. Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.5. Class I: Bottom Up Model and T = 0 Shooting Problem . . . . . . . 81

2.5.1. T=0 Infrared Expansions . . . . . . . . . . . . . . . . . . . . . 82

2.5.2. Overview of the Numerical Shooting Problem . . . . . . . . . 89

2.5.3. T=0 Fractionalisation Transition . . . . . . . . . . . . . . . . 91

2.5.4. Comments on Finite Temperature . . . . . . . . . . . . . . . . 96

2.6. Class Ib: Bottom - Up Model . . . . . . . . . . . . . . . . . . . . . . 97

6

2.7. Class II: M Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.7.1. Phase Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2.7.2. Neutral Top-Down Solutions . . . . . . . . . . . . . . . . . . . 99

2.8. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3. Stationary Elliptic Numerical Relativity 104

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.2. Static Spacetimes from a Lorentzian Perspective . . . . . . . . . . . . 105

3.3. Stationary Spacetimes with Globally Timelike Killing Vector . . . . . 110

3.4. Stationary Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.4.1. Ellipticity of the Harmonic Einstein Equation . . . . . . . . . 116

3.4.2. Reduced Stationary Case . . . . . . . . . . . . . . . . . . . . . 118

3.4.3. Horizon and Axis Boundaries . . . . . . . . . . . . . . . . . . 119

3.5. Example of Boundary Conditions: Kerr . . . . . . . . . . . . . . . . . 123

3.6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4. Black Holes in Einstein-Aether Theory 128

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.2. Structure and Regularity of Einstein-Aether Black Holes . . . . . . . 131

4.3. Ingoing Stationary Methods . . . . . . . . . . . . . . . . . . . . . . . 133

4.4. Spherically Symmetric Black Hole Solutions . . . . . . . . . . . . . . 135

4.5. Stationary Black Hole Solutions . . . . . . . . . . . . . . . . . . . . . 148

4.6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163