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Thèse de doctorat/ PhD Thesis Citation APA:
Lindman Hornlund, J. (2011). Sigma-models and Lie group symmetries in theories of gravity (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des Sciences – Physique, Bruxelles.
Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/209911/4/0faeae68-51c0-4d17-91b1-e17eeea151f0.txt
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D 03827
Sigma-models and Lie group symmetries in théories of gravity
Thèse présentée par Josef Lindman Hôrnliind
En vue de l’obtention du grade de Docteur en Science
Service de Physique Mathématicjue des Interactions Fondamentales Faculté des Sciences
de l’Université Libre de Bruxelles Bruxelles, Belgique
2011
décoctions.
Secondly to my mother for her neverending patience and support.
Finally, to Laurent Houart, my supervisor who sadly passed away during the
completion of this thesis, and is deeply missed.
11
Acknowledgments
A few months before this thesis was complétée!, my superviser Doctor Laurent Houart sadly passed away. He had been an invaluable guide to me in my research and will be deeply missed. He was always enthusiastic and encouraging and this thesis would hâve been very different without him.
Throughout my work I hâve also had a ‘second’ superviser, Doctor Axel Klein- schmidt, who has inspired and aided me in ail I hâve done. Always having the time to try to answer my never ending flow of questions, his help has been indispensable.
I wish him the best of luck in future journeys into the world of unknown physics and Kac-Moody algebras.
I am also greatly indebted to Professer Marc Henneaux for generously giving me the possibility of completing a doctorate at his department. Although I still hâve a hard time understanding why I of ail people had the fortune, I am infinitely grateful.
During my master thesis at Chalmers University of Technology, I received great encouragement from Professer Bengt E.W. Nilsson, who I thank for contacting ULB and recommending me to Professer Marc Henneaux. This thesis would not hâve existed without Professer Nilsson’s support and I owe this amazing time to him.
I also thank my great-aunt Ulla Hôrnlund for a careful proof-reading of this thesis.
Prom the world outside of physics, I hâve always been supported and encouraged by my dear mother. This thesis is above ail an humble attempt by her son to make her proud. I would hâve been nothing without her.
Finally I thank ail my stringy colleagues and friends here at the ULB and
elsewhere: Riccardo Argurio, Alice Bernamonti, Guillaume Bossard, Cyril Clos-
set, François Dehouck, Jarah Evslin, Federico Galli, Ella Jasmin, Sung-Soo Kim,
Chetan Krishnan, Carlo Maccafferi, Fredrik Ohlsson, Jakob Palmkvist, Daniel Pers-
son, Diego Redigolo, Antonin Rovai, Nassiba Tabti, Bogdan Teaca, Cédric Troes-
saert, Amitabh Virmani and many many others.
Abstract
By using non-linear sigma-models of maps into symmetric spaces GjH we discuss black hole and black brane solutions in various supersymmetric théories of gravity.
The Kac-Moody/supergravity dictionary and dimensional réduction allow us to de-
scribe cohomogeneity-one black holes as geodesics on G/H. Extremal black holes
or black branes are null geodesics and corresponds to a nilpotent element of the
Lie algebra g of G. Using the mathematical theory of nilpotent orbits we classify
ail extremal black holes in Af = 2 pure four-dimensional supergravity, A/ = 2
four-dimensional supergravity and minimal supergravity in five dimensions. Sim-
ilarly, when G is a subgroup of a very- or over-extended Kac-Moody group, we
map the minimal nilpotent orbit, using highest weights of g, to supersymmetric
brane solutions and non-supersymmetric solutions in ten- and eleven-dimensional
supergravity.
Contents
1 Mathematical physics - simplifying the world since 1687 1
1.1 Pourquoi devrais-je m’inquiéter? ... 2
1.2 Going top-down... 3
1.3 String theory and Kac-Moody symmetry... 4
1.4 Lie groups... 6
1.4.1 Symmetries and difFerential équations ... 7
1.5 Layout of the thesis ... 10
1 Background 13 2 Black holes - Singularly unexplained 15 2.1 Structure of black holes space-times... 17
2.2 Particularities... 21
2.3 The positive mass theorem... 26
2.3.1 Généralisations with matter... 27
2.4 Extremality... 27
2.5 Taub-NUT space-times... 29
2.6 Branes in higher dimensions... 31
2.7 Attractive black holes ... 32
3 Symmetric spaces - Group actions in transit 35 3.1 Symmetric homogeneous spaces... 36
3.2 Examples... 42
3.2.1 Matrix-representation of the involution... 42
3.2.2 Example 1: s((3,M)... 43
3.2.3 Example 2; su(2,1)... 44
3.3 Berger’s classification... 44
3.4 Global geometry of symmetric spaces... 45
3.5 Global décompositions... 46
4 Hidden symmetries for the scholar of scalars 49 4.1 Supergravity and non-linear sigma-models . ... 51
4.2 Compact or non-compact ... 54
4.3 Cohomogeneity-one black holes as geodesics... 55
4.4 Iwasawa co-ordinates... 58
4.4.1 Geodesics on SL(2, M)/SO(l, 1)... 61
4.4.2 Time-like geodesics on SU(2,1)/(SL(2, M) X U(l)) ... 63
4.4.3 Null geodesics on SL(3,M)/SO(2,1)... 63
4.5 Example: Pure gravity... 66
4.5.1 Construction of the sigma-model... 66
4.5.2 Some extremal solutions... 69
5 Uniqueness theorems and the characteristic équation 71 5.1 The BMC theorem... 72
5.1.1 A suitable Harrison-transfonnation does not always exist . . . 73
5.1.2 Remédiés and an assumption... 74
5.2 The characteristic équation ... 76
5.3 Nilpotency of the géodésie tangent vector... 76
5.4 Classifications using supersymmetry... 77
6 Nilpotent orbits - extremality defined 79 6.1 Complex nilpotent orbits... 79
6.2 Sekiguchi’s bijections and dual and associated symmetric spaces ... 81
6.3 Masse diagrams and the space of solutions... 82
6.4 if-orbits... 83
6.5 Example 1; SL(3,M)/SO(2,l)... 86
6.6 Example 2: G2(2)/SOo(2,2)... 87
6.6.1 Generalities on G2(2) ... 6.6.2 G2-orbits... 89
6.6.3 G2(2)-orbits... 90
6.6.4 SOo(2,2)-orbits... 92
II Results 95 7 Einstein-Maxwell and electromagnetic and gravitational duality 97 7.1 Pure A7 = 2 supergravity in four dimensions... 98
7.2 A généralisation of the Reissner-Nordstrom black hole... 100
7.3 Action of SU(2,1) on BPS solutions ... 102
7.3.1 Action of the nilpotent generators ... 102
7.3.2 Action of the non-compact Cartan generator...103
7.3.3 Action of ... 103
7.3.4 Describing H as a subgroup of SO(2,2) ... 104
7.3.5 The space of BPS solutions...105
8 Extremal black holes in the supergravity theory 107 8.1 Single modulus S^ AT = 2, Z) = 4 supergravity... 109
8.1.1 The theory... 109
8.1.2 Réduction on time and the sigma-model... 111
8.1.3 Five-dimensional lift and h3q)er-Kâhler base-space ... 113
8.1.4 Asymptotic frame ...115
8.1.5 Nilpotent orbits...116
8.1.6 Generating orbits in practice... 116
8.2 Supersymmetric orbits... 117
Contents vu
8.2.1 TheOiorbit... ... 117
8.2.2 The O2 orbit... 120
8.2.3 The O3H orbit... 122
8.2.4 Truncation to the Einstein-Maxwell theory ... 128
8.3 Non-supersymmetric orbit... 129
8.3.1 The orbit... 129
8.4 Nilpotency and the Gibbons-Hawking form ...133
9 Extremal five-dimensional black holes in minimal supergravity 135
9.1 The Cvetic-Youm family and its extremal limits...136
9.1.1 Preliminaries... 136
9.1.2 Cvetic-Youm, BMPV, and extremal non-supersymmetric black holes ... 137
9.2 Describing five-dimensional black holes as geodesics...138
9.3 Nilpotent orbits and phases of extremal black holes...141
9.3.1 Supersymmetric branch... 142
9.3.2 Non-supersymmetric branch... 143
9.4 Discussion...143
10 Membrane bound States from geodesics 145
10.1 Branes in the E\\ sigma-model... 147
10.1.1 The supergravity/Kac-Moody dictionary...150
10.1.2 Embedding of subalgebras... 153
10.1.3 The SL(2, M)/SO(l, 1) (T-model and half-BPS branes... 154
10.2 Bound States of two branes ... 158
10.2.1 An SL(3, M)/SO(2,1) sigma-model... 158
10.2.2 The dyonic membrane... 160
10.2.3 The M2 with magnetic Kaluza-Klein charge...163
10.2.4 The M5 with magnetic Kaluza-Klein charge...166
10.2.5 A (D6,D8) bound state ... 166
10.2.6 The SL(3, M) sigma-model solution as an orbit space... 167
10.2.7 Intersection rules... 169
10.3 Bound States of three or more branes... 170
10.3.1 An SL(4, E)/SO(3,1) sigma-model...170
10.3.2 An SL(4, E)/SO(2,2) sigma-model...176
10.3.3 Intersection rules... 178
11 An example of a géodésie solution on a non-symmetric space 181
11.1 A null root solution... 181
11.2 Solving the sigma-model équations of motion... 183
11.3 Checking supersymmetry ... 186
12 Future work and outlook 187
A Some solutions and supergravity Lagrangians 189
A.l Eleven-dimensional supergravity ...189
A. 2 Cvetic-Youm metric ... 189
B Lie algebra details 191
B. l Canonical décomposition of sl(3, R)...191
B.2 Décomposition of the sl(4,R) algebra - 5o(3,1) case... 192
B.3 Décomposition of the sl(4,M) algebra - so(2,2) case... 193
B.4 Five-dimensional canonical décomposition of 92(2) 195 B.5 Canonical décompositions of su(2,1)...196
B.5.1 su(2,1): définitions...196
B.5.2 The Cartan décomposition...198
B.5.3 The restricted root System of su(2,1)...199
B.5.4 The canonical décomposition...200
B. 6 The generators of so(2,2) ...201
C Details on üf-orbits in 92(2) 203 C. l Oi... 203
C.2 O2... 203
C.3 O3/4... 204
Bibliography 205
List of Figures
2.1 A rough/conceptual sketch of a black hole space-time... 20
4.1 A collection of geodesics on SL(2, E)/SO(l, 1) projected onto two- dimensional Minkowski space... 62
4.2 A collection of null geodesics on SL(3,R)/SO(2,1) projected onto three-dimensional Minkowski spaces... 65
6.1 The roots of 02 given as vectors in the two-dimensional root space. . 89
6.2 Hasse diagram for the partial ordering of the nilpotent orbits in 02(2)- 93 9.1 Two type of geodesics on the coset manifold, corresponding to fiat black holes in five and four dimensions... 139
9.2 A phase diagram of the extremal limits of the cohomogeneity-one Cvetic-Youm black hole... 144
10.1 The Dynkin diagram of Cl 1...147
10.2 Dynkin diagram for A3, with the black node indicating time...171
10.3 Different limits of the (M2,M5,KK6) bound state... 175
10.4 Dynkin diagram for A3, with the black node indicating time...176
10.5 The four different single brane limits of the M2cM5^ C KK6 bound State... 177
B.l Diagram of m' as a so(3,1) représentation...193
B.2 Diagram over m as a so(2,2) représentation...194
B.3 The Tits-Satake diagram of su(2,1). . ...197
B.4 The restricted root System of su(2,1)... 200
6.1 The five non-zero G2(2)-orbits... 92
6.2 The four /f-orbits O^
h, ^3//) CI4//, O'^fj within the two G2(2)-orbits C>3 and O4... 94
10.1 Level décomposition of en under sl(ll,M) up to level t = 3...148
10.2 The space-time position of a single M2 brane... 155
10.3 The space-time position of a single M5 brane... 156
10.4 The space-time position of a single KK6 brane...157
10.5 The space-time positions of a M2 and a M5 brane...160
10.6 The space-time positions of the two branes in the bound state between a M2 and a KK6...164
10.7 The space-time positions of the two branes in the bound state between an M5 and a KK6...166
10.8 The space-time positions of the two branes in the bound state between a D6 and a D8...167
10.9 The space-time positions of the three intersecting bound-states. . . . 169
lO.lOThe space-time positions of the three branes in the bound state be tween an M2, an M5 and a KK6...172
lO.llThe space-time positions of the three branes in the bound state be tween a IF, a D6 and a D8... 176
10.12The space-time positions of the four branes in the bound state be tween an M2, two M5 and a KK6... ... 177
10.13The space-time positions of the four branes in the bound state be
tween a D4, two D6 and a D8... 178
List of Symbols
R
dThe Ricci scalar in D dimensions.
Rico The Ricci tensor in D dimensions.
Riemo The Riemann curvature tensor in D dimensions voln The volume-form in D dimensions.
TM Tangent bundle for the manifold M TpM Tangent space at point p € M üÇS) Sections for the fibre bundle H
A"(At) Bundle of n-forms over the manifold M G A Lie group
Go The component of G connected to the identity
g A Lie algebra
V The
*D The
V* The
V* The
H The
The
LieG The
Ad The
ad The
K The
The
exPAip The
The End{V) The Ham{V, W) The
At The
U
dThe
A"(M) of a map V : M Af
TAf of a map V \ M —* Af
Mathematical physics - simplifying the world since 1687
The western scientific tradition started, by some accounts, with the lonian philoso- phers in Miletus in Asia Minor, such as Thaïes, Anaximander and Anaximenes.
These wise men from Miletus tried to explain natural phenomena such as lightning and earthquakes from a purely naturalistic perspective, without imposing divine intent or supernatural influences. Although their scientific methodology may hâve been found lacking at a Popperian trial, it contained the basic ingrédients of modem science: a critical discourse and deductive reasoning. What lacked in the work of the Milesian philosophers, compared to modem day physics, was however the use of mathematics as the guiding principle when developing their théories. Although the starting point of the tradition of mathematical physics is inherently arbitrary, a reasonable choice is with Newton’s Philosophiae Naturalis Principia Mathemat- ica published in 1687. When describing gravitational interactions and guided by a mathematical relationship between the strength of the interaction and the distance between the involved objects. Newton postulated a gravitational force, invisible and acting from afar. His contemporaries objected strongly to this mystical force but physical prédictions derived from the assumption were in accordance with obser
vations not explained in theory before. In this sense Newton applied a classical équivalent of the famous gratis dictum ‘Shut up and calculate’, coined by Richard Feynman when questioned about the interprétation of quantum mechanics. We can think of Newtonian gravity as a first prime example of how the équations and the mathematical conséquences of a postulated law were considered to be more im
portant than our human intuition or conception about reality. Although Newton’s force, acting at a distance, was later proved unphysical with the introduction of Ein- stein’s theory of general relativity, which we corne back to soon, the methodology of modem physics has since then been such that mathematics has occupied a spécial and unquestionable position in its centre.
A modem équivalent to Newton’s équations describing gravity, in which theory encourages us to make seemingly unphysical conclusions, is string theory. Under the assumption that the fondamental degrees of freedom are tiny strings, vibrating and interacting in an essentially infinité number of ways, the équations tell us that reality is ten-dimensional, an idea that seems at first sight to be completely outrageons.
In principle however, this is not in any way different from Newton’s mystical force,
and only history and experiment will tell us if it helps us in our understanding of
the universe.
2 Chapter 1. Mathematical physics - sîmplifying the world sînce 1687
If we put experiment aside for a moment, mathematical principles thus replace physical preconception or intuition, when constructing and developing physical thé
ories. In term of scientific methodology this must be considered a very important step, since mathematical consistency is arguably more objective than our individual perceptions. It might of course be that the ‘unreasonable’ effectiveness of math- ematics in physics cornes to an end one day, but so far it has been surprisingly useful.
Let us corne back to Einstein’s theory of gravity. Taking his spécial relativity to its logical end point, Einstein postulated that we could think of space-time as a four- dimensional manifold that curves and bends around the matter and energy of our physical reality. This implies that we can restate gravitational problems in terms of the généralisation of Newton’s and Leibniz’s calculi to curved manifolds: the discipline of mathematics called differential geometry. This becomes in particular true with respect to so-called black holes; the ‘icebergs of the sky’ to any careless trekker of the stars. The équations of general relativity tell us, under quite general assumptions, that these same équations break down in the gravitational collapse of stars or star Systems for example. The collapse results in singular cosmological configurations; black holes, which are the main topic of this thesis and introduced in detail in chapter 2. In terms of differential geometry and algebra, we can understand and develop the theory of relativity and its modem généralisations, in ways that perhaps would be inconceivable without the tools of mathematical physics.
This is the approach we take in this thesis. In an attempt to understand and classify black hole solution, we develop and use techniques from differential geometry and Lie group theory. In part I we introduce our methods and the background: the theory of black holes; symmetric spaces; non-linear sigma-models; and nilpotent orbits. In part II we use these methods to investigate extremal black holes and black branes in generalised théories of gravity, such as M-theory and supergravity in higher dimensions. In the rest of this chapter we give a general introduction to the contemporary théories of gravity, string theory and introduce the theory of Lie groups. In section 1.1 we make an attempt to motivate the work we hâve undertaken when writing this thesis. In section 1.2 we discuss the ‘top-down’ approach of the modem theoretical physicist, of which this thesis is an exemple. In section 1.3 we recall the framework of string theory and in section 1.4 we introduce Lie groups and the use of Lie groups when solving differential équations. Finally in section 1.5 we give a general overview of the contents of this thesis with emphasis on ail original results. We also allow ourselves to indulge in some (more or less) philosophical musings.
1.1 Pourquoi devrais-je m’en préoccuper?
To repeat the eternal cri-de-coeur of the frustrated high school student always seems
appropriate, when motivating a study such as the one in this thesis. The pursuit
of theoretical physics could easily be questioned from the perspective of usefulness.
Although general relativity has found an important application in satellite position- ing and quantum mechanics in the theory of lasers or nuclear energy for example, the nature of the big bang or the microscopie degrees of freedom of a black hole, are likely never to be directly relevant to the technological advancements of human soci
ety. As we discuss in the next section, this objection might be especially important, due to the current state of theoretical physics, where theory is maybe historically uniquely far from experiment, and even reality. The mathematician G.S. Hardy was proud about the fact that the number theory he developed was completely disjoint with respect to practical application and would surely hâve been quite upset, when it turned out to be relevant for cryptology and computer science. For a physicist this can of course never be a suitable point of view. Our aim must always be to affirm or falsify théories about the world we perceive, in ways that increase our understanding about ourselves and our place in the universe.
The content of this thesis is however not particularly concerned with realistic théories. The reason is simple. Realistic théories are at this stage too complicated.
To apply the technology we introduce in later chapters to an even remotely realistic model would be immensely complicated, if not perhaps impossible. We concern ourselves instead with toy models in order to focus on paxticular aspects that are likely to be relevant for the more realistic théories. The argument would hence go as follows. Assume we would like to understand quantum gravity, the combination of general relativity with quantum mechanics. It is likely that a quantum theory of gravity is going to change our idea about the universe in fundamental ways, so it is a reasonable aim. For this we would need to understand classical gravity and plausibly its supersymmetric généralisations. To understand classical gravity, we need to understand the kind of solutions it allows, in particulax black holes.
To understand black holes, maybe with respect to quantum gravity, we need to understand simple examples of black holes, such as supersymmetric and extremal black holes. To understand supersymmetric or extremal black holes, we need to classify them. In this thesis we therefore develop and apply a method of classifying and analysing extremal and supersymmetric black holes.
1.2 Going top-down
The motivation we just gave is an example of a typical ‘top-down’ approach to physics. Here ‘up’ can be either towards higher energies, or towards higher ab
straction. If we start at the top with an abstract principle such as the équivalence principle of Einstein, or a string theory, we try to work our way downwards towards falsifyable prédictions. Einstein succeeded in this attempt in his resolution of the anomaly in the procession of the perihelion of mercury. String theory still needs to converge to a point where the contact with more concrète physics is clearer. In this thesis, as indicated above, we take the top-down point of view.
The other approach is thence ‘bottom-up’. A ‘bottom-up’ physicist tries to
investigate anomalies in observation that current théories so far cannot solve, in
4 Chapter 1. Mathematical physics - simplifying the world since 1687
a more ad hoc manner instead of assuming a mathematical or physical principle.
The construction of the standard model of particle physics can be interpreted as resulting from a ‘bottom-up’ approach. The forces between particles in the standard model are governed by gauge theory, but the model wasn’t developed originally from the assumption of gauge symmetry, or even more ‘top-down’; by starting with a connection on a principal G-bundle, but instead from small pièces added when needed. Scientific progress probably needs both methods, although contemporary discourse yields arguably more prestige to the followers of an Einsteinean ‘top-down’
approach.
We can also think of the différence between ‘top-down’ and ‘bottom-up’ as the directions on a vertical energy scale; the ‘top-down’ approach of string theory is mainly concerned with energies approaching the Planck-scale for which quantum effects of gravity dominate, whereas the ‘bottom-up’ approach is relevant for physics at the electroweak scale, or at energies probed by contemporary particle accelerators.
The huge séparation between these two energy scales is clearly a problem, especially from a ‘top-down’ point of view. Let us assume we hâve an experimental anomaly observed in some current experiment. In an attempt to explain this anomaly from a string theory scénario, we start at the Planck-scale and set up a framework or model describing the physics of the experiment. The energy gap is however so big that many models produce the same resuit, and also such that the dérivation of the phenomena at lower energies is not well-défined or trustworthy, with regards to stability for example. This has motivated some agitators to give string theory a Popperian red card.
It has, in defence of the current state of methodology in theoretical physics, therefore been argued that internai mathematical consistency is to be the guiding principle in research, temporarily until experiment and theory again meet on some common ground. To compare with Newton, we hâve thus reached a point where mathematics, not only is put before physical intuition, but has become the only means of progress for the ‘top-down’ theorist. Hence; the methods of the mathe- matician must also be the methods of the physicist. Proving theorems is the obvions candidate. In the vast no mans land of supergravity and string theory, where there is no contact with experiment in sight, what other means does one hâve to judge a resuit? Taking this point view, we hâve therefore chosen to use a very mathematical formalism and method. The slight digression in this section, into the epistemol- ogy of fondamental physics, has thus emerged as a motivation for this somewhat mathematical treatise.
1.3 String theory and Kac-Moody symmetry
String theory, or rather the assumption that ail particles at high enough energies are
accurately described as vibrating strings, is an extensive source of models relevant for
understanding quantum gravity. One of the most important questions of quantum
gravity is a description of the microscopie degrees of freedom, at the horizon or
elsewhere, of a black hole, and string theory seems to give satisfactory hints to a way of answering this question. In the low energy limit of string theory, the fundamental strings can be approximated again by particles and we arrive at théories of gravity coupled to some matter. If supersymmetry, the conjectured symmetry between bosonic and fermionic degrees of freedom, is présent, we call the resulting theory a supergravity theory. Supergravity contains a wide range of interesting black hole solutions, and is the source of ail models we use in the following chapters. Let Al be a ten-dimensional manifold, S a Lorentzian surface and X : H —* M the string embedding map. String theory dynamics is governed by how X (and its fermionic counterpart ’î') embed into Af, in a sigma-model with background fields for example. Quantizing this theory leads to a set of création and annihilation operators derived from the maps X and acting on a Hilbert-space. In a low energy limit, the quantum states with the lowest mass dominate and become thus the fields describing the supergravity theory. We do not however know how to quantise the theory for a general manifold M and a general set of background fields; the only case for which this has been accomplished is fiat space. For many classical and semi-classical questions it is however enough to consider supergravity only. Let Ai = x Y where F is a compact Calabi-Yau manifold for example. In order to describe the low energy dynamics in four-dimensional Minkowski-space it is enough to know the massless fields in ten dimensions and expand these in terms of Fourier-modes on Y.
Physical théories hâve in the twentieth century been guided heavily by symme
try principles, general covariance in general relativity and gauge symmetry in the standard model for example. Also string theory and supergravity seem to be based on a fundamental symmetry but we hâve yet to understand how. Consider the four- dimensional supergravity theory resulting from a string theory on Ad = x Y.
Internai symmetries of Y combine with symmetries of the four-dimensional La- grangian to form a continuons group G, a Lie group acting on the space of solutions to the supergravity. Equivalently, G appears as a symmetry of the differential équa
tions governing the dynamics of the theory. Lie groups are briefly introduced below in section 1.4. Could it be that the symmetries revealed at low energies by com
pactification also are symmetries of the higher-dimensional quantised theory? One approach to solving this deep problem is via evidence of a hidden infinité symmetry in M-theory or eleven-dimensional supergravity in terms of the Kac-Moody algebras cio or eu [1, 2]. Infinite-dimensional symmetries hâve a long history in théories of gravity, starting with Geroch (3) in pure gravity, and generalised to various super
gravity théories by Julia [4, 5]. Since their discoveries various attempts hâve been made to understand how, and if, these conjectured infinité symmetries are realised.
Near a space-like singularity one can describe the billiard like dynamics in terms of Weyl reflections in ejo [6] (see |7, 8] for reviews). This limit hence describes in
teractions at the Planck-scale and if eio appears in a natural way, it is reasonable
to think that this algebra has something do with the fundamental dynamics of M-
theory. Purther evidence cornes from the fact that one can reproduce parts of the
action of eleven-dimensional supergravity via a level expansion of eio, deriving a
6 Chapter 1. Mathematical physics - simplifying the world since 1687
Kac-Moody/supergravity dictionary [2, 9]. So far, the investigations of Kac-Moody symmetries of M-theory hâve not resulted in any definite conclusions, many open problems remain unsolved and M-theory remain as elusive as ever.
We do not delve deeper into Kac-Moody algebras and the infinité symmetry approach to M-theory for now, but refer to the excellent PhD theses of Sophie de Buyl [10], Nassiba Tabti [11], D. Persson [12] and Ella Jamsin [13| for more information. However, we do return to it in the last two chapters.
1.4 Lie groups
The basic and most important concept of this thesis is that of a Lie group G. A Lie group is both a smooth manifold and a group, combining two old branches of mathematics, those of geometry and algebra. Lie groups where discovered by the Norwegian mathematician Sophus Lie (b. 1842 - d. 1899) in his research on differ- ential équations. Inspired by the work of Évariste Galois on the discrète symmetries of polynomial équations, Sophus Lie aspired to develop a similar theory for partial differential équations (PDE’s) or ordinary differential équations (ODE’s). In his paper [14] he formulâtes the question as ^How can the knowledge of a stability group for a differential équation be utilised towards its intégration?' The aim was to find symmetries that could help solve and classify PDE’s and ODE’s. The idea was con- cretised iii Lie’s famous theorem constructing the integrating faotor of a differential équation, when given a vector field generating a local stability group of the équation (see e.g. [15]). In subsection 1.4.1 we show how this ambition of Lie’s leads to the discovery of Lie groups, and their infinitésimal versions, the Lie algebras, by using Lie’s method to find the solution to an ODE. Our main method of constructing and classifying black hole solutions in varions supergravity théories in part I and II are in many ways a straightforward application of Lie’s methods but in a more modem context. A short review on the work by Lie is given by Helgason in [16].
Although the programme of Lie was eventually largely abandoned, as the in
terest in explicit solutions to differential équations shifted towards questions about existence and uniqueness, his ideas turned out to be very important and led to the development of a huge mathematical theory that bears his name. One of the most impressing results is that a complété classification of the so-called ‘simple’ Lie groups and algebras is possible, a somewhat surprising resuit as complété classifi
cations are quite rare in mathematics. Lie’s approach was originally local in nature and not particularly rigourous, but via the French mathematician Élie Cartan and his German colleague Hermann Weyl, the theory of Lie groups was put on a more solid ground.
Elie Cartan is in some sense one of the most important sources of inspiration to the physics we concern ourselves with in later chapters. Building on the work of Killing, he completed the classification of complex Lie algebras in his thesis [17].
Cartan also introduced a spécial family of manifolds called ‘symmetric spaces’ that
are of a spécial significance in supergravity models and introduced in chapter 3.
Symmetric spaces are described as quotients of two Lie groups and their theory was developed in detail by Élie Cartan in his monograph [18], For a short survey of Élie Cartan’s and Hermann Weyl’s work see for example the essays by Armand Borel [19]. We will unfortunately not, in this thesis, be able to delve deeper into the other aspects of the work of Cartan but for the interested reader we recommend Sharpe’s excellent book [20] on differential geometry, from the point of view of Cartan.
1.4.1 Symmetries and differential équations
Let us now turn to the historical inspiration for the theory of Lie groups. In terms of what Sophus Lie called ‘transformation groups’, he used a systematic approach in terms of symmetries to integrate a wide range of differential équations. As a teaser of what is to corne, we walk through an example of how to solve a differential équation using, in a very simple case, Lie’s theory (the example is taken from Gilmore [21]).
For a modem exposition of how to use Lie groups to solve differential équations we recommend the book by Olver [22]. It is in fact exactly this method that is used in later chapters, when considering the symmetries of the space of solutions to varions supergravity théories.
Consider the differential équation
(1.1) Let P ~ and think of p as an independent parameter. The surface
F{x,y,p) = xp + y-xy'^ = 0 (1.2)
now define a submanifold A4 in In fact this submanifold exhibits a symmetry, by rescaling x, y and p, as
au 9 _
—\-y — = 0.
dx
X
5(A) : y P
leave A4 invariant, or more explicitly; the rescaling acts as
g O F e~^F, (1.4)
so the surface F = 0 is unchanged. We call this transformation g and although it seems very trivial now, it obeys the important property that
g{s) O p(f) O F = e~^e~^F = e~^~^F = g{t + s) o F. (1-5) The symmetry g tells us that A4 has a symmetric direction and by changing co- ordinates we can make this manifest. To find this change of variables is the basic trick of Lie, and it has deep conséquences. Consider the infinitésimal version of g.
—> \x
- \-^y (1.3)
—> X~^p
8 Chapter 1. Mathematical physics - simplifying the world since 1687
and call it X. In fact, X is a vector-field in generating the symmetry g. So, if we define the action of g on functions / in as
go f{x) = f{g-x) (1.6)
the infinitésimal version of g is
= dt{g{t)f{x,y,p)\t=Q
= df(/(e‘x,e'‘y,e“^‘p)|t=o
= xd^f - ydyf - 2pdpf (1.7)
and we find that the vector field generating g is
X = xdx - ydy - 2pdp. (1.8)
We would like to find variables such that X is only depending on one direction on M.. This would imply that the direction of symmetry is exactly along that variable.
We achieve this by choosing co-ordinates S{x,y)^ R{x,y) and T{x,y,p) such that XR = 0,
XS = -1, (1.9)
XT = 0.
As the action of A on T and R is zéro, it is clear that S, and only S, parametrise the symmetry direction on M. Note that T is a function of S and R as p — We choose
S = logy
(1.10)and
and via the relation
R = xy
^ _ g25__dfi_
dx 1-R^
we find the transformed difïerential équation
(1.11)
(1.12)
i-R^\R l + R
dR
0.(1.13)
As long as 1 — R^ ^ 0 this équation is straightforwardly integrated. Note that the explicit dependence on S has disappeared, as we chose co-ordinates such that S became the symmetry direction. The généralisation that Lie continued with was to look at more complicated difïerential équations with more than one symmetry.
The infinitésimal versions of these transformations generated an algebra, what we
now call a Lie algebra, and using this algebra we can choose co-ordinates so that
the symmetries become manifest. The transformation g above is what we now call
a one-parameter Lie group, and X is called the (differential) Lie algebra generator corresponding to the Lie group generated by g.
Interestingly enough, Lie did not make a big différence between the local and global theory, especially not a big différence between the algebra and the group.
Also in the physics literature the différence is sometimes ignored. We hope that in this thesis, the reader will learn to appreciate the différence. One must however remark that Lie’s method of transformation groups does not necessarily generate groups that are well-defined globally, and this turn out to be of spécial importance for the regularity of the black holes we describe, as discussed in some detail in section 4.4.
When Lie discovered the importance of the infinitésimal transformations that could be used to solve differential équations, he started to try to classify and gén
éralisé these groups on multi-dimensional spaces. Let us assume that Xi is a set of infinitésimal vector fields generating the symmetries of a differential équation. Lie showed that these vector fields obeyed the relations
\Xi,Xj] = Cij>^Xk (1.14)
where the Cij^ become the structure constants of ‘Lie’s algebras’, antisymmetric in i and j and obeying the Jacobi identity
Cij Cmn 4“ Cim ^nj 4” ^in ^jm ~ (l'I^) The reason this identity is called the Jacobi identity is that Jacobi discovered it in his development of Hamiltonian mechanics, as the Poisson brackets of conserved quantifies obey this relation. Lie thus faced the problem if he could classify the algebras obeying the relations (1.14) and (1.15). Although Lie tried he did not suc- ceed, apart frorn some low-dimensional cases. It was instead Killing in four papers from 1886 to 1890 who succeeded, for Lie algebras, over the complex numbers, in finding the standard infinité sériés sl(n,C),so(n,C),sp(2n,C) and the five excep- tional algebras e6,e7,C8,f4 and Q2- His dérivation contained however many errors, which were later filled in and corrected by Élie Cartan in 1894. The classification is nowadays therefore called the Killing-Cartan classification. It can also be noted that Killing was very disappointed with his classification, as he had tried to find the classification over the real numbers. The papers however gave him a professorship and financially secured he did not achieve much during the rest of his academie career. The classification of real Lie algebras was completed by Cartan.
Example 1.4.1. (Non-abelian symmetry group). Let us consider the case of a dif
ferential équation with a non-abelian symmetry group. Take the differential équation 2xy" + y' - a(y'Ÿ = 0, (1.16) where ' dénotés dérivatives with respect to x. This équation has two ‘manifest’
symmetries:
51 (0 : y y 4-1 (1.17)
10 Chapter 1. Mathematical physics - simplifying the world since 1687
and
g2{t) : Xtx,yty. (1.18)
To see the Q2 symmetry, one has to calculate how y' and y" transform. A few Unes will show that y' does not transform at ail, and y" as y" —» t~^y". In fact, the differential équation (1-16) has a third non-linear symmetry
53 : a: ^ -» (1-19)
although to see that this transformation leaves (1-16) invariant on the finite level takes a considérable amount of computation (the transformations of y' and y" tum out to be quite complicated and depending on a). Hence gi,i = 1,2,3 generate a three-dimensional symmetry group, in fact the spécial linear Lie group SL(2,M).
The finite versions of gi,i = 1,2,3, written in terms of vector flelds in R'*, would generate the Lie algebra sI(2,M), a real forrn of the complex Lie algebra 0[(2,C) in the Cartan-Killing classification. To see that the action of g\ and 52 does not commute, consider
9
f^{t)
9f^is)
9iit)
92{s){x,y)
= 5f^(i)5^^(s)(sx + i, sy)= {x + s~^t-t,y)
=
- t)(x,y), (1.20)
i.e. trying to undo the shift and the rescaling, we end up with a shift.
In chapter 4 we again encounter the SL(2,M)-group as a symmetry of a set of partial differential équations in the context of pure four-dimensional gravity, and Work out the details more explicitly.
1.5 Layout of the thesis
We hâve divided this thesis in two parts, part I in which we review the literature and introduce important terminology, and part II in which one finds our original research. Some original results are also spread out over part I.
In chapter 2 we introduce and discuss black holes and their varions characteristics that are important in the following chapters. In chapter 3 we introduce symmetric spaces G/H where G is a connected semi-simple Lie group and H a certain subgroup.
We discuss these spaces in detail from the perspective of differential geometry and
Lie algebras. In chapter 4 we combine the theory of black holes with the theory
of symmetric spaces, when we describe black hole solutions as sigma-model maps
to G/H. In particular we discuss cohomogeneity-one black holes which become
geodesics on the symmetric space. This chapter contains the original results lemma
4.4.4 and lemma 4.4.5 which concern geodesics on G/H with respect to a certain
co-ordinate patch on G/H, the ‘Iwasawa patch’. In chapter 5 we discuss uniqueness
theorems of black holes, using the sigma-model machinery of chapter 4. The main
resuit is that we can describe extremal black holes in terms of nilpotent orbits. This
chapter contains original results in subsection 5.1.1 and 5.1.2 in which we ‘correct’
an old resuit by Breitenlohner, Maison and Gibbons [23]. We end part I in chapter 6 where we discuss the classification of nilpotent orbits. This chapter contains an original discussion of nilpotent /f-orbits, section 6.4 where we develop a technique of finding ail such orbits.
Part II starts in chapter 7 with a discussion of AA = 2 pure supergravity, based on the paper [24], although somewhat rewritten. One of the main results is theorera 7.3.1 in which we prove that ail extremal black holes of this theory are in the same if-orbit. In chapter 8 we discuss the extremal black holes of the S^-theory using the machinery of nilpotent orbits. This chapter is based on the paper |25|.
We continue in chapter 9 to discuss the lifting of the S^-theory to five dimensions and five-dimensional extremal black holes with respect to the same nilpotent orbits as in chapter 8. This chapter is based on [26]. In chapter 10 we introduce the Kac-Moody approach to black holes and black branes; it is based on [27]. One of the surprising conclusions of chapter 10 is that we can apply the nilpotent orbit technique almost immediately in the Kac-Moody case, although the method and présentation in chapter 10 is somewhat different. We end part II with chapter 11 with a discussion based on [28] of a sigma-model, derived from a subalgebra of the Kan-Moody algebra eio but not based on a symmetric space. We dérivé the corresponding space-time solution which is of a very different type compared to the other solutions we hâve described thusfar. In the appendices we summarise some information, mostly algebraic details, that we use throughout the thesis.
Proofs are in general not included if they are to long or easily available in the
literature. We hâve however included them in some cases, where they are quite
short and illustrative or hard to find in the literature. We also assume quite some
knowledge about (co-ordinate free) differential geometry. Lie algebra theory and
string theory/supergravity among other things. In order to keep this thesis as
short as possible this has been necessary. The reader is encouraged to consult
the references for more details if needed. A short list of some symbols we use are
listed after the table of contents for quick references.
Part I
Background
Black holes - Singularly unexplained
The term ‘black hole’ originates from the 60’s and was probably coined by John Wheeler to describe a then well-known solution of Einstein’s theory of gravity. This solution was originally discovered by Karl Schwarzschild in 1915 [29], during his ser
vice as an officer in the German army during the first world war. Karl Schwarzschild tried to describe the gravitational field generated by a massive sphere in the recently formulated theory of general relativity, published by Einstein only months before [30]. In the early history of general relativity, Schwarzschild’s solution was mainly used as an approximation of the gravitational field of a cosmological body such as our sun or the planets. However, the solution contains some surprising properties that was not fully understood and appreciated until much later. Before we dis- cuss the formalities of general black holes in section 2.1 and section 2.2, let us first consider the metric of the four-dimensional Schwarschild black hole,
dsl = + V-^dr"^ -1- r^dn|, (2.1) where
V = 1- 2m
r (2.2)
and dr22 is the metric on the two-sphere S^. The constant m parametrises this family of solutions and is interpreted as the mass of the black hole as measured by an observer at spatial infinity, the définition given by équation (2.18) below. Note first that P 1 as r oo, implying that far away from the centre of the blaek hole, the metric reduce to the metric of fiat Minkowski space, i.e. the solution is asymptotically fiat. Furthermore, at r = = 2m the solution as it is written above is singular, due to the appearance of V~^ in front of dr^, which blows up at This singularity is however superficial since it can be removed by the co-ordinate transformation
dt = du + P Mr.
This yields the metric
(2.3)
ds4 = —Pdu^ — 2dudr -t- r^dn^i (2-4)
which is the Schwarzschild solution in so-called retarded null co-ordinates, and this
metric is non-singular at r-u- The point rji is nonetheless very important as it is
16 Chapter 2. Black holes - Singularly unexplained
the position of the so-called event horizon of the Schwarzschild black hole. The event horizon is defined rigourously in définition 2.1.4 below. If an observer falling into the black hole pass this point she will not be able to get ont again, implying the name ‘event horizon’. In other words, whatever happens within the horizon haa (classically) no implications to the whereabouts of observers on the outside. As information at most travel at the speed of light, this implies that no light càn escape from behind the horizon. This effect is due to the extreme gravitational forces at r = Oand make the surface at completely black. That gravity gives rise to such phenomena was realised already by, for example, Laplace in the eighteenth century (see e.g. [31] for a brief discussion of the results by Laplace).
When considering the Schwarzschild solution as an approximation for the gravi
tational field of a cosmological body, there is no problem as long as the radius of the body is larger than the Schwaxschild radius r-^. Under, for example, a gravitational collapse of a heavy star however, the surface of the star might eventually disappear behind the Schwarszchild radius and a black hole is formed.
The contraction of the Riemann curvature tensor, corresponding to the solution (2.1), with it self goes as
O
48m^ .
Rterrii = —(2.5) At r = 0 we hence hâve a real curvature singularity, not removable by any co- ordinate transformation. This singularity signais a breakdown of Einstein’s classical theory, since curvatures when r —> 0 approach the Planck-scale and we expect quantum effects to become important.
Let us now give the outline of this chapter. As an exposition on some basic aspects of black holes it is based on the books by Hawking and Ellis [31], Wald [32], Heusler [33] and to some degree O’Neill [34]. In section 2.1 we follow [31] to define the casual structure of a black hole space-time and regularity conditions such as asymptotic fiatness and predictability. In section 2.2 we introduce notions such as stationarity, staticity, mass and surface gravity that we use in later chapters in order to characterise the black holes we encounter. The important positive mass theorem of vacuum gravity and its généralisations are reviewed in section 2.3. The generalised version of this theorem has significant conséquences for what kind of physical black holes a given theory contain, in particular for the so-called extremal black holes introduced in section 2.4. In section 2.5 we briefiy review the Taub-NUT space-time, famously called ‘the counterexample to everything’ by Misner [35]. The généralisation of black hole like solutions to théories of gravity in higher dimen
sions, so-called branes, are discussed briefiy in section 2.6. Finally, we introduce the concept of attractor black holes in section 2.7.
It is also important to remark that although we hâve the ambition of being rigourous, there are many open problems in the theory of black holes. A particularly troubling problem is the question of analyticity of the metric, and many of the theorems we review are strictly speaking not on completely solid ground (see [33]).
In what follows D is the dimension of space-time. Note that the considérations in
the following sections are valid for black holes in any space-time dimension. The
original formalism was however developed in the context of gràvitational collapse in four dimensions and to our knowledge this bas not been studied in detail in bigber dimensions. It migbt be tbe case tbat tbe définitions in this chapter are too strict or too loose when considering gravitational phenomena, in particular gravitational collapse, in an increasing number of spatial dimensions.
2.1 Structure of black holes space-times
After the conceptual introduction of Schwarzschild’s solution above we aim in this section to formalise the notion of a black hole, with particular emphasis on the asymptotic structure and the horizon. Let us start with the asymptotic casual structure of the solution (2.1). As already mentioned the solution approaches pure Minkowski-space when r oo. We will in fact require this to be the case for ail black holes to be considered in this thesis. The casual structure of Minkowski is therefore important to investigate in detail for an understanding of the casual structure of the full Schwarzschild solution, and more general black holes. We must however take some care in order to find a suitable co-ordinate indépendant way of defining asymptotic flatness. Following (3lj (and originally Penrose) we hâve the following définition;
Définition 2.1.1. A space-time M with metric g is called asympotically simple and fiat if there exists a manifold A4 with metric g and an embedding
l\ A4 AA which embeds A4 as a submanifold with smooth boundary dA4 in A4 such that the following properties holds:
• AA is conformai to AA, Le. there exists an everywhere positive smooth function n e C°°{AÀ) such that
g = i*(n-^g), (2.6)
i.e. i*{g) is conformally équivalent to g with conformai factor fl^.
• The function fl is such that = 0 and dfî|gx ^ 0.
• Every null géodésie in AA has two endpoints on dAA.
• Rie = 0 on an open neighbourhood of dAA.
We call AA the conformai compactification of Al as it is similar to the compacti
fication of the complex line to the projective complex line for example, since we add
‘points’ at infinity. The manifold JA can be thought of as Ad plus the boundary dAA of points where null geodesics begin and end. Contrary to the projective com
plex line however, we add two ‘points’ instead of one, due to the casual nature of
Lorentzian spaces and the implied différence between past and future. A particular
set of compactifications as in définition 2.1.1 are sometimes illustrated in terms of-
so called Penrose-diagrams. These diagrams give us a clear way of illustrating the
casual structure of a given space-time.
18 Chapter 2. Black holes - Singularly unexplained
Hawking and Ellis further show that for a four-dimensional space-time, the boundary dM is a null surface and has two disconnected components with topology 5^ X M. That dM. is a null surface is ensured by the two conditions on the fonction Q. The two components of dM are I'^ where null geodesics ‘end’ and I~ where null geodesics ‘begin’. We call the surface /■*" future null infinity and I~ past null infinity.
Information is ultimately transmitted by light, and light follows null geodesics so according to définition 2.1.1, is clearly the ‘point where ail information eventually ends up’. If an observer has the ambition to measure ‘everything’, is the place to be.
Example 2.1.2. Let us illustrate définition 2.1.1 by àdding future null infinity to the Schwarzschild solution in retarded null co-ordinates (2.^). Consider the co-ordinate transformation
l^l. (2.7)
r This makes us able to rewrite the metric (2.4) cis
ds^ = ( - l'^Vdu^ - 2dudl + dQ2 (2.8)
and we can identify Çl = l in définition 2.1.1. Future null infinity is at the hyper-surface l = 0 and it is obvions that fl = 0 and that dQ ^ 0 at l — 0.
The condition that every null géodésie has endpoints on the boundary at infinity excludes black holes, null geodesics inside the horizon being the obvions counterex- ample as they converge at the singularity. Instead we define a weakly asymptotically simple and fiat space-time as a Lorentzian manifold M with metric g such that there exists an asymptotically simple and fiat space M' and an open setU G M such that U is isométrie to an open neighbourhood V of dM', where dM' is defined according to définition 2.1.1 with respect to M'. For a black hole, the neighbourhood U is taken as some open région near future and past null infinity outside the horizon.
In this sense the notion of a weakly asymptotically simple space-time become our sought after co-ordinate independent and more formai way of simply saying that the metric should reduce to Minkowski when r —> oo. When we talk about asymptotic flatness in what follows, it is always weakly asymptotically fiat and simple we mean.
Let us continue with the définition of an event horizon. For the Schwarzschild
space-time to make sense, a physical observer must be protected from the break-
down of Einstein’s équations that dccur at the singularity at r = 0. As discussed
above this is accomplished by the existence of the surface at r-^. The séparation
of the singularity from null infinity by the horizon then ensures the predictability
of the theory. It is therefore appealing to try to find suitable physical conditions
such that singularities always are accompanied by horizons. This physical idea is
formalised through définition 2.1.3 below in terms of a ‘future asymptoticallly pre-
dictable space-time’. Before we State this définition let us recall some technical
terminology regarding the casual structure of a Lorentzian manifold.
We call a surface S a partial Cauchy surface if it is a space-like D—1 dimensional hypersurface such that no non-space-like curve intersects the surface twice. We call a surface S' closed and trapped if it is a space-like compact D — 2 dimensional surface such that ail null geodesics orthogonal to the surface converge within it. Let {J~{S)) be the set of ail points in the ambient manifold which can be reached from a given subset 5 C by a future (or past) directed non-space-like curve. Given a space-like co-dimension one surface S let D'^{S) {D~(S)) be the set of ail points q E M such that ail future (or past) directed non-space-like curves intersecting q pass through S. The set hence contain the casual future of the points in S, i.e. the set of points where events can be affected by past events on S. The set D'^{S) is called the Cauchy development of S and is the set of points where events are completely determined by past events on S. In a deterministic theory, it is hence sufficient to hâve the initial data on S to détermine everything on
A space-time for which there exists a surface S such that the Cauchy developments and D~{S) cover the space-time is called globally hyperbolic. Hawking and Ellis [31] define a future predictable space-time as a metric manifold containing a partial Cauchy surface S
CA4 such that is contained in the closure of the future development D~^{S) of S. In more colloquial terms, this thus ensures that given ail available information at a fixed point in time we can détermine the infinité future completely. They go on to prove that under some suitable physical condition on the matter in the theory (see définition 2.3.1 below) it follows under these assumptions that closed trapped surfaces cannot be seen by an observer at future null infinity in a future predictable space-time. This means that the closed trapped surface cannot intersect the open set J~ (/■•■), and hence must be contained in the open set If we turn this resuit around, it implies that a singularity not protected by a horizon might lead to events we cannot predict (there exists non-space-like curves between and the singularity). The Schwarzschild solution contains a partial Cauchy surface and observers outside the horizon are hence protected from the singularity.
Although this shows that the existence of a partial Cauchy surface leads to a physical space-time, it is not clear that it is stable against perturbations. It is therefore common to introduce another notion of predictability:
Définition 2.1.3. An asymptotically fiat and simple space-time A4 with metric g is called strongly asymptotically predictable if there exists an open set V C. A4 such that the closure of in A4 is contained in V and V is globally hyperbolic.
The manifold Ai in this définition is the conformai compactification of A4 from définition 2.1.1. The existence of the partial Cauchy surface S is strictly speaking not sufficient to ensure that we can predict a neighbourhood of the horizon |31|.
Définition 2.1.3 is therefore stronger and such that it ensures such a predicability.
Définition 2.1.3 furthermore leads us to the following définition of the event horizon:
Définition 2.1.4. Let A4 be a space-time with metric g. The event horizon 7ï+ of
20 Chapter 2. Black holes - Singularly unexplained
Figure 2.1: A rough/conceptual sketch of a black hole space-time with horizon, singularity, ‘warped’ light cônes and null infinities. The singularity at r = 0 is indicated by the pointed line. Light cônes inside the event horizon eventually always intersect this ‘point’, since ail light-like geodesics converge there. The horizon H shield off the singularity from future and past null infinity, and , indicated by grey lines. A photon emitted at the horizon lias its light-cone completely inside the event horizon. The circle <S is a spatial cross-section of the horizon.
the space-time Ai is defined as the boundary
n+ = J-{I+)n{M - J-{I+)), (2.9)
where (/■*■) is the casual past of future null infinity /■*“, i.e. points in Ai that can be reached by a past-directed non-space-like curve from The set V dénoté the closure of any set V.
The set V from définition 2.1.3 is roughly speaking conformai to the région outside of the horizon, and V being hyperbolic ensures that given a sufficient set of initial data, i.e. a Cauchy surface, the future and the past can be predicted, outside of the horizon. We call the région the domain of outer communication V.
If we return to our example of the Schwarzschild black hole, it is not yet clear how the horizon in définition 2.1.4 coincide with the surface at In the next section however, we make the connection apparent.
Définition 2.1.4 give what we could call the ‘future’ event horizon and we can
similarly define a ‘past’ event horizon, by considering the boundary to the casual
future of past null infinity, denoted H-. The famous topology theorem of Hawking States that in four dimensions, spatial cross-sections of the event horizon must be homeomorphic to two-spheres [36]. With the concept of an event horizon and of a weakly asymptotically flat and simple space-time we can finally say that a strongly asymptotically predictable space-time is a black hole space-time if (Ad —
is non-empty. Note that this définition of the black hole is not depending on the existence or non-existence of a singularity inside the event horizon. Since we hâve defined a predictable space-time as insensible to such a singularity, a définition of a black hole space-time should not dépend on something we cannot measure.^
Finally, we note this very useful lemma [37]:
Lemma 2.1.5. A weakly asymptotically flat, simple and strongly predictable space-
time is simply-connected in the domain of outer communication T>.
The most important conséquence of this lemma is that together with Poincare’s lemma it implies that every closed one-form on T> is exact. For a bona-fide asymp
totically flat and simple four-dimensional black hole, the topology of T> is in general X 5^. A conceptual sketch of a black hole space-time is given in figure 2.1.
2.2 Particularities
Let us discuss various particular properties of the black hole space-times defined in the previous section. These properties are related to isométries of the metric p on a manifold Ad containing a black hole. The Schwarzschild metric (2.1) is for example a spherically sjnnmetric solution and static in time. This implies that the metric defined by (2.1) is invariant under the action of four Killing vector fields ù and [Çÿ,Ce], i.e. vector fields X G fl(TAd) such that the Lie dérivative of the metric with respect to X vanish :
Cxg = 0.
(2.10)We dénoté by TAd the tangent bundle of Ad and by f2(TAd) the space of sections of TM. For the black hole space-time (2.1) we hâve in partiular that = dt is a Killing vector field, since the space-time geometry does not change in time. More generally, a black hole space-time is called stationary if there exists a Killing vector field that is time-like in an open neighbourhood of I'^ and I~. If there furthermore exists a space-like co-dimension one surface E c Ad such that a time-like Killing vector field is hypersurface orthogonal to this surface, the solution is called static (see e.g. [32]). The condition
^tAd^,' = 0, (2.11)
on the Killing vector field implies via Frobenius’s theorem the existence of such a surface E. We use ‘harmonie notation’ to dénoté the dual one-form of under the metric ÿ on Ad. In the static case Is null at the black hole horizon and in the
'it may however be noted that to experimentally establish the existence of a cosmological black hole is for similar reasons very difflcult.
