Sigma-models and Lie group symmetries in theories of gravity

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Sigma-models and Lie group symmetries in theories of gravity

Thèse présentée par Josef Lindman Hörnlund

En vue de l’obtention du grade de Docteur en Sciences

Service de Physique Mathématique des Interactions Fondamentales Faculté des Sciences

de l’Université Libre de Bruxelles Bruxelles, Belgique

2011

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This thesis is first and foremost dedicated to coffee, the most divine of earthly decoctions.

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.

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Acknowledgments

A few months before this thesis was completed, my supervisor 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 have been very different without him.

Throughout my work I have also had a ‘second’ supervisor, Doctor Axel Klein- schmidt, who has inspired and aided me in all I have 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 Professor Marc Henneaux for generously giving me the possibility of completing a doctorate at his department. Although I still have a hard time understanding why I of all people had the fortune, I am infinitely grateful.

During my master thesis at Chalmers University of Technology, I received great encouragement from Professor Bengt E.W. Nilsson, who I thank for contacting ULB and recommending me to Professor Marc Henneaux. This thesis would not have existed without Professor Nilsson’s support and I owe this amazing time to him.

I also thank my great-aunt Ulla Hörnlund for a careful proof-reading of this thesis.

From the world outside of physics, I have always been supported and encouraged by my dear mother. This thesis is above all an humble attempt by her son to make her proud. I would have been nothing without her.

Finally I thank all 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.

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Abstract

By using non-linear sigma-models of maps into symmetric spaces G/H we discuss black hole and black brane solutions in various supersymmetric theories of gravity.

The Kac-Moody/supergravity dictionary and dimensional reduction allow us to describe 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 all extremal black holes in N = 2 pure four-dimensional supergravity, N = 2 S³ 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.

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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,R)/SO(1,1) 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 ofg₂ given as vectors in the two-dimensional root space. . 89

6.2 Hasse diagram for the partial ordering of the nilpotent orbits ing₂₍₂₎. 93 9.1 Two type of geodesics on the coset manifold, corresponding to flat black holes in five and four dimensions.. . . 139

9.2 A phase diagram of the extremal limits of the cohomogeneity-one Cvetič-Youm black hole. . . 144

10.1 The Dynkin diagram of e₁₁. . . 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 M2⊂M5² ⊂KK6 bound state.. . . 177

B.1 Diagram of m⁰ as a so(3,1)representation. . . 193

B.2 Diagram over mˆ as a so(2,2)representation. . . 194

B.3 The Tits-Satake diagram of su(2,1). . . 197

B.4 The restricted root system of su(2,1). . . 200

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List of Tables

6.1 The five non-zeroG₂₍₂₎-orbits. . . 92

6.2 The fourH-orbits O_3H,O_3H⁰ ,O_4H,O_4H⁰ within the two G₂₍₂₎-orbits O₃ and O₄. . . 94

10.1 Level decomposition of e₁₁ undersl(11,R) up to level `= 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

10.10The space-time positions of the three branes in the bound state between an M2, an M5 and a KK6. . . 172

10.11The space-time positions of the three branes in the bound state between a 1F, a D6 and a D8. . . 176

10.12The space-time positions of the four branes in the bound state between an M2, two M5 and a KK6. . . 177

10.13The space-time positions of the four branes in the bound state between a D4, two D6 and a D8.. . . 178

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List of Symbols

R_D The Ricci scalar in Ddimensions.

Ric_D The Ricci tensor inD dimensions.

RiemD The Riemann curvature tensor inD dimensions vol_n The volume-form in Ddimensions.

TM Tangent bundle for the manifold M TpM Tangent space at pointp∈ M Ω(Ξ) Sections for the fibre bundle Ξ

Λⁿ(M) Bundle ofn-forms over the manifold M

G A Lie group

G₀ The component ofGconnected to the identity

g A Lie algebra

V The topological closure of a subset V

?_D The Hodge dual inD dimensions

V^∗ The pull-backV^∗ : Λⁿ(N)→Λⁿ(M) of a mapV :M → N V_∗ The push-forwardV_∗ :TM →TN of a mapV :M → N i_ξ The interior product with respect to a vector fieldξ L_ξ The Lie derivative with respect to a vector field ξ LieG The Lie algebra of a Lie groupG

Ad The adjoint representation ofG on its Lie algebrag ad The adjoint representation ofg on itself.

κ The Killing-form of a Lie algebra g

R^p,q The vector space R^p+q with metric with signature (p, q) exp_Mp The exponential map fromTpMinto the manifoldM

ξ^[ The one-form dual to the vector field ξ with respect to a given metric g End(V) The space of endomorphisms on a vector space V

Hom(V, W)The space of linear mappings fromV to W A^† The conjugate transpose of a matrixA 1l_D The identity matrix inD dimensions

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Chapter 1

Mathematical physics - simplifying the world since 1687

The western scientific tradition started, by some accounts, with the Ionian philosophers in Miletus in Asia Minor, such as Thales, 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 have been found lacking at a Popperian trial, it contained the basic ingredients of modern science: a critical discourse and deductive reasoning. What lacked in the work of the Milesian philosophers, compared to modern day physics, was however the use of mathematics as the guiding principle when developing their theories. 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 predictions derived from the assumption were in accordance with obser- vations not explained in theory before. In this sense Newton applied a classical equivalent of the famous gratis dictum ‘Shut up and calculate’, coined by Richard Feynman when questioned about the interpretation of quantum mechanics. We can think of Newtonian gravity as a first prime example of how the equations and the mathematical consequences of a postulated law were considered to be more important 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 come back to soon, the methodology of modern physics has since then been such that mathematics has occupied a special and unquestionable position in its centre.

A modern equivalent to Newton’s equations describing gravity, in which theory encourages us to make seemingly unphysical conclusions, is string theory. Under the assumption that the fundamental degrees of freedom are tiny strings, vibrating and interacting in an essentially infinite number of ways, the equations tell us that reality is ten-dimensional, an idea that seems at first sight to be completely outrageous.

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.

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2 Chapter 1. Mathematical physics - simplifying the world since 1687

If we put experiment aside for a moment, mathematical principles thus replace physical preconception or intuition, when constructing and developing physical theories. 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 mathematics in physics comes to an end one day, but so far it has been surprisingly useful.

Let us come back to Einstein’s theory of gravity. Taking his special 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 generalisation 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 equations of general relativity tell us, under quite general assumptions, that these same equations 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 chapter2. In terms of differential geometry and algebra, we can understand and develop the theory of relativity and its modern generalisations, 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 partIwe 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 theories 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 theories of gravity, string theory and introduce the theory of Lie groups. In section1.1we make an attempt to motivate the work we have undertaken when writing this thesis. In section 1.2 we discuss the ‘top-down’ approach of the modern theoretical physicist, of which this thesis is an example. In section 1.3we recall the framework of string theory and in section1.4we introduce Lie groups and the use of Lie groups when solving differential equations. Finally in section 1.5we give a general overview of the contents of this thesis with emphasis on all 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 eternalcri-de-coeurof 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.

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1.2. Going top-down 3

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 microscopic 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 have 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 theories 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 theories. The reason is simple. Realistic theories 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 particular aspects that are likely to be relevant for the more realistic theories. 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 generalisations. To understand classical gravity, we need to understand the kind of solutions it allows, in particular 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 equivalence principle of Einstein, or a string theory, we try to work our way downwards towards falsifyable predictions. 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 concrete 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 theories so far cannot solve, in

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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 pieces 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 difference 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 separation between these two energy scales is clearly a problem, especially from a ‘top-down’ point of view. Let us assume we have an experimental anomaly observed in some current experiment. In an attempt to explain this anomaly from a string theory scenario, 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 result, and also such that the derivation of the phenomena at lower energies is not well-defined 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 internal 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 have 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 mathematician must also be the methods of the physicist. Proving theorems is the obvious 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 have to judge a result? Taking this point view, we have therefore chosen to use a very mathematical formalism and method. The slight digression in this section, into the epistemol- ogy of fundamental 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 all 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 microscopic degrees of freedom, at the horizon or

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1.3. String theory and Kac-Moody symmetry 5

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 theories of gravity coupled to some matter. If supersymmetry, the conjectured symmetry between bosonic and fermionic degrees of freedom, is present, we call the resulting theory a supergravity theory. Supergravity contains a wide range of interesting black hole solutions, and is the source of all models we use in the following chapters. Let M be a ten-dimensional manifold, Σ a Lorentzian surface and X : Σ → M the string embedding map. String theory dynamics is governed by how X (and its fermionic counterpart Ψ) embed intoM, in a sigma-model with background fields for example. Quantizing this theory leads to a set of creation and annihilation operators derived from the maps X and Ψ, 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 flat space. For many classical and semi-classical questions it is however enough to consider supergravity only. LetM=R^3,1×Y whereY 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 theories have in the twentieth century been guided heavily by symmetry 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 have yet to understand how. Consider the four- dimensional supergravity theory resulting from a string theory on M =R^3,1×Y. Internal symmetries of Y combine with symmetries of the four-dimensional La- grangian to form a continuous groupG, a Lie group acting on the space of solutions to the supergravity. Equivalently,Gappears as a symmetry of the differential equations 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 compactification also are symmetries of the higher-dimensional quantised theory? One approach to solving this deep problem is via evidence of a hidden infinite symmetry in M-theory or eleven-dimensional supergravity in terms of the Kac-Moody algebras e₁₀ or e₁₁ [1,2]. Infinite-dimensional symmetries have a long history in theories of gravity, starting with Geroch [3] in pure gravity, and generalised to various supergravity theories by Julia [4,5]. Since their discoveries various attempts have been made to understand how, and if, these conjectured infinite symmetries are realised.

Near a space-like singularity one can describe the billiard like dynamics in terms of Weyl reflections in e₁₀ [6] (see [7,8] for reviews). This limit hence describes interactions at the Planck-scale and if e₁₀ appears in a natural way, it is reasonable to think that this algebra has something do with the fundamental dynamics of M- theory. Further evidence comes from the fact that one can reproduce parts of the action of eleven-dimensional supergravity via a level expansion of e₁₀, deriving a

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Kac-Moody/supergravity dictionary [2,9]. So far, the investigations of Kac-Moody symmetries of M-theory have 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 infinite 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 differential equations. Inspired by the work of Évariste Galois on the discrete symmetries of polynomial equations, Sophus Lie aspired to develop a similar theory for partial differential equations (PDE’s) or ordinary differential equations (ODE’s). In his paper [14] he formulates the question as ‘How can the knowledge of a stability group for a differential equation be utilised towards its integration?’ The aim was to find symmetries that could help solve and classify PDE’s and ODE’s. The idea was con- cretised in Lie’s famous theorem constructing the integrating factor of a differential equation, when given a vector field generating a local stability group of the equation (see e.g. [15]). In subsection1.4.1 we show how this ambition of Lie’s leads to the discovery of Lie groups, and their infinitesimal 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 various supergravity theories in part I andII are in many ways a straightforward application of Lie’s methods but in a more modern 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 equations 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 complete classification of the so-called ‘simple’ Lie groups and algebras is possible, a somewhat surprising result as complete 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.

Élie 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 special family of manifolds called ‘symmetric spaces’ that are of a special significance in supergravity models and introduced in chapter 3.

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1.4. Lie groups 7

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 equations

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 equations. As a teaser of what is to come, we walk through an example of how to solve a differential equation using, in a very simple case, Lie’s theory (the example is taken from Gilmore [21]).

For a modern exposition of how to use Lie groups to solve differential equations 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 various supergravity theories.

Consider the differential equation xdy

dx+y−xy² = 0. (1.1)

Letp= ^dy_dx, and think of pas an independent parameter. The surface

F(x, y, p) =xp+y−xy²= 0 (1.2) now define a submanifold MinR³. In fact this submanifold exhibits a symmetry, by rescaling x, yand p, as

x →λx

g(λ) : y →λ⁻¹y (1.3)

p →λ⁻²p

leaveMinvariant, or more explicitly; the rescaling acts as

g◦F →e^−λF, (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)◦g(t)◦F =e^−te^−sF =e^−t−sF =g(t+s)◦F. (1.5) The symmetry g tells us that M 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 consequences. Consider the infinitesimal version of g,

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and call it X. In fact, X is a vector-field in R³ generating the symmetry g. So, if we define the action of g on functionsf inR³ as

g◦f(x) =f(g·x) (1.6)

the infinitesimal version of gis

Xf = ∂t(g(t)f(x, y, p)|_t=0

= ∂t(f(e^tx, e^−ty, e^−2tp)|_t=0

= x∂xf−y∂yf −2p∂pf (1.7)

and we find that the vector field generatingg is

X=x∂x−y∂y −2p∂p. (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 X on T and R is zero, it is clear that S, and onlyS, parametrise the symmetry direction onM. Note thatT is a function ofS andR asp= ^dy_dx. We choose

S = logy (1.10)

and

R =xy (1.11)

and via the relation

dy dx =e^2S

dS dR

1−R_dR^dS (1.12)

we find the transformed differential equation 1

1−R^dS_dR 1

R −1 +RdS dR

= 0. (1.13)

As long as 1−R^dS_dR 6= 0 this equation 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 generalisation that Lie continued with was to look at more complicated differential equations with more than one symmetry.

The infinitesimal 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

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1.4. Lie groups 9

a one-parameter Lie group, andX is called the (differential) Lie algebra generator corresponding to the Lie group generated by g.

Interestingly enough, Lie did not make a big difference between the local and global theory, especially not a big difference between the algebra and the group.

Also in the physics literature the difference is sometimes ignored. We hope that in this thesis, the reader will learn to appreciate the difference. 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 special 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 infinitesimal transformations that could be used to solve differential equations, he started to try to classify and gen- eralise these groups on multi-dimensional spaces. Let us assume that X_i is a set of infinitesimal vector fields generating the symmetries of a differential equation. Lie showed that these vector fields obeyed the relations

[Xi, Xj] =cijkXk (1.14) where thecijk become the structure constants of ‘Lie’s algebras’, antisymmetric in iand j and obeying the Jacobi identity

c_ij^kc_mnⁱ+c_im^kc_njⁱ+c_in^kc_jmⁱ = 0. (1.15) 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 quantities 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 from 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 infinite series sl(n,C),so(n,C),sp(2n,C) and the five excep- tional algebras e₆,e₇,e₈,f₄ and g₂. His derivation 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 academic 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 differential equation with a non-abelian symmetry group. Take the differential equation 2xy⁰⁰+y⁰−α(y⁰)³= 0, (1.16) where ⁰ denotes derivatives with respect to x. This equation has two ‘manifest’

symmetries:

g1(t) :y→y+t (1.17)

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and

g2(t) :x→tx, y→ty. (1.18) To see the g₂ symmetry, one has to calculate how y⁰ and y⁰⁰ transform. A few lines will show that y⁰ does not transform at all, and y⁰⁰ as y⁰⁰ → t⁻¹y⁰⁰. In fact, the differential equation (1.16) has a third non-linear symmetry

g3 :x→e^2ytx, y→e^yty (1.19) although to see that this transformation leaves (1.16) invariant on the finite level takes a considerable amount of computation (the transformations of y⁰ and y⁰⁰ turn out to be quite complicated and depending on α). Hence gi, i = 1,2,3 generate a three-dimensional symmetry group, in fact the special linear Lie group SL(2,R).

The finite versions of g_i, i = 1,2,3, written in terms of vector fields in R⁴, would generate the Lie algebra sl(2,R), a real form of the complex Lie algebra sl(2,C) in the Cartan-Killing classification. To see that the action of g₁ and g₂ does not commute, consider

g₁⁻¹(t)g₂⁻¹(s)g1(t)g2(s)(x, y) = g⁻¹₁ (t)g⁻¹₂ (s)(sx+t, sy)

= (x+s⁻¹t−t, y)

= g₁(ts⁻¹−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,R)-group as a symmetry of a set of partial differential equations in the context of pure four-dimensional gravity, and work out the details more explicitly.

1.5 Layout of the thesis

We have 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 chapter2we introduce and discuss black holes and their various characteristics that are important in the following chapters. In chapter 3 we introduce symmetric spacesG/HwhereGis a connected semi-simple Lie group andHa 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 onG/H, the ‘Iwasawa patch’. In chapter5we discuss uniqueness theorems of black holes, using the sigma-model machinery of chapter 4. The main result is that we can describe extremal black holes in terms of nilpotent orbits. This

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1.5. Layout of the thesis 11

chapter contains original results in subsection 5.1.1 and 5.1.2 in which we ‘correct’

an old result by Breitenlohner, Maison and Gibbons [23]. We end partI in chapter 6 where we discuss the classification of nilpotent orbits. This chapter contains an original discussion of nilpotent H-orbits, section6.4 where we develop a technique of finding all such orbits.

Part II starts in chapter 7 with a discussion ofN = 2pure supergravity, based on the paper [24], although somewhat rewritten. One of the main results is theorem 7.3.1 in which we prove that all extremal black holes of this theory are in the same H-orbit. In chapter 8 we discuss the extremal black holes of the S³-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 S³-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 presentation 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 Kac-Moody algebra e₁₀ but not based on a symmetric space. We derive the corresponding space-time solution which is of a very different type compared to the other solutions we have 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 have 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.

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Part I

Background

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Chapter 2

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 service 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 discuss the formalities of general black holes in section 2.1and section2.2, let us first consider the metric of the four-dimensional Schwarschild black hole,

ds²₄ =−Vdt²+V⁻¹dr²+r²dΩ²₂, (2.1) where

V = 1− 2m

r (2.2)

and dΩ²₂ is the metric on the two-sphere S². 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 definition given by equation (2.18) below. Note first that V → 1 as r → ∞, implying that far away from the centre of the black hole, the metric reduce to the metric of flat Minkowski space, i.e. the solution is asymptotically flat. Furthermore, atr=rH= 2mthe solution as it is written above is singular, due to the appearance of V⁻¹ in front of dr², which blows up at rH. This singularity is however superficial since it can be removed by the co-ordinate transformation

dt= du+V⁻¹dr. (2.3)

This yields the metric

ds²₄ =−Vdu²−2dudr+r²dΩ²₂, (2.4) which is the Schwarzschild solution in so-called retarded null co-ordinates, and this metric is non-singular at rH. The point rH is nonetheless very important as it is

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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 definition 2.1.4 below. If an observer falling into the black hole pass this point she will not be able to get out again, implying the name ‘event horizon’. In other words, whatever happens within the horizon has (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 can escape from behind the horizon. This effect is due to the extreme gravitational forces at r = 0and make the surface at rH 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 gravitational field of a cosmological body, there is no problem as long as the radius of the body is larger than the Schwarschild radiusrH. 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

Riem42= 48m²

r⁶ . (2.5)

At r = 0 we hence have a real curvature singularity, not removable by any co- ordinate transformation. This singularity signals 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 section2.1 we follow [31] to define the casual structure of a black hole space-time and regularity conditions such as asymptotic flatness 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 generalisations are reviewed in section 2.3. The generalised version of this theorem has significant consequences for what kind of physical black holes a given theory contain, in particular for the so-called extremal black holes introduced in section2.4. In section2.5we briefly review the Taub-NUT space-time, famously called ‘the counterexample to everything’ by Misner [35]. The generalisation of black hole like solutions to theories of gravity in higher dimensions, so-called branes, are discussed briefly in section2.6. Finally, we introduce the concept of attractor black holes in section 2.7.

It is also important to remark that although we have 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 considerations in the following sections are valid for black holes in any space-time dimension. The

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2.1. Structure of black holes space-times 17

original formalism was however developed in the context of gravitational collapse in four dimensions and to our knowledge this has not been studied in detail in higher dimensions. It might be the case that the definitions 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 R^3,1 when r → ∞. We will in fact require this to be the case for all 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 independent way of defining asymptotic flatness. Following [31] (and originally Penrose) we have the following definition:

Definition 2.1.1. A space-timeMwith metricgis called asympotically simple and flat if there exists a manifoldM˜ with metricg˜and an embeddingι:M →M˜ which embedsMas a submanifold with smooth boundary∂MinM˜ such that the following properties holds:

• M˜ is conformal toM, i.e. there exists an everywhere positive smooth function Ω∈C^∞( ˜M) such that

g=ι^∗(Ω⁻²g),˜ (2.6)

i.e. ι^∗(˜g) is conformally equivalent to g with conformal factor Ω².

• The function Ω is such thatΩ|_∂M= 0 anddΩ|_∂M6= 0.

• Every null geodesic inM has two endpoints on ∂M.

• Ric= 0 on an open neighbourhood of ∂M.

We callM˜ the conformal compactification ofMas it is similar to the compactification of the complex line to the projective complex line for example, since we add

‘points’ at infinity. The manifold M˜ can be thought of as M plus the boundary

∂Mof points where null geodesics begin and end. Contrary to the projective complex line however, we add two ‘points’ instead of one, due to the casual nature of Lorentzian spaces and the implied difference between past and future. A particular set of compactifications as in definition2.1.1 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.

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18 Chapter 2. Black holes - Singularly unexplained

Hawking and Ellis further show that for a four-dimensional space-time, the boundary∂Mis a null surface and has two disconnected components with topology S²×R. That∂Mis a null surface is ensured by the two conditions on the function Ω. The two components of∂MareI⁺where null geodesics ‘end’ andI⁻where null geodesics ‘begin’. We call the surfaceI⁺future null infinityandI⁻past null infinity.

Information is ultimately transmitted by light, and light follows null geodesics soI⁺, according to definition 2.1.1, is clearly the ‘point where all information eventually ends up’. If an observer has the ambition to measure ‘everything’, I⁺ is the place to be.

Example 2.1.2. Let us illustrate definition2.1.1by adding future null infinity to the Schwarzschild solution in retarded null co-ordinates (2.4). Consider the co-ordinate transformation

l= 1

r. (2.7)

This makes us able to rewrite the metric (2.4) as ds² =l⁻²

−l²Vdu²−2dudl+ dΩ²

(2.8) and we can identify Ω = l in definition 2.1.1. Future null infinity I⁺ is at the hyper-surface l= 0 and it is obvious that Ω = 0and that dΩ6= 0 atl= 0.

The condition that every null geodesic has endpoints on the boundary at infinity excludes black holes, null geodesics inside the horizon being the obvious counterexample as they converge at the singularity. Instead we define aweakly asymptotically simple and flatspace-time as a Lorentzian manifoldMwith metricgsuch that there exists an asymptotically simple and flat spaceM⁰and an open setU ⊂ Msuch that U is isometric to an open neighbourhoodV of∂M⁰, where∂M⁰ is defined according to definition 2.1.1 with respect to M⁰. For a black hole, the neighbourhood U is taken as some open region 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 formal way of simply saying that the metric should reduce to Minkowski when r→ ∞. When we talk about asymptotic flatness in what follows, it is always weakly asymptotically flat and simple we mean.

Let us continue with the definition of an event horizon. For the Schwarzschild space-time to make sense, a physical observer must be protected from the breakdown of Einstein’s equations that occur at the singularity at r = 0. As discussed above this is accomplished by the existence of the surface at rH. The separation 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 definition 2.1.3 below in terms of a ‘future asymptotically predictable space-time’. Before we state this definition let us recall some technical terminology regarding the casual structure of a Lorentzian manifold.

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2.1. Structure of black holes space-times 19

We call a surfaceSapartial Cauchy surfaceif it is a space-likeD−1dimensional hypersurface such that no non-space-like curve intersects the surface twice. We call a surfaceS⁰ closed and trappedif it is a space-like compactD−2dimensional surface such that all null geodesics orthogonal to the surface converge within it. Let J⁺(S) (J⁻(S)) be the set of all points in the ambient manifold which can be reached from a given subset S ⊂ M 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 all points q ∈ M such that all future (or past) directed non-space-like curves intersecting q pass through S. The set J⁺(S) 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 completelydetermined by past events onS. In a deterministic theory, it is hence sufficient to have the initial data on S to determine everything on D⁺(S).

A space-time for which there exists a surfaceS such that the Cauchy developments D⁺(S) andD⁻(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 surfaceS ⊂ Msuch thatI⁺is contained in the closure of the future developmentD⁺(S) ofS. In more colloquial terms, this thus ensures that given all available information at a fixed point in time we can determine the infinite future completely. They go on to prove that under some suitable physical condition on the matter in the theory (see definition 2.3.1below) 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 setJ⁻(I⁺), and hence must be contained in the open set M−J⁻(I⁺). If we turn this result 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 I⁺ 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:

Definition 2.1.3. An asymptotically flat and simple space-time M with metric g is called strongly asymptotically predictable if there exists an open set V ⊂M˜ such that the closure of J⁻(I⁺) in M˜ is contained in V and V is globally hyperbolic.

The manifold M˜ in this definition is the conformal compactification ofMfrom definition 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].

Definition 2.1.3 is therefore stronger and such that it ensures such a predicability.

Definition2.1.3furthermore leads us to the following definition of the event horizon:

Definition 2.1.4. Let M be a space-time with metric g. The event horizon H₊ of

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20 Chapter 2. Black holes - Singularly unexplained I⁺

I⁻ r= 0

H

S H

Spatial infinity

Figure 2.1: A rough/conceptual sketch of a black hole space-time with horizon, singularity, ‘warped’ light cones and null infinities. The singularity at r = 0 is indicated by the pointed line. Light cones inside the event horizon eventually always intersect this ‘point’, since all light-like geodesics converge there. The horizon H shield off the singularity from future and past null infinity,I⁺ andI⁻, indicated by grey lines. A photon emitted at the horizon has its light-cone completely inside the event horizon. The circleS is a spatial cross-section of the horizon.

the space-time Mis defined as the boundary

H₊=J⁻(I⁺)∩(M −J⁻(I⁺)), (2.9) where J⁻(I⁺)is the casual past of future null infinity I⁺, i.e. points inMthat can be reached by a past-directed non-space-like curve from I⁺. The set V denote the closure of any set V.

The set V from definition 2.1.3 is roughly speaking conformal to the region 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 region J⁻(I⁺) the domain of outer communication D.

If we return to our example of the Schwarzschild black hole, it is not yet clear how the horizon in definition 2.1.4 coincide with the surface atrH. In the next section however, we make the connection apparent.

Definition 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

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2.2. Particularities 21

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 (M −J⁻(I⁺)) is non-empty. Note that this definition of the black hole is not depending on the existence or non-existence of a singularity inside the event horizon. Since we have defined a predictable space-time as insensible to such a singularity, a definition of a black hole space-time should not depend on something we cannot measure.¹

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 D.

The most important consequence of this lemma is that together with Poincare’s lemma it implies that every closed one-form onD is exact. For a bona-fide asymptotically flat and simple four-dimensional black hole, the topology ofDis in general R²×S². 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 isometries of the metricgon a manifoldMcontaining a black hole. The Schwarzschild metric (2.1) is for example a spherically symmetric solution and static in time. This implies that the metric defined by (2.1) is invariant under the action of four Killing vector fields ξt, ξ_φ,ξ_θ and[ξ_φ, ξ_θ], i.e. vector fieldsX ∈Ω(TM)such that the Lie derivative of the metric with respect to X vanish :

L_Xg= 0. (2.10)

We denote by TMthe tangent bundle of Mand by Ω(TM) the space of sections of TM. For the black hole space-time (2.1) we have in partiular that ξt =∂t is a Killing vector field, since the space-time geometry does not change in time. More generally, a black hole space-time is calledstationaryif there exists a Killing vector field that is time-like in an open neighbourhood ofI⁺and I⁻. If there furthermore exists a space-like co-dimension one surface Σ ⊂ M such that a time-like Killing vector fieldξt is hypersurface orthogonal to this surface, the solution is calledstatic (see e.g. [32]). The condition

ξ_t^[∧dξ_t^[= 0, (2.11)

on the Killing vector field ξ_t implies via Frobenius’s theorem the existence of such a surfaceΣ. We use ‘harmonic notation’ξ_t^[ to denote the dual one-form ofξ_t under the metricg onM. In the static caseξt is null at the black hole horizon and in the

1It may however be noted that to experimentally establish the existence of a cosmological black hole is for similar reasons very difficult.

Sigma-models and Lie group symmetries in theories of gravity