Lovelock gravity, black holes and holography

Departamento de F´ısica de Part´ıculas LOVELOCK GRAVITY, BLACK HOLES AND HOLOGRAPHY Xi´an Otero Cama˜no TESE DE DOUTORAMENTO. UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas LOVELOCK GRAVITY, BLACK HOLES &HOLOGRAPHY Xi´an Otero Cama˜no Compostela, maio do 2013. UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas LOVELOCK GRAVITY, BLACK HOLES &HOLOGRAPHY Tese presentada para optar ao grao de Doutor en F´ısica por: Xi´an Otero Cama˜no Compostela, maio do 2013 vii UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Departamento de F´ısica de Part´ıculas Jos´e D. Edelstein Glaubach, Profesor Titular de F´ısica Te´orica da Universidade de Santiago de Compostela, CERTIFICO: que a memoria titulada Lovelock gravity, black holes and holography foi realizada, baixo a mi˜na direcci´on, por Xi´an Otero Cama˜no, no departamento de F´ısica de Part´ıculas desta Universidade e constit´ue o traballo de Tese que presenta para optar ao grao de Doutor en F´ısica. Asinado: Jos´e D. Edelstein Glaubach Compostela, maio do 2013 ix “Unha serea que canta de noite polos tellados, e un astronauta en bicicleta que no asomar das estrelas ven beijarlle as palmas das mans. Nese intre, no espello fr´ıo da l´ua acendes a noite, e ardemos. Eu creo que foi as´ı como naceu o Universo” [Patrieira, Big Bang] A todos os que confiaron en min, m´ais do que eu mesmo. xi “We are like dwarfs standing upon the shoulders of giants, and so able to see more and see farther than the ancients.” Bernard of Chartres Agradecementos . . . e at´e aqu´ı chegou o cami˜no, un ronsel entre tantos, fin de etapa, porto de abrigo. Tempo de ollar atr´as antes do seguinte paso, a pr´oxima traves´ıa. Porque un nunca est´a s´o na s´ua viaxe, porque cada persoa que cruzou o noso carreiro, cada compa˜neiro de andaina, deixa pegada e leva un anaco de n´os. Ubuntu, eu son porque n´os somos. Un non ´e, non se pode entender, sen todas as persoas que vai atopando ao cami˜nar. Mari˜neiro son, coma meu bisav´o, meu av´o e meu pai; por´en un non pode navegar s´o. Nom pode un sair ao mar sen tribo, sen porto, sen barco e mans amigas, sem miolos e sem forza. ´ E tempo de lembrar e adicar unha verba a todos os que fixeron que eu hoxe poida estar escribindo estas li˜nas. A todos os moitos e bos mestres que tiven. Eles ensin´aronme que vivir ´e procurar o propio cami˜no. Hoxe que inventar novos cami˜nos ´e m´ais importante ca nunca, hoje que nos est´an a retirar o enlousado de baixo os p´es. ´ E tempo de voltar cami˜nar sobre a herba. A eles por soprar as velas da mi˜na curiosidade insaci´abel. A toda a xente de Compostela, xa mi˜na segunda aldea, campo base. Aos compa˜neiros dos anos da carreira: Patxi, Patri, Edu, Gonza, Luc´ıa, Xe, Celes, Vane, Jesus, Lionel, Brais, Vero, Meri, Rub´en, ´ Angel, Gemma... e tantas e tantos outros con quen compart´ın conversas, ceas, troulas... e mesmo alg´un escenario do QMF (que ousad´ıa!). Todos me vistes medrar para ser quen son hoxe. A todos os f´ısicos, Paolo, John, Jose, Ricardo, Josi˜no, Alfonso, Javier, Tarr´ıo, Daniel... ´a FROGsS! Mesmo os meteor´ologos. Todos me arroupastes nos meus primeiros pasos coma f´ısico? te´orico? Literatos do m´ais mi´udo e o m´ais grande, de todo o que escapa aos sentidos, a´ında aos m´ais sofisticados aparellos. Quen dixo que poderiamos sequera enxergar tales cousas? Pouco m´ais do que simios ollando ao ceo, do cabo do mundo escoitar o universo. Que poder´ıa aportar eu? Mais houbo quen confiou en min, deume unha palmadi˜na nas costas e dixo – ti podes. Grazas Jose por ensinarme novos mares, na f´ısica e na vida, polo teu entusiasmo contaxioso, pola t´ua paciencia e o teu apoio. Grazas por compartires as t´uas grandes ideas e por escoitar sempre as mi˜nas1. 1Creo que desta non me libro, para o pr´oximo asado conta co tiramis´u. xiii Moitas veces o cami˜no levoume lonxe da costa, Chamb´ery, Cambridge, Porto, Waterloo, Amsterdam, Buenos Aires, Santiago de Chile ou Princeton. De todos estes lugares gardo lembranzas imborr´abeis, amigos que a´ında lonxe me acompa˜nar´an sempre. A Sonya e Letizia, a Stephan, Savan, Pilar ou Valentin, a toda a familia do outro lado do mar. A todos mil milleiros de grazas, por acollerme cos brazos abertos, por cami˜nar comigo, por darme un empurr´on no momento certo. Grazas tam´en a todos os grand´ısimos f´ısicos cos que tiven a sorte de traballar, todos contribu´ıron enormemente a expandir os meus horizontes cient´ıficos. A Miguel, Rob, Alex, Jan, Juan e Sasha. E menci´on especial para Gast´on e Andy, por axudarme a co˜necer algunhas das moitas marabillas que o cono sul ten para ofrecer. Foi todo un pracer poder colaborar con todos v´os e espero que poidamos seguir traballando xuntos no futuro. Deixo para o final a raiz, o cerne, a madeira m´ais dura, a que aguanta o peso. Sempre ser´a pouco o que poida agradecer ´a familia e os amigos, por confiar sempre en min m´ais do que eu mesmo, pola paciencia e o apoio que sempre me brindaron. ´ A mi˜na nai, pola s´ua forza sosegada, o son dos seus pasos ao longo dun cami˜no que nen sempre soubo onde ´ıa levar. E grazas a mares mi˜na Patrieira, meu rumbo, por cami˜nares comigo, por ollares ao futuro aos ollos. Grazas porque sen ti non dar´ıa chegado at´e aqu´ı, sen o teu ollar sobre o mundo, sen o teu pulo creador de universos, mi˜na inspiraci´on. Grazas a todos! xiv Contents Foreword xix I LOVELOCK THEORIES &BLACK HOLES 1 1Lovelock theories of gravity 3 1.1 Lovelock gravity .................................. 7 1.2 Gauss-Bonnet theorem and boundary terms .................. 12 2Lovelock black holes 21 2.1 Black holes in Lovelock gravity ......................... 23 2.2 Branches ...................................... 25 2.3 Singularities and horizons ............................ 27 2.4 Black holes in third order Lovelock theory ................... 36 2.5 Charged and rotating solutions ......................... 43 2.6 Lovelock cosmologies ............................... 49 3Black hole thermodynamics 55 3.1 The path-integral approach to Quantum Gravity ................ 58 3.1.1 Spacetime complexification and thermodynamics ............ 58 3.1.2 AdS spacetime and the canonical ensemble ............... 60 3.2 Lovelock black holes and thermodynamics ................... 62 3.2.1 Vacuum horizons and Einstein-Hilbert gravity ............. 67 3.2.2 Black hole entropy at extremality .................... 69 3.3 Taxonomy of Lovelock black holes ........................ 71 3.3.1 The Einstein-Hilbert branch ....................... 71 3.3.2 AdS (other than EH) branches ...................... 74 3.3.3 dS branches ................................ 77 3.4 Heat capacity and local thermodynamic stability ................ 79 3.4.1 Black holes in the EH-branch ...................... 79 3.4.2 Hyperbolic black holes in the AdS-branches .............. 83 3.4.3 Spherical black holes in the dS-branches ................ 83 3.5 Hawking-Page-like phase transitions ....................... 84 3.5.1 Spherical black holes ........................... 85 3.5.2 Hyperbolic black holes .......................... 86 xv xvi CONTENTS 3.6 Discussion ..................................... 87 4Metric perturbations and stability 91 4.1 Graviton potentials ................................ 92 4.2 Black hole instabilities .............................. 94 4.3 Stability of new uncharged extremal black holes ................ 98 4.4 The Cosmic Censorship Conjecture and stability ................ 100 4.5 Instabilities and black hole evaporation ..................... 103 4.6 Discussion ..................................... 104 5Bubbles and new phase transitions 107 5.1 Higher order free particle ............................. 109 5.2 Generalized Hawking-Page transitions ...................... 112 5.3 Junction conditions ................................ 113 5.3.1 Thermalon configuration ......................... 114 5.3.2 Junction conditions and bubble dynamics ................ 116 5.3.3 Bubbles, horizons and the cosmic censor ................ 122 5.4 Thermalons and thermodynamics ........................ 123 5.4.1 LGB thermalons ............................. 128 5.5 Generalized Hawking-Page transitions in Lovelock theories of gravity . . . . 130 5.5.1 Spherical LGB bubbles .......................... 134 5.5.2 Hyperbolic LGB bubbles ......................... 135 5.6 Discussion ..................................... 140 II HIGHER CURVATURE GRAVITY &HOLOGRAPHY 145 6AdS/CFT and Lovelock gravity 147 6.1 The Maldacena conjecture ............................ 149 6.1.1 Correlation functions ........................... 152 6.1.2 Confinement/deconfinement phase transition .............. 153 6.1.3 Effective Conformal Theories and local AdS dynamics ......... 154 6.2 Lovelock theories and holography ........................ 156 6.2.1 CFT unitarity and 2-point functions .................. 156 6.2.2 Three-point function and conformal collider physics .......... 157 6.3 Shock waves in higher curvature gravity .................... 162 6.3.1 Probing shock waves in general theories of gravity ........... 165 7Causality and positivity of energy 169 7.1 Holographic causality and black holes ...................... 173 7.2 Gravitons and shock waves ............................ 175 7.3 Causality constraints in cubic Lovelock theory ................. 179 7.4 Causality for higher curvature vacua ...................... 183 7.5 Discussion ..................................... 186 Part I LOVELOCK THEORIES &BLACK HOLES 1 Chapter 1 Lovelock theories of gravity “Imagination will often carry us to worlds that never were. But without it we go nowhere” Carl Sagan The General Theory of Relativity [13] is one of the greatest scientific accomplishments of the XXth century. It was born from the need to reconcile the Newtonian laws of the gravitational interaction with the new paradigm of the Special Theory of Relativity [14]. It was pursued independently at the same time by two of the greatest minds of that time, Albert Einstein and David Hilbert, reason of the name of the action giving rise to the equations of the theory. Two basic ideas stand behind this extrordinary mathematical construction, the Special Theory of Relativity mentioned before and the Principle of Equivalence. On its weakest version the latter is just the observation of the exact equivalence between inertial and gravitational mass, two very different concepts with exactly the same value. This equality allows, at any point of spacetime, to choose a locally inertial reference frame such that the effect of the gravitational force is completely screened at that point by an equal and opposite acceleration. This in turn motivated the Strong Equivalence Principle that moreover asserts that in a small enough neighbourhood of that point the laws of nature, not just those of dynamics, take the well known form of Special Relativity without gravity. In other words it is impossible to tell the difference locally between an experiment in the presence of gravitational forces and the same experiment in an accelerated laboratory. Gravity cannot be avoided globally in this way as we would need to give different accelerations to different points. The view of spacetime from this logic is that of a curved manifold whose metric parameterizes the gravitational interaction. The spacetime is no longer the inert scene for all physical phenomena to become the dynamical fabric of the universe. The set of transformations under which the laws of Physics must be invariant are enlarged to general changes 3 4CHAPTER 1. LOVELOCK THEORIES OF GRAVITY of coordinates that include, of course, Lorentz transformations as a particular example. In a sense the Strong Equivalence Principle states that the laws of physics are independent of the coordinates chosen to describe them, whether they correspond to inertial or non-inertial observers. The source of the gravitational field is the matter content of the spacetime or, more specifically, the induced stress energy tensor. The mass that entered the Newtonian theory, equivalent to energy through the celebrated E=mc2, is just one of its components. Once the field carrying the gravitational force and its sources have been identified, the final piece of information needed for a complete description of the classical theory is the choice of action that encodes the dynamics of the interaction. There is `a priori a plethora of lagrangians that realize the requirement of general covariance and are viable candidates. Nonetheless if we restrict the possibilities to those yielding second order equations of motion the choice becomes almost unique in four dimensions. In particular we require the field equations to be of the form Gµν(gαβ, gαβ,γ, gαβ,γλ) = Tµν (1.0.1) where the left hand side is a tensor valued local functional of its local arguments, symmetric and conserved Gµν ;ν= 0 .(1.0.2) in agreement with the analogous property for the stress energy tensor. Then Lovelock’s theorem [15] states that the possible equations reduce to Rµν −1 2gµνR+ Λgµν = 8πGNTµν (1.0.3) where the constant of proportionality is chosen in order to reproduce the correct Newtonian limit. These equations of motion arise from the Einstein-Hilbert (EH) action with cosmological constant coupled to matter, that can be nicely written in terms of differential forms as Id=4 =1 16πGNZ√−gR−2ˆ Λ+Imat .(1.0.4) The cosmological constant, ˆ Λ, was first introduced by Einstein [16] in order to describe a stationary universe, what he later referred to as his greatest blunder, once the observation of the Hubble redshift made clear the Universe is actually expanding. The actual value of ˆ Λ in the observed Universe is not zero though, and a number of observations, including the discovery of cosmic acceleration, have revived the cosmological constant. The ΛCDM model of the Universe, the most accepted modern cosmological model to date, asserts that ˆ Λ is positive, although negligible even by the scale of our galaxy the Milky Way. In a much more general context, this constant will also play a very important rˆole in our discussion although the most interesting case for us will be that of negative ˆ Λ. Another possible characterization of the EH lagrangian, valid as well in higher dimensions, is that of the corresponding field equations being linear in second order derivatives of the metric [17–19]. This restriction together with the above requirements singles out, in any dimension, the action of General Relativity. In dimensions greater than four, there are however other tensors admissible if this linearity condition is relaxed, the Lovelock lagrangians. The modified equations of motion will then just be quasi-linear (see [20] for a 5 detailed definition), quasi-linearity implying the absence of squared or higher order terms in second derivatives of the metric with respect to a given direction. This is important in order to have a well defined initial value problem for gravity. The coefficient of this second derivatives can depend however on first derivatives of the metric and may vanish for this reason, leaving the second derivative in question indetermined. In [20] some of the problems which may arise because of the quasi-linearity of the Lovelock equations are discussed. Lanczos [21,22] in 1932 found a generalization of the EH lagrangian quadratic in the Riemann tensor and whose equations of motion are symmetric, conserved and second order in the metric. Yet another property of the EH lagrangian is that it is a pure divergence in two dimensions and the Einstein tensor vanishes identically in one and two dimesnions. Similarly, the Lanczos, or Lanczos-Gauss-Bonnet (LGB), lagrangian is a pure divergence and that the corresponding equations are identically zero in four or less dimensions. The LGB term is the Euler density appearing in the Gauss-Bonnet theorem [11] in four dimensions. Lovelock [15] generalized these results in 1971 and obtained, for any dimension, a formal expression for the most general, symmetric and conserved tensor which is quasi-linear in the second derivatives of the metric without any higher derivatives. He also found the lagrangian from which that tensor derives: in d-dimensions it corresponds to a linear combination of the d−1 21dimensionally continued Euler densities. In dimensions 5 and 6, the explicit form of the Lovelock Lagrangian reduces to a linear combination of the EH and LGB lagrangians (with the possible addition of the cosmological constant). The Lovelock lagrangians are, due to their properties, the most natural generalization of that of Einstein and Hilbert to describe pure gravity in dimensions larger than four. In physics, actions are built based on general principles, such as symmetry, causality and other consistency requirements. All terms satisfying these and built from the appropriate fields should then be included in the Lagrangian. In this sense, there is no `a priori reason2, why higher order Lovelock terms should be excluded from the action. The dimensionful couplings of the theory increase their length dimension with the order in curvatures in such a way that higher order contributions become important at short distances (or high energies) while solutions of Lovelock gravity reduce to those of General Relativity asymptotically. In this thesis we will be mainly concerned with gravity theories of the Lovelock family. As mentioned before these only contribute to the gravitational dynamics in dimensions five or higher3so that we need to first answer a pressing question. Why should we be interested in spacetimes with dimensions different from the four known to our experience? The idea of higher dimensional spacetimes goes back to the groundbreaking papers of Kaluza [24] and Klein [25] but most of the present renewed interest comes from the advent of string theory. Inspired by the physics of strings and other motivations, much effort has been devoted in the last quarter of a century in high energy physics dealing with scenarios involving higher dimensional gravity, and it is fair to say that at present it is unclear if gravity is a truly 1Square brackets denote the integer part of a number. 2Even for general covariant actions, the issue of causality in gravity is a non-trivial one, as we will analyze in the second part of this thesis. The same happens for non-gravitational theories [23] where covariance does not imply causality in relativistic quantum field theories with higher dimensional terms. 3In some cases they may contribute to the equations of motion when coupled to other fields in lower dimensions. 6CHAPTER 1. LOVELOCK THEORIES OF GRAVITY four-dimensional interaction. On the other hand, the inclussion of terms non-linear in the curvature modifying the EH lagrangian is an idea first proposed by Weyl [18] and Eddington [26]. Of course, these extra terms introduce contributions with derivatives of the metric up to the fourth. In the seventies and early eighties such quadratic Lagrangians were exploited in view of renormalizing the quantized theory of linearized general relativity (see e.g. [27] for a review of that period) as well as to renormalize the stress-energy tensor of quantized matter fields in classical, curved, backgrounds, see [28] for a review. They however made their most forceful entrance again when it was shown that they should arise as next-to-leading corrections to the low energy limit of string theory. In particular, the simplest Lovelock lagrangian, the LGB term, has been explored to a large extent, mainly as a consequence of its appearance in this context [29]. Since their inception, a steady attention has been devoted to scrutinize the main properties of Lovelock theories of gravity, their vacuum structure, induced cosmologies, hamiltonian formalism, dimensional reduction, wormhole configurations and, most importantly, their black hole solutions, including their formation, stability and thermodynamics. In spite of the abundant literature on the subject, most articles deal with particular cases of the general Lovelock formalism due to the intricacy endowed by the increasing number of coupling constants: there are [d−3 2] dimensionful quantities (alongside the Newton and cosmological constants) in a d-dimensional theory. For this reason, many investigations on black hole solutions of Lovelock gravities are restricted to one-parameter (zero measure) subspaces in the space of couplings. It is the aim of the first three chapters of this thesis to tackle the existence and main features of Lovelock black holes for arbitrary values of the full set of gravitational couplings. We will be dealing with arbitrary orders in the Lovelock action and arbitrary dimensions, most of our results being completely general. We will just concentrate in specific examples for illustrative purposes or when the intricacy of the equations requires. Despite its debatable phenomenological interest, Lovelock gravities provide an interesting framework from a theoretical point of view for several reasons. As higher dimensional members of Einstein’s general relativity family, they allow to explore several conceptual issues of gravity in depth in a broader setup. Among these, we can include features of black holes such as their existence and uniqueness theorems, their thermodynamics, the definition of their mass and entropy, etc. Lovelock theories are perfect toy models to contrast our ideas about gravity. A final piece of motivation comes from the theoretical framework proposed by Juan Maldacena [30]. This will be the object of the second part of the thesis. The AdS/CFT correspondence establishes a holographic identification between conformal field theories and quantum gravities in higher dimensional AdS spaces. Besides its original maximally supersymmetric formulation, the correspondence seems robust enough to survive its generalization to less supersymmetric scenarios [31], and even non-supersymmetric [32], as well as nonstringy realizations [33] (see also the seminal paper [34]). In particular, even if some caution remarks should be quoted at this point, the AdS/CFT correspondence seems to apply in higher dimensions too. We know very little about non-trivial conformal field theories in higher dimensions (see [35] for a recent discussion). The interest of the AdS/CFT correspondence in this context is twofold. It provides an effective definition of higher dimensional CFTs from the gravity 1.1. LOVELOCK GRAVITY 7 side, whereas, in the opposite sense it opens new perspectives in the gravitational dynamics and its quantization. In the particular case of Lovelock gravity this effective approach has yielded some unsuspected surprises in the form of very interesting connections. These will be reviewed in the second part of the thesis and involve some central concepts in physics, such as positivity of energy and causality [1,2,36–38]. This results also motivated the discovery of new relations between unitarity and causality in CFTs [39]. Applications of AdS/CFT towards the understanding of the hydrodynamics of CFT plasmas in arbitrary dimensions demand a proper understanding of Lovelock black holes in AdS. This provides the final bit of motivation to pursue the present investigation. Regardless of the phenomenological dimensionality required by these applications, it is customarily the case that pushing some ideas to their extremes, besides verifying their robustness, allows to discover novel features that are hidden in the somehow simpler original formulation (see, for instance, [40] for a beautiful recent example of this statement). 1.1 Lovelock gravity Lovelock theories of gravity are the most general second order gravity theories in higherdimensional spacetimes. They have the same degrees of freedom as General Relativity and it is free of higher derivative ghosts [15,41]. The bulk action has a very complicated form in terms of the Riemann tensor and its contractions, nonetheless it can also be written very simply in terms of differential forms as I=1 16πGN(d−3)! K X k=0 ck d−2kZLk,(1.1.1) GNbeing the Newton constant in dspacetime dimensions. {ck}is a set of couplings with length dimensions L2(k−1),Lbeing a length scale related to the cosmological constant, while Kis a positive integer, K≤d−1 2,(1.1.2) labeling the highest non-vanishing coefficient, i.e.,ck>K = 0. Lkis the exterior product of k curvature 2-forms with the required number of vielbein, ea, to construct a d-form, Lk=Rked−2k=f1···fdRf1f2...f2k−1f2k∧ef2k+1...fd.(1.1.3) The zeroth and first term in (1.1.1) correspond, respectively, to the cosmological term and the Einstein-Hilbert action. It is fairly easy to see that c0=L−2and c1= 1 correspond to the usual normalization of these terms, the cosmological constant having the customary value 2ˆ Λ = −(d−1)(d−2)/L2. Either a negative (c0=−L−2) or a vanishing (c0= 0) cosmological constant can be easily incorporated as well. The first non-trivial Lovelock term contributes just for dimensions larger than four and corresponds to the LGB coupling c2=λL2. The Lovelock action written in this way has the advantage that it can be equivalently considered in first order formalism, i.e. we can consider the vielbein and the spin connection as independent variables. We then have two equations of motion, one for each field. First, 8CHAPTER 1. LOVELOCK THEORIES OF GRAVITY varying the action with respect to the connection 1-form the resulting equation is proportional to the torsion. We may use all the technology of exterior algebra and treat exterior covariant derivatives as normal derivatives inside the brackets. We can then integrate by parts to show, δωLk=k  D(δω)Rk−1ed−2k(1.1.4) =kd (δωRk−1ed−2k)−k(d−2k)(δωTRk−1ed−2k−1) where we have used that δωRab =D(δωab) and the Bianchi identity DRab = 0. The first term in the above variation is a total derivative and does not contribute to the equations of motion whereas the second is proportional to the torsion. We may safely restrict to the torsionless sector as usual, allowing us to compare our results with those coming from the tensorial formalism based on the metric. The second equation of motion is obtained by varying the action with respect to the vielbein. It can be cast into the form Ea≡af1···fd−1cKFf1f2 (1) ∧···∧Ff2K−1f2K (K)∧ef2K+1...fd−1= 0 ,(1.1.5) where Fab (i)≡Rab −Λiea∧eb. This expression involves just the curvature 2-form and no extra covariant derivatives, making explicit the two derivative character of the Lovelock equations of motion. Also, for the critical dimension d= 2k, the kth term contribution to the equations vanishes. In our approach this is simply due to the absence of vielbeins in the corresponding action term, thus yielding zero upon variation. More generally, the integral of that term becomes a topological invariant, the Euler number for that particular dimension. We will comment more on this in the next section. In dimensions lower than the critical one the corresponding Lovelock term exactly vanishes and we are led to the restriction (1.1.2). Besides, (1.1.5) makes manifest that, in principle, this theory admits Kconstant curvature vacuum solutions, Fab (i)=Rab −Λiea∧eb= 0 .(1.1.6) Inserting Rab = Λ ea∧ebin (1.1.5), one finds that the Kdifferent cosmological constants are the solutions of the Kth order characteristic polynomial Υ[Λ] ≡ K X k=0 ckΛk=cK K Y i=1 (Λ −Λi) = 0 ,(1.1.7) each one corresponding to a different vacuum, positive, negative or zero for dS, AdS and flat spacetimes. The effective cosmological constants correspond to the possible radii of these (A)dS spaces and should not be confused with the bare cosmological constant, ˆ Λ appearing in the action. The theory will have degenerate behavior whenever two or more effective cosmological constants coincide. This is captured by the discriminant, ∆ = K Y i<j (Λi−Λj)2,(1.1.8) 1.1. LOVELOCK GRAVITY 9 that vanishes in a certain locus of the parameter space corresponding to the coupling constants of Lovelock theory where some special features arise. The discriminant can be written as well in terms of the first derivative of the Lovelock polynomial, Υ, as ∆ = 1 cK K K Y i=1 Υ0[Λi].(1.1.9) As we move forward through the text it will become clear the preeminent rˆole played by this polynomial in the most diverse situations. For the sake of clarity let us briefly consider the K= 2 case. In LGB gravity there are two possible values of the effective cosmological constant Λ±=−1±√1−4λ 2λL2,(1.1.10) and they agree when the discriminant ∆≡(Λ+−Λ−)2=1−4λ λ2L4= 0 for λ=1 4,(1.1.11) vanishes. This implies that, for 1 −4λ > 0, there are two (A)dS vacua around which we can define our theory. If 1 −4λ < 0, there is no constant curvature vacuum. For the exact value 4λ= 1, the theory displays a degenerate behavior due to symmetry enhancement. In the particular case of d= 5, the symmetry enhances to the full SO(4,2) group and the expression (1.1.1) gives nothing but the Chern-Simons Lagrangian for the AdS group [42] (see also [43]). It is a well-known fact in LGB gravity that one of the vacua, the one with the + sign in front of the square root, leads to negative mass black holes with a naked singularity that signals the instability of the vacuum [44]. We are thus led to the remaining branch of solutions, so called EH-branch as it is continuously connected to the solution of General Relativity as λ→0. Another property of any degenerate vacuum is the absence of linearised gravitational degrees of freedom about it. The equations of motion for a metric perturbation, hab, around a given vacuum, Λ1, are easily obtained from the perturbation of the curvature Rab = Λ1eab +δgRab (1.1.12) yielding Ea= Υ0[Λ1]af1···fd−1δgRf1f2ef3···fd−1(1.1.13) to the linear level, thus exactly zero as the first derivative of Υ vanishes for a degenerate vacua, Υ0[Λ1] = cKY i6=1 (Λ1−Λi)=0.(1.1.14) Moreover, it is easy to verify that the equations of motion around a non-degenerate vacuum are exactly the same as for Einstein-Hilbert gravity multiplied by a global factor proportional to Υ0[Λi]. The propagator of the graviton corresponding to the vacuum Λiis then proportional to Υ0[Λi] in such a way that when Υ0[Λi]<0 it has the opposite sign with 10 CHAPTER 1. LOVELOCK THEORIES OF GRAVITY respect to the Einstein-Hilbert case and thus the graviton becomes a ghost. This generalizes the observation first done by Boulware and Deser [44] in the context of LGB gravity. Thus, a given vacuum of Lovelock gravity, Λ1, must satisfy Υ0[Λ1]>0 (1.1.15) in order to correspond to a vacuum that hosts gravitons propagating with the right sign of the kinetic term. See [45] for a recent discussion on the subject. In the non-degenerate case the number of degrees of freedom about any of these vacua is exactly the same as in General Relativity. Curiously enough, most of the studies in the context of Lovelock theory have been performed within the degenerate locus, ∆ = 0. In this thesis however we will aim at making the complementary effort of digging into the non-degenerate case Υ[Λk] = 0, Υ0[Λk]6= 0, where Λkis the vacuum under consideration. We will eventually see that, among the branches of solutions of (1.1.7), only one would end up being physically relevant, say Λ = Λ?. Degeneracies that do not involve Λ?are harmless, our analysis being thus valid for the whole parameter space, except the zero measure set Υ[Λ?]=Υ0[Λ?] = 0. As the simplest examples of the Lovelock family, we will later focus in the LGB and third order Lovelock lagrangians. Let us discuss in some detail the cubic case. The lowest dimensionality where this term arises is 7d reducing in lower dimensions to LGB gravity. Consider the following action, I=1 16πGNZddx√−gR−2ˆ Λ + (d−5)! (d−3)! λ L2L2+(d−7)! (d−3)! µ 3L4L3,(1.1.16) where the quadratic and cubic Lagrangians are L2=R2−4RµνRµν +RµνρσRµνρσ ,(1.1.17) L3=R3+ 3RRµναβRαβµν −12RRµνRµν + 24RµναβRαµRβν + 16RµνRναRα µ + 24RµναβRαβνρRρ µ+ 8Rµν αρRαβ νσRρσ µβ + 2RαβρσRµναβRρσ µν .(1.1.18) This complicated tensorial expression can be casted very simply in the language of our previous discussion as (c2=λL2and c3=µ 3L4) I=µ L4 3(d−6) Zabcdefg1···gd−6Rabcdef +d−6 d−4 3λ µ L2Rabcd ∧eef +d−6 d−2 3 µ L4Rab ∧ecdef +d−6 d 3 µ L6eabcdef ∧eg1···gd−6,(1.1.19) whose equations of motion, once the torsion is again set to zero, can be written as: a···fg1···gd−6Rab −Λ1eab∧Rcd −Λ2ecd∧Ref −Λ3eef ∧eg1···gd−7= 0 .(1.1.20) 1.2. GAUSS-BONNET THEOREM AND BOUNDARY TERMS 17 Another important rˆole of the boundary terms in General Relativity is that they give rise to the so called Israel junction conditions that govern the dynamics of shells separating different bulk domains [49] K+ AB −K− AB = 8πGNSAB −1 2ηABS(1.2.35) By performing an integration of Einstein’s equations accross the shell and taking the thickless limit it is possible to show that the jump in the extrinsic curvature is related to the surface stress-energy tensor, SAB. It can be shown that Israel’s method for singular hypersurfaces is equivalent to an action principle with boundary terms at the hypersurface. In this case the variation of the metric evaluated there should not be fixed, its variation giving the junction conditions [50]. The Lovelock action in the presence of singular hypersurfaces can also be written in terms of smooth bulk integrals plus a boundary term [51–53], that in the case of n= 2 LGB theory was first written down by Davis [54]. One may consider the spacetime manifold as the union of two submanifolds with a commom boundary Σ in such a way that in addition to the boundary term at infinity we get two extra surface terms at the boundary between the two. The action in this case would be written as Itot = (I−−I− ∂)+(I++I+ ∂)−I∂(1.2.36) where −(+) denote the inner (outer) region. Remark that the contribution of the hypersurface Σ can be written as the sum of two boundary terms of the same form seen above IΣ=−I− ∂+I+ ∂(1.2.37) the plus sign in the second term coming from the fact that the bulk regions induce opposite orientations on the wall5. In case the spin connection is continuous in Σ the surface terms cancel and we recover the usual Lovelock action. This way of writting the action is useful in order to find solutions where the spin connection is discontinuous across some co-dimension one hypersurface (the metric being continuous). The variation of the bulk terms on each side yield the usual Lovelock equations of motion while the junction conditions arise from the variation of the boundary terms with respect to the induced vielbein field or equivalently the pullback of the metric on Σ. We will comment more on this on section 5where we will be interested in distributional solutions. The issue of finding equations with singular sources is a non-trivial one in non-linear gravity theories as many operations with distributions are not unambiguously defined. In order to take the variation of the action written in this way we have to also vary with respect to the metric and the spin connection induced in the intermediate surface. The variation with respect to the intrinsic spin connection is again zero as it cancels the boundary term coming from the bulk integrals (this is the reason we introduced these terms 5One can also construct solutions with the same orientation on both sides leading in turn to wormholes and Randrall-Sundrum-like models [55,56] 18 CHAPTER 1. LOVELOCK THEORIES OF GRAVITY to begin with) whereas the variation with respect to the intrinsic metric is proportional to the canonical momenta in such a way that the variation of each term may be written as δI∂=−Z∂M dd−1x πABδhAB .(1.2.38) Therefore, the equations of motion for the surface Σ amount just to the continuity of the canonical momenta [57–59]. The analogous term from the boundary does not contribute as we keep the boundary metric fixed, δh = 0. The canonical momenta in Lovelock gravity can be expressed as πB C=−1 2 δI∂ δeC∧eB(1.2.39) = K X k=1 k ckZ1 0 dξ KA1∧RA2A3 0(ξ)∧. . . ∧RA2k−2A2k−1 0(ξ)eA2k···Ad−2BA1···Ad−2C, The generalization of the Israel junction conditions to Lovelock gravity being π+ AB −π− AB = 8πGN(SAB −1 2ηABS) (1.2.40) that reduces to (1.2.35) in the case of EH gravity. Junction conditions can also be seen to arise in our 1-dimensional example. First of all, the same kind of boundary term appears if we split the action, or the interval of integration, in two. The variation of each term in that case is δI−=Zt? t1 dt ∂L ∂q −d dt ∂L ∂˙qδq +∂L ∂˙qt? δq?(1.2.41) The first term vanishes because of the equations of motion but the second does not as the position is not fixed for t=t?. That term combines from an analogous one coming from the other part of the action, I+, to yield the junction condition p−−p+= 0 (1.2.42) i.e. continuity of the canonical momentum. For univalued momentum such as in (1.2.1) this in turn implies the continuity of the velocity. The free particle is necessarily continuous without any need of imposing this continuity `a priori. There is no discontinuous solution of ¨x= 0. In the same way as for EH gravity, in order to add a discontinuity we have to include new source terms. In the particle example the analogous of a dust shell would be localized at a given time, t= 0 for simplicity. ˜ I=I+Zt2 t1 dtλxδ(t) = I+λx(0) (1.2.43) When varying the action, fixing xin the borders, x(t1,2) = x1,2we get the equation d dt ∂L ∂˙x−∂L ∂x =λδ(t) (1.2.44) 1.2. GAUSS-BONNET THEOREM AND BOUNDARY TERMS 19 This equation includes junction conditions that can be found by integrating in infinitesimal region around t= 0, entre t= 0−and t= 0+. In this way we find p+−p−=λ(1.2.45) where p±=p(0±). We could also have started with the splitted action in which case the junction conditions may arise from the variation on the boundary, in this case the variation of x(0) which is not fixed by the boundary conditions. In order to keep the discussion as general as possible we will consider the action written in terms of the canonical variables qand pinstead of the velocity L(q, p) = p˙q−H(q, p) (1.2.46) In this way the lagrangian can be varied independently with respect to the two canonical variables, qand p, yielding the well known Hamilton equations, ˙p=−∂H ∂q ; ˙q=∂H ∂p (1.2.47) In the same way as before this action is prepared to fix the value of the position at the extrema, δq = 0. However we can also use a different lagrangian L0(q, p) = −˙pq −H(q, p) (1.2.48) that in turn is prepared to fix the momenta pinstead. This can be easily understood as a result of the symmetry q↔p, ˙q↔ −˙p. However we can supplement this lagrangian (1.2.48) with a boundary term in such a way that it is equivalent to (1.2.46) L(q, p) = L0(q, p) + d dt(pq) (1.2.49) Obviously the right hand side is obtained from the original lagrangian just by integrating by parts. The nice thing about this way of writting the lagrangian is that now we can split a given interval in two pieces and vary the action not just with respect to the bulk variables but also with respect to the ones at the splitting surface,t= 0, I0=Zt? t1 dt L0(q, p)+(p−−p+)q?+Zt2 t? dt L0(q, p) (1.2.50) The variation with respect to the boundary momentum vanishes automatically and so do the variations inside the intervals (t1, t?) and (t?, t2) due to the equations of motion. The only contribution to the variation thus comes from the intermediate boundary term at t=t? δI0= (p−−p+)δq?(1.2.51) again implying the continuity of the canonical momentum accross the discontinuity. In the same way as before we may add source terms that induce jumps in the canonical momentum in the same way as matter in GR. As we mentioned above, for univalued momentum continuity of momentum implies the continuity of the the velocity as well. In more general cases however, the momentum may be multivalued this becoming a non-trivial equation. The velocity may jump as long as the canonical momentum is conserved. We will comment more on this on chapter 5, with a specific example. Chapter 2 Lovelock black holes “To myself I am only a child playing on the beach, while vast oceans of truth lie undiscovered before me.” Isaac Newton The concept of singularity is central to General Relativity. Due to the attractive and universal nature of the gravitational interaction, the theory predicts that these kind of objects inevitably form, either in the form of black hole or as a cosmological singularity such as the Big Bang. The first to describe a singular solution was Karl Schwarzschild [60] as soon as 1916, a little more than a month after the publication of Einstein’s original paper. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I. The singular character of the solution he found was at first considered just as a mathematical curiosity, of none physical relevance, until it was realized quite a long time afterwards that such objects actually do generally form from the collapse of matter [61,62] such as that of a dying star. Any physical object whose radius Rbecomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole. However it was not until the sixties, with the advent of the singularity theorems of Hawking and Penrose [63,64], that the debate was definitely settled. In short, a black hole is a self-gravitating object so densely packed that nothing, not even light, can scape its gravitational attraction. Nowadays black holes are thought to be quite commom objects in the universe being generally present at the center of galaxies such as the Milky Way. They cannot be directly seen but their presence is detected through the trajectories of stars on their vicinity or radiation coming from their accretion disks. The fact that the gravitational field can affect the trajectory of light rays is well known. 21 22 CHAPTER 2. LOVELOCK BLACK HOLES In fact it was the way Eddington proved Einstein theory right in his famous 1919 expedition to Africa. The concept of black hole as that of an object from which not even light can scape is much older though. We can trace it back as far as 1783, to a letter [65] John Michell sent to Henry Cavendish, his fellow at the Royal Society of London. In that letter, using just Newtonian gravity, Michell describes the hypothetical case of a heavenly object massive enough to prevent light from scaping. Michell calculated that when the escape velocity at the surface of a star was equal to or greater than lightspeed, the generated light would be gravitationally trapped, so that the star would not be visible to a distant astronomer. He named his discovery dark star, the precursor of black holes. For an object leaving the surface of a dark star of mass Mwith some speed vto reach infinity we need the sum of its kinetic and gravitational energy to be equal or greater than zero, 1 2mv2−GNMm R≥0⇒v2 s=2MGN R(2.0.1) in such a way that the radius of the dark star has to be smaller than R≤RS≡2MGN c2(2.0.2) which, curiously enough, is independent of the mass of the object and actually coincides with the Schwarzschild radius of General Relativity, r= 2Min geometric units. In the context of GR, this particular radial position is named event horizon. The concept of black hole is quite different from that of a dark star. Nothing sent from the dark star can reach infinity but it can leave the star and even reach infinity if we furnish some extra acceleration. The black hole however is provided with an event horizon that acts as a one way membrane in the sense that objects can get into the horizon but they cannot get back out. More precisely, consider the form of the Schwarzschild metric in General Relativity ds2=−1−2M rdt2+dr2 1−2M r+r2dΩ2.(2.0.3) The time and radial variables exchange their rˆoles beyond r= 2Min such a way that the rcoordinate becomes timelike and tspacelike. Being timelike, rhas to increase along any timelike trajectory in the same way as the time ticks forward outside the black hole. In fact, any object falling through the event horizon will reach the central singularity in finite proper time, it would be inevitably driven there. The event horizon plays yet another very important rˆole. As it prevents anything from leaving the black hole, it effectively divides the spacetime in two. Nothing happening inside the horizon can ever influence the dynamics of the exterior region. This is essential in order to have a well defined initial value problem in the presence of a singularity. The singularity represents a break down of the theory, in a sense, it is the place where General Relativity shows its failure. It is also the place where quantum effects become dramatically important so that we would need a quantum theory of gravity to disclose the dynamics at the singularity. The existence of the event horizon protects the exterior region from this unknown dynamics, the exterior evolution being always well defined. 2.1. BLACK HOLES IN LOVELOCK GRAVITY 23 We will study the analogous solution to that of Schwarzschild in the context of Lovelock theories of gravity in general. Finding an analytic black hole solution requires to explicitly solve a polynomial equation and we are certainly restricted by the implications of Galois theory; meaningly, quartic is the highest order polynomial equation that can be generically solved by radicals (Abel-Ruffini theorem). However, an implicit but exact solution can be found, and we develop some tools to extract all relevant information, mainly their horizon structure and thermodynamics. We will devote this chapter to present our proposal to deal with generic black holes in Lovelock theory, focusing in the case of LGB and cubic Lovelock for a detailed description. In the next chapter we perform a classiffication of all possible black hole solutions, including the case of asymptotically dS solutions, and all possible horizon topologies within maximally symmetric conffigurations. The analysis of these solutions for the Lovelock family may also provide some useful information about the dynamics of black holes in more general gravity theories. This is specially important due to the high nonlinearity of the field equations that make very difficult finding nontrivial exact analytical solutions of Einstein’s equation with higher derivative terms. In most cases, one has to adopt some approximation methods or find solutions numerically. In the last few months there were some papers constructing gravitational theories that share some compelling properties with Lovelock lagrangians [66,67]. In particular, these are lower dimensional theories displaying black hole solutions whose profile precisely correspond to Lovelock black holes [68,69]. In particular these theories allow for the addition of an extra term of degree K=d+1 2in odd dimensions [69], contributing in every way as the corresponding Lovelock term in higher dimensions. Some other higher order terms may be also added that do not change the form of the black hole solution. Some of the results of this paper are therefore of direct application to those cases as well. This is particularly interesting due to the fact that quasi-topological gravities are higher curvature theories in dimensions lower than their corresponding Lovelock cousins, thus the results are of interest in more ‘physical’ setups of AdS/CFT [70]. 2.1 Black holes in Lovelock gravity It has been shown in [71] that Lovelock theories admit asymptotical (A)dS solutions with non-trivial horizon topologies. We can consider for instance solutions with a planar or hyperbolic symmetry as a straightforward generalization of the usual spherically symmetric ansatz, ds2=−A(t, r)dt2+dr2 B(t, r)+r2 L2dΣ2 d−2,σ ,(2.1.1) where dΣ2 d−2,σ =dρ2 1−σρ2/L2+ρ2dΩ2 d−3,(2.1.2) is the metric of a (d−2)-dimensional manifold of negative, zero or positive constant curvature (σ=−1,0,1 parameterizing the different horizon topologies), and dΩ2 d−3is the metric of the unit (d−3)-sphere. This does not imply that the horizon is just spherical or non-compact. By means of the Killing-Hopf theorem [72], any complete connected Riemannian manifold of 24 CHAPTER 2. LOVELOCK BLACK HOLES Euclidean signature and constant curvature σcan be written as a quotient space Σd−2,σ/Γ, where Γ is a discrete subgroup of the isometry group of Σd−2,σ. Thus, even in (what we shall call) the spherical case, we have non-spherical possibilities; for example, one may take the horizon to be a lens space. Besides, planar or hyperbolic horizons can be made compact in this way. It has been proven in [73] that these black holes admit a version of Birkhoff’s theorem, in such a way that in addition to the SO(d−1), Ed−2or SO(1, d−2) isometry groups, these spacetimes admit an extra timelike killing vector (for A, B > 0). This means that these solutions of the field equations are locally isometric to their corresponding static counterparts, which can be found by means of the ansatz ds2=−f(r)dt2+dr2 f(r)+r2 L2dΣ2 d−2,σ .(2.1.3) There are extra solutions with different functions in the timelike and radial direction but they are just valid for degenerate values of the cosmological constant [74]. In that case, the most general solution is ds2=−f(r)dt2+dr2 (σ−Λr2)+r2 L2dΣ2 d−2,σ ,(2.1.4) for any function f(r). This allows in particular Lifshitz-like solutions f(r)∼r2zfor any value of the critical exponent z. These black hole solutions are all three asymptotic to a maximally symmetric space. Thus, when considering the same curvature for all of them they are locally asymptotically equivalent, but globally different. They are often referred to as topological black holes for this reason. Indeed, there are global changes of coordinates that relate the sets of coordinates corresponding to the three topologically different vacuum solutions associated with a given Λ [75]. Each set covers a different patch of AdS and has a different time coordinate. Thus, we can also look at the different topologies as static black holes for different classes of observers. Using the natural frame, e0=pf(r)dt , e1=1 pf(r)dr , ea=r L˜ea,(2.1.5) where a= 2, . . . , d −1, and ˜ Rab =σ˜ea∧˜eb. The Riemann 2-form reads R01 =−1 2f00(r)e0∧e1, R0a=−f0(r) 2re0∧ea, R1a=−f0(r) 2re1∧ea, Rab =−f(r)−σ r2ea∧eb.(2.1.6) If we insert these expressions into the equations of motion, we get d dlog r+ (d−1) K X k=0 ckgk!= 0 ,(2.1.7) 2.2. BRANCHES 25 where g= (σ−f)/r2. This can be readily solved as Υ[g] = K X k=0 ckgk=κ rd−1,(2.1.8) where κis an integration constant related to the mass of the spacetime [76,77], M=(d−2)Vd−2 16πGN κ , (2.1.9) Vd−2being the volume of the unit (d−2)-dimensional horizon. Notice that the polynomial giving the implicit black hole solution is the same as the one defining the possible vacua of the theory. This is not surprising as maximally symmetric spaces appear as massless solutions, M= 0. This can also be understood as follows. If there is actually a mass source for the gravitational equations of motion, ρ=Mδ(d−1)(r), therefore E0∧e0∼T0 0⇒d dlog r+ (d−1)κ rd−1∼ρ , (2.1.10) and, as the right hand side of the equation does not depend on the Lovelock theory we are considering, the left hand side cannot either. Thus, the relation between κand the mass must be the same as in Einstein-Hilbert gravity (2.1.9). This assertion can be made precise using the Hamiltonian formalism [77]. For arbitrary dimension, the spherically symmetric solutions where found in [44,78,79] and their extension to planar and hyperbolic symmetry was given in [71,80]. 2.2 Branches Notice that (2.1.8) leads to Kdifferent roots for every value of the radius and, thus, to K different branches associated to each of the cosmological constants (1.1.7) (some of them may be imaginary, though), in such a way that gi(r→ ∞)=Λi. For instance, in LGB gravity there are two branches that read g(±)=−1 2L2λ 1±s1−4λ1−κL2 rd−1!,(2.2.1) each one associated with a different cosmological constant. As for the corresponding vacua we need λ≤1/4 in order to have real solutions, otherwise the argument of the square root may become negative at some finite radius. Only one of the solutions, g(−), is connected to the standard Einstein-Hilbert gravity, in the sense that it reduces to it when λ→0, g(−)≈ − 1 2L2λ1−1−2λ1−κL2 rd−1=−1 L21−κL2 rd−1,(2.2.2) while g(+) blows up in that limit. It will be referred to as the EH-branch. It can be seen that this is the branch corresponding to the intersection of Υ[g] with the vertical axis, g= 0. The 26 CHAPTER 2. LOVELOCK BLACK HOLES -1.5 -1.0 -0.5 0.5 g 0.5 1.0 1.5 U@gD ¥ r+ r4 r3 r2 r1 Figure 2.1: A branch of the polynomial Υ[g] for the case K= 2, i.e., GB theory (with λ= 0.2 and L= 1), in arbitrary spacetime dimension, for different values of the radius ranging from ∞to r+,r1> r2> . . . > r+. The projection of the depicted points give g(ri) for the EH-branch in the planar case (σ= 0). Kdifferent branches of (2.1.8) are continuous functions of the radial coordinate, as long as the roots of a polynomial equation depend continuously on its coefficients [81], and renters monotonically in the zeroth order coefficient ˜c0(r)≡c0−κ/rd−1. When r→ ∞, (2.1.8) is nothing but the expression leading to the Kcosmological constants. The different Lovelock couplings ck>1fix the shape of the polynomial Υ[g]. While varying rfrom ∞to r+(see figure 2.1), the function g(r) is given by the implicit solution of equation (2.1.8) that graphically corresponds to climbing up (down for negative masses) a given monotonic part of the curve Υ[g] starting from one of its roots (tantamount of a given comological constant). The metric function gis a monotonic function of rsince ˜c0(r) is so and the remaining coefficients are frozen. Then each branch can be identified with a monotonic section of the polynomial Υ[g], and can easily be visualized graphically. As discussed above, the propagator of the graviton corresponding to the vacuum Λiis proportional to Υ0[Λi] in such a way that we will restrict to positive values of that derivative. It is zero just for degenerate vacua where there are actually no linearized degrees of freedom. In the present context the restriction Υ0[Λi]>0 is just verified by positive slope branches thus we will only consider those in the future. All the relevant or BD-stable branches correspond then to positive slope sections of the polynomial and, therefore, gwill be considered a monotonically decreasing function of r. For positive κthe solution runs over the points with positive value for Υ[g] while for negative mass it is the other way around. Either way, every branch always encounters a maximum/minimum, or it grows unboundedly. For the sake of clarity and the ease of reading, let us first classify the different types of branches that one may encounter when dealing with a Lovelock theory of gravity. The appearance of a given type of branch will depend, in general, on the specific theory considered and on the values of the different coupling constants. On the one hand, we may classify the branches by their asymptotics: AdS, flat or dS branches. In the particular case we are considering, with c0=L−2, there are no asymptotically flat branches. The sign of the 2.3. SINGULARITIES AND HORIZONS 27 cosmological constant corresponding to the EH-branch (when real) is the opposite to c0(or, equivalently, the same as the explicit cosmological constant, as in standard Eintein-Hilbert gravity); thus, the EH-branch is asymptotically AdS. Due to the particular features and relevance of this branch, we will consider it separately. Some of the branches (monotonic sections of the polynomial) may also be associated to complex values of Λ. Therefore, they do not correspond to real metrics and should be disregarded as unphysical. We will refer to these as excluded branches, and to the sector of the parameter space where the EH-branch is excluded as the excluded region. We will then exhaustively classify branches on (non-EH) AdS (i.e., not crossing g= 0), EH, dS and excluded branches. The latter, being unphysical, do not need further discussion. The AdS-branches must end at a maximum of the polynomial in order not to cross g= 0. The other two cases may end at a maximum or, else, continue all the way up to g→ ∞. We will then consider two subclasses of branches: those (a) continuing all the way to infinity or (b) ending at a maximum. For the AdS-branches we will also consider two subclasses: (a) positive mass and (b) negative mass. 2.3 Singularities and horizons Where are the singularities of these spacetimes located? The simplest way to answer this question is to calculate the curvature scalar and see where it diverges. As it depends on the metric and its derivatives, these divergences can be traced back to those of the first derivative of g, g0=−(d−1)κ rdΥ0[g]−1.(2.3.1) Then, the metric is regular everywhere except at r= 0 and at points where Υ0[g] = 0; that is, whenever the branch we are looking at coincides with any other. In such case, Υ0[g] = K X k=1 k ckgk−1= 0 .(2.3.2) These are precisely the maxima/minima at which all branches end, except those growing unboundedly (that also approach asymptotically to a singularity located at r= 0). The values of rwhere this happens exhibit a curvature singularity that prevents from entering a region where the metric becomes complex. Type (a) branches correspond to solutions with a singularity at r= 0 whereas those of (b) type display the singularity at finite radius. It can be easily seen that the mass parameter κmust be positive in the planar case (σ= 0) in order for the spacetime to have a well defined horizon. We can actually rewrite equation (2.1.8) as K X k=1 ckgk=κ rd−1−1 L2,(2.3.3) and realize that the equation admits a vanishing gonly when r=r+≡(κL2)1 d−1. In the planar case, furthermore, only one branch has a horizon at r+and all the rest display naked 34 CHAPTER 2. LOVELOCK BLACK HOLES -5 -4 -3 -2 -1 Re@xD -0.10 -0.05 0.05 0.10 Ám@xD ¥ r+ ¥ r+ Figure 2.7: The AdS black hole solution of the Gauss-Bonnet theory for λ= 0.2. For λ > 0, the (still real) red branch jumps to the left of the green one, without flipping its orientation. For bigger values of λ, it slides to the right, approaching the green branch. Needless to say, their infinities collide at λ= 1/4. -4 -3 -2 -1 Re@xD -0.4 -0.2 0.2 0.4 Ám@xD ¥ r+ ¥ r+ Figure 2.8: The AdS black hole solution of the LGB theory for λ= 0.26. For λ > 1/4 the branches become complex conjugates to each other. The critical value, indeed, is the singular locus of the theory (see figure 1.1). 2.3. SINGULARITIES AND HORIZONS 35 -1.0 -0.5 0.5 1.0 1.5 2.0 Λ -1 1 2 3 4 5 Μ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.02 0.04 0.06 0.08 0.10 0.12 Figure 2.9: Vanishing locus of ∆(r) for different values of r, spanning from ∞(thickest red and blue curve in the left) to r= (20/17)1/(d−1) r+(thinest red and blue curve in the right). The vertex of the curve slides up along the parabola µ=λ2(dashed curve). Points in the brown region (as the one depicted in yellow) belong to two curves. The black curve is µ=3 4λ2(that corresponds to ∆(r+) = 0), which bounds the regions colored in light blue, where ∆(r) only vanishes for one value of the radius between infinity and the horizon. The zoom in the right allows to better understand the structure where the different regions merge. more involved in the third order Lovelock theory. The relevant polynomial is pLL3[x;r] = 1−rd−1 + rd−1+x+λ x2+µ 3x3= 0 ,(2.3.11) whose discriminant is ∆(r) = a0(r)λ(6µ−4λ2)−3a2 0(r)µ2+λ2−4µ/3. As a function of the radius, ∆(r) = 0 spans a family of curves in the (λ, µ) plane that go from µ=3 4λ2(namely, ∆(r+) = 0) to the singular locus Γ depicted in figure 1.1. It is not difficult to see that for intermediate values of r, the curves look roughly like Γ, the singular vertex sliding up the parabola µ=λ2(see figure 2.9). Thus, the (λ, µ) plane has two regions, M(±), where there is a value r?∈[r+,∞) such that ∆(r?) = 0, M(+) =λ≤0, µ+(λ)≤µ≤3 4λ2∪0< λ < 8 27 ,3 4λ2≤µ≤µ+(λ) ∪8 27 ≤λ≤1 3, µ−(λ)< µ ≤µ+(λ),(2.3.12) M(−)=λ < 8 27 , µ ≤µ−(λ)∪λ≥8 27 , µ < 3 4λ2,(2.3.13) and one region, M(2), where there are two values r± ?∈[r+,∞) where ∆(r± ?) = 0, M(2) =8 27 < λ ≤1 3,3 4λ2≤µ≤µ−(λ)∪λ > 1 3,3 4λ2≤µ < λ2.(2.3.14) 36 CHAPTER 2. LOVELOCK BLACK HOLES -5 5 10 -5 5 10 15 20 N xN yN ∆ h Figure 2.10: Geometrical meaning of the parameters of the cubic. Recall also that the cubic is symmetric with respect to the point of inflexion. Everywhere else, ∆(r) does not vanish for real values of r(unless they are hidden by the horizon). The previous analysis suggests that the space of parameters is divided into different regions that should be treated separately. Depending on the sign of the discriminant we will have one or three real solutions (cosmological constants). As we shall see, for ∆ >0 we have three real solutions while for ∆ <0 we will have just one. The algebraic approach portrayed in this section can be extended to arbitrary higher order Lovelock theory in higher dimensions. It can be surely dealt with analytically up to the fourth order gravity while higher order cases may require numerical analysis since the quartic is the highest order polynomial equation that can be generically solved by radicals (Abel-Ruffini theorem). Despite these difficulties most of the relevant information about black holes in a given Lovelock theory can be extracted from the implicit solution (2.1.8), as we will see in the next chapter. This effective approach will allow the discussion of completely general Lovelock theories, namely their thermodynamic properties. We will focus from here on in the case of cubic Lovelock theory. This particular case has taught us that it is a subtle issue to determine which is the branch of solutions connected to Einstein-Hilbert and, thus, presumably stable. From the point of view of a black hole background, the relevant question is whether one can find an asymptotically AdS black hole with a well-defined horizon. This is a delicate problem that can be amusingly casted in terms of a purely algebraic setup. 2.4 Black holes in third order Lovelock theory As discussed in the preceding paragraphs, most of the information needed to clarify the existence of black hole solutions for different values of the Lovelock couplings relies in the behavior of a cubic polynomial. There is a very convenient way of parametrizing the cubic, suitable for analyzing the EH branch of the black solution in terms of the geometry of the polynomial. We can characterize a cubic in terms of four fundamental parameters [84], δ, h, xNand yN(see figure 2.10). Let us show how this works in our case. 2.4. BLACK HOLES IN THIRD ORDER LOVELOCK THEORY 37 If we start from our cubic polynomial, pLL3[x;r] = µ 3x3+λx2+x+a0(r), a shift on the variable x=z+xN, where xN=−λ µ, leads to a polynomial, pLL3[z;r], that is known as the reduced cubic. Nis the point of inflexion, and it can be shown that his a simple function of δ, namely h=−2 3µ δ3,where δ2=λ2−µ µ2.(2.4.1) Thus, the shape of the cubic is completely characterized by the parameter δ. Either the maximum and minimum are different (δ2>0), or they coincide at N (δ2= 0), or there are no turning points (δ2<0). We choose the sign of δso that δ > 0 corresponds to the situation depicted in the figure (µ < 0) and h(when δis real) is a positive quantity, δ=−1 µpλ2−µ⇒h=2 3µ2λ2−µ3/2.(2.4.2) Consider the usual form of the cubic equation pLL3[x;r] = 0, with roots α,βand γ, and obtain the reduced form by the substitution x=xN+z. The equation has now the form, µ 3z3−µ δ2z+yN(r) = 0 ,(2.4.3) with roots α−xN,β−xNand γ−xN. The parameter yN(r) is obviously the only one that depends on the radial coordinate through a0(r), yN(r) = a0(r) + λ 3µ22λ2−3µ.(2.4.4) This form allows us to use the identity (p+q)3−3p q(p+q)−(p3+q3) = 0 .(2.4.5) Thus z=p+qis a solution where p q =δ2and p3+q3=−3yN µ.(2.4.6) Solving these equations by cubing the first and substituting into the second, and solving the resulting quadratic in p3gives p3=3 2µ−yN±qy2 N−h2.(2.4.7) The discriminant of the polynomial reads ∆(r) = −3µ2y2 N−h2=−3µ2(y+y−),(2.4.8) where y±=yN±hare the y-coordinates of the maximum and the minimum respectively. ∆(r) acquires a neat geometrical meaning in the light of this expression. We can see that the sign of the discriminant is determined by the position of the maximum and the minimum 38 CHAPTER 2. LOVELOCK BLACK HOLES with respect to the x-axis. When h∈C,y+, y−∈C, this corresponding to the case µ>λ2. The singular locus ∆ = 0 corresponds to y±= 0 at r=∞(see figure 1.1). We can treat separately the different regions. For y2 N> h2(y+y−>0), there are just one real root that can be easily obtained as1 α=xN+3 r−yN+qy2 N−h2+3 r−yN−qy2 N−h2.(2.4.9) For y2 N=h2(y+y−= 0), there are three real roots two of them equal. The roots are α=xN+ 2˜ δand β=γ=xN−˜ δ, where the sign of ˜ δdepends on the sign of yNand has to be determined from ˜ δ=3 r−yN 2a=±3 r−h 2a=±δ(2.4.10) If yN=h= 0, then δ= 0, in which case there are three equal roots at x=xN. For y2 N< h2(y+y−<0), all three roots are real and distinct. The easiest way to proceed, without having to find the cube root of a complex number, is to use trigonometry to solve the reduced form with the substitution z= 2δcos θ, that gives cos 3θ=yN h.(2.4.11) The three roots are therefore given by α=xN+ 2δ cosθ , β=xN+ 2δ cos(θ+ 2π/3) ,(2.4.12) γ=xN+ 2δ cos(θ+ 4π/3) . The important point to notice here is that, as yNvaries (or equivalently rsince yN(r) is monotonic), the angle θcan run from zero (where β=γand αis the real branch parameterized by (2.4.9)) and π/3 (where α=γand βis the real branch parameterized by (2.4.9)). We can follow the real root(s) as yNchanges. For yN> h we have just (2.4.9) as a real root. For yN=hwe have α=xN+ 2δand β=γ=xN−δ. For −h < yN< h, the angle θis monotonic with yNgoing from θ= 0 to θ=π/3. Beyond that point, two roots become complex again but not the same two that were imaginary for yN> h. Thus we have β=xN−2δand α=γ=xN+δfor yN=−hthe latter two roots becoming complex for yN<−hwhere βcorrespond to (2.4.9). Each of this solutions is continuous with yNand the same can be checked with the other parameters of interest, µand λ. The only presumably 1The complex branches being, β=xN+−1 + i√3 2 3 r−yN+qy2 N−h2+−1−i√3 2 3 r−yN−qy2 N−h2, γ=xN+−1−i√3 2 3 r−yN+qy2 N−h2+−1 + i√3 2 3 r−yN−qy2 N−h2. 2.4. BLACK HOLES IN THIRD ORDER LOVELOCK THEORY 39 -1.5 -1.0 -0.5 0.5 1.0 1.5 -4 -2 2 4 ¥ r+ ¥ r+ ¥ r+ Figure 2.11: ∆(r)<0∀r∈[r+,∞) corresponding to M(0) −. There is a single branch which is real. As we will see later, it is the EH-branch for the black hole solution. singular point is µ= 0 where the degree of the equation changes, but we can take a well defined limit where one of the solutions diverges and the other two coincide with those of the quadratic polynomial. These solutions are continuous, but the ‘always real branch’ as usually parameterized in (2.4.9) is discontinuous since there is an interval of values of xthat cannot be taken by the αnor the βbranches (see figure 2.16). From now on we will refer to as α(β) the branch real for yN→ ∞ (−∞). Just by analyzing the shape of the cubic we can understand which one is the branch connected to the horizon (x= 0 for r=r+(a0= 0)). We have given values of a1and a0 and so the polynomial at the origin and its first derivative there. The first derivative of the polynomial at x= 0 is always 1, and so the polynomial is growing at that point. Also, we know the sign of xNwhen λand µare given. We can distinguish several different cases. We know from our previous analysis that we can take a representative point in each of the relevant regions and analyze the behavior of the solutions. For µ≥λ2(δ2<0) there are no turning points (or they coincide at the inflexion point) and the polynomial is monotonically growing. There is just one real branch of solutions for all values of rand it has then no discontinuity. The same happens in the region contained in between the curve µ= 3/4λ2and µ=λ2for λ < 0 and in between µ=µ+(λ) and µ=λ2 for λ > 0. In this whole region we have ∆(r)<0∀r∈[r+,∞). This happens for any point (λ, µ) lying in the upper white part of figure 2.9. Let us call this region M(0) −, M(0) −=λ≤0, µ > 3 4λ2∪0< λ ≤1 3, µ > µ+(λ)∪λ > 1 3, µ > λ2.(2.4.13) In this case, the roots of (2.3.11) behave the same all along the radial flow; there is a single real branch which is the relevant one for asymptotically AdS black hole solutions with a well-defined horizon, and two complex conjugate unphysical branches (see figure 2.11). The µ=λ2line for λ > 1/3 is excluded from M(0) −since in this case the inflection point is situated above the x-axis yN(∞) = 1 λλ−1 3>0 (2.4.14) 40 CHAPTER 2. LOVELOCK BLACK HOLES -4 -2 2 4 -2 -1 1 2 ¥ r+ ¥ r+ ¥ r+ Figure 2.12: ∆(r)>0∀r∈[r+,∞) corresponding to M(0) +. This situation is very similar to LGB gravity. One of the solutions diverges as µ→0 with xN<0, and then, when the inflection point crosses the x-axis (r=r?where ∆(r?) = 0) a (naked) singularity shows up even if the spacetime is perfectly regular for r6=r?. The lower white part of figure 2.9, instead, has ∆(r)>0∀r∈[r+,∞). Let us call this region M(0) +, M(0) +=λ≤0, µ−(λ)< µ < µ+(λ)∪0<λ< 8 27 , µ−(λ)<µ<3 4λ2.(2.4.15) In this case, again, the roots of (2.3.11) behave the same all along the radial flow (see figure 2.12); there are three real branches but only one is relevant for asymptotically AdS black hole solutions with a well-defined horizon (see figure 2.12), the EH one. This is reminiscent of the LGB case where there are two real solutions for λ < 1/4. One of the branches diverge as µ→0. For µ<λ2we have in general two values of r(or yN) for which ∆(r) = 0, but they can be included or not in the interval r∈[r+,∞) (in the previously analyzed regions these values lie outside this interval). For µ > 0 (δ < 0) λ < 0 (xN>0) maximum and minimum are located at positive values of xand so the branch connected to the horizon has no problems of reality or continuity even if eventually (in the subregion contained in M(+)) ∆(r∗)=0 for some value r∗∈[r+,∞) (see figure 2.13). For µ < 0 (δ > 0) the only growing part of the polynomial is the one in between the minima and the maxima, corresponding to the branch γ. The maximum is located to the right of the origin x= 0 and does not pose any problem regarding the EH branch, but the minimum is located in negative values of xand then we must have y−(∞)<0 (yN(∞)< h) in order to have a real cosmological constant. The singular curve y−= 0 correspond in this region to µ=µ−(λ). The region with real cosmological constant correspond to M(0) +while the region with complex cosmological constant correspond to the region M(−)(see figure 2.14). The γbranch for µ < 0 is continuously deformed into αor βwhen crossing the µ= 0 line. For λ < 0 the αbranch diverges and γand βinterchange their rˆoles. For λ > 0 is βthe diverging branch and γis deformed into α. For µ= 0 we can identify the remaining 2.4. BLACK HOLES IN THIRD ORDER LOVELOCK THEORY 41 -2 2 4 6 8 -2 -1 1 2 ¥ r+ ¥ r+ ¥ r+ Figure 2.13: ∆∞<0 and ∆(r+)>0 corresponding to M(+). -3 -2 -1 1 2 3 -0.6 -0.4 -0.2 0.2 0.4 0.6 ¥ r+ ¥ r+ ¥ r+ Figure 2.14: ∆∞<0 and ∆(r+)>0 corresponding to M(−). Regarding the behaviour of ∆(r) this region seems similar to the previous one but in this case one of the degenerate branches when ∆(r) = 0 is the one connected to the horizon. Thus, there is a real cosmological constant but it does not correspond to the branch with horizon. 42 CHAPTER 2. LOVELOCK BLACK HOLES -8 -6 -4 -2 2 4 -3 -2 -1 1 2 3 ¥ r+ ¥ r+ ¥ r+ Figure 2.15: ∆∞>0 and ∆(r+)<0 corresponding to M(+). -8 -6 -4 -2 2 4 -2 -1 1 2 ¥ r+ ¥ r+ ¥ r+ Figure 2.16: ∆(r)<0∀r∈[r+, r− ?)∪(r+ ?,∞) corresponding to M(2). In this case as in the figure 2.14 there is no way of continuously connect the real cosmological constant to the horizon branches as the solutions for the LGB case. The remaining case to be discussed is µ > 0 (δ < 0) and λ > 0 (xN<0). In this region (except as already discussed for µ≥λ2) the two critical points are located at negative values of x, and they can have positive or negative y-values. For the subregion contained in M(+), see figure 2.15. As in the previous case, in order to have a real value for the relevant cosmological constant the value y−for the minimum must be negative. Again, the limiting case (y−= 0) corresponds to µ=µ−(λ) as can be seen in figures 2.16 and 2.14 In any case the necessary and sufficient condition for the cosmological constant to be real is y−(∞)<0 or yN(∞)< h. Therefore we have an excluded region of parameters below µ=µ−(λ). The other curve µ=µ+(λ) does not affect the qualitative behaviour of the EH solution since it is just indicating the appearance of two new real cosmological constants. Thus, in most of the space of parameters the existence of two extra solutions does not qualitatively affect the solution with horizon, except in the excluded region where the cosmological constant connected to the horizon is not real. This excluded region reduces to λ > 1/4 in the LGB limit, as it should be expected. The remaining boundary of the 2.5. CHARGED AND ROTATING SOLUTIONS 43 excluded region, for λ > 1/3, is the µ=λ2line. Notice that the well behaved solution is regular when crossing the curve µ=µ+(λ); the would be symmetry enhancement affects the other two cosmological constants. In other words, the two cosmological constants that agree over this curve are not those connected to the horizon, and then the theory has propagating linear perturbations when expanded about the EH vacuum. Symmetry enhancement for this vacuum occurs at µ=µ−(λ) and this is the boundary of the excluded region. The usual parameterization of the black hole solution [85,86] shall make manifest the different properties portrayed in the previous section. In fact, the usual parametrization of the three, in general complex, solutions for the function fin third order Lovelock gravity is fi=r2 L2 λ µ1 + αiJ(r) + pΩ(r)1/3+ ¯αiJ(r)−pΩ(r)1/3,(2.4.16) where Ω(r) = J(r)2+ Γ3,(2.4.17) with J(r)=1−3µ 2λ2+3µ2 2λ31−r6 + r6=3µ2 2λ3yN(r),(2.4.18) Γ≡µ λ2−1⇒Γ3=3µ2 2λ3h2.(2.4.19) αiare the three cubic roots of unity, α0= 1, α±=−1±i√3 2, and the bar indicates complex conjugation. Each of these solutions is associated with one possible value of the cosmological constant, and so, fixing the value of the cosmological constant fixes the function. Notice that fis directly related to our previous parameterization as f=−r2 L2x , (2.4.20) where xis the relevant branch (α,βor γ) in each region. 2.5 Charged and rotating solutions One obvious extension of Lovelock gravity, probably the simplest one, is that of LovelockMaxwell theory. Solutions charged under both Maxwell and Born-Infeld electrodynamics have been known for long time [87,88], and were reconsidered recently [89–103]2. Most of these efforts have been devoted to the simpler LGB case and a complete classification of all possible black hole solutions in Lovelock theories is still missing. Even though LovelockMaxwell solutions have in general a more complex structure, it is quite straightforward how to modify the general approach outlined throughout this chapter to the charged case. 2The analogous solutions in quasi-topological gravity have been considered in [104] 50 CHAPTER 2. LOVELOCK BLACK HOLES the argument function Gbeing in this case G=σ+ ˙a2 a2, the rˆole of the ffunction of the black hole being played by −˙a2. The vacuum solutions are in this way exactly the same ones we found using the black hole ansatz, i.e. maximally symmetric solutions with σ+ ˙a2 a2=¨a a= Λ (2.6.6) If we source these equations by a homogeneous isotropic perfect fluid filling the whole universe, then we have that the polynomial is basically the energy density, ρand the second equation of motion defines the corresponding pressure, p, subject the energy conservation equation. That system of equations can be equivalently written as Υ[G] = 8πGN d−1ρ(2.6.7) ˙ρ+ (d−1) ˙a a(ρ+p) = 0 (2.6.8) We can even introduce a third equation for G d dlog a+ 2G= 2¨a a(2.6.9) in such a way that σappears as an integration constant, thus set by the initial conditions. We thus have a complete set of three differential equations that fully determine the dynamics of our solution. The only missing piece is the equation of state of the fluid, consider for simplicity a single species fluid with a linear equation of state, p= (ω−1)ρ. Then we can use the conservation equation together with this last expression to get a density function, ρ(a), in our case ρ=ρ0 a(d−1)ω,(2.6.10) effectively reducing the number of equations to two; one conservation equation (2.6.7) and one acceleration equation (2.6.9). ω= 0 is equivalent to the cosmological constant and can be reabsorved into Υ, we will not consider this case and asume 0 < ω ≤2 which satisfies the dominant energy condition, ρ > kpk. We can make an analogy with a one-dimensional system of a particle on a potential, although with a non-canonical kinetic term. The conservation equation is in general of the form E(˙a2, a) = 0 (2.6.11) whereas the acceleration equation derives from the first one deriving with respect to time ∂E ∂(˙a2)¨a=−1 2 ∂E ∂a ,(2.6.12) analogous to the 1 2˙x2+V(x) = E0and ¨x=−V0(x) equations of a particle in a potential. Due to the higher order form of the equations of motion the form of these is more complicated but the logic is the same. In our case the conservation and acceleration equations are independent 2.6. LOVELOCK COSMOLOGIES 51 because we have an extra variable G. Still we may integrate (2.6.9) in order to get G=σ+ ˙a2 a2 and everything will then reduce to the conservation equation Υσ+ ˙a2 a2=8πGN d−1ρ(a) (2.6.13) We just introduced the acceleration equation (2.6.9) in order to make clear that σmay be considered an integration constant instead of something of our choice. Even though we can rescale this constant to the three 0,±1 usual values, it may take in principle any value. σ plays the same rˆole as the energy, E0, of the one-dimensional particle. We have effectively changed from a system in real space (t, a(t)) to a problem in phase space (a, ˙a(a)). We can even write the original metric in those variables as ds2=−da2 ˙a(a)2+a2dΣ2 d−1(2.6.14) Making the analogy with the black hole solution much more transparent. This parallelism is even more explicit taking into account that R2 AH =1 G(a,˙a)is the radius of the apparent horizon of that spacetime [109] and has associated thermodynamic variables in very much the same way as the black hole. Once the conservation equation (2.6.7) has been stablished the dynamics of the system is completely determined by two initial conditions a(0) = a0; ˙a(0) = v0(2.6.15) and the choice of a given branch of the polynomial. In the general relativistic case we would just plug ρ(a) into the constraint in order to find a=a(t). In this more general case this cannot always be done and even when we can the equation for the different branches would be very complicated. Nevertheless we can still use graphical techniques in order to analyze the qualitative behavior of the solutions. We can determine the position of the turning points as we did for the black hole positions in the black hole case. This is even clearer taking into account that the rˆole of fis now played by −˙a2, thus horizons,f= 0, are now turning points, ˙a= 0. There is a crucial difference however as the cosmological evolution corresponds to ˙a2≥0, hence the trapped region in black hole language. The regions of interest are exactly complementary to those analyzed in the black hole case. The turning points G+=σ/a2, are the roots of the following equation Υ[G+] = 8πGN d−1ρpσ/G+(2.6.16) For pressureless matter (ω= 1) this reduces to exactly the same equation as for black hole horizons with κ=8πGN d−1ρ0. Remark that considering positive energy, ρ0>0 density the behavior of ρ(a) is monotonic even for several species. This follows from conservation of the energy and the energy conditions. Notice also that we recover the vacuum solutions in the limit of infinite expansion, a→ ∞, at least for dS branches. For AdS branches the vacuum is in the untrapped region and thus we always encounter a turning point before reaching that point. Remarkably, in the presence of radiation (ω= 2) the right hand side of (2.6.16) grows 52 CHAPTER 2. LOVELOCK BLACK HOLES at least as Gd−1 +whereas the polynomial grows at most as G d−1 2 +. The Big Bang singularity a= 0 is then always in the ˙a2>0 (trapped) region and for low enough densities there is also always at least one turning point. As we have seen, the analogy with the black hole solution is very useful, vacua and branches are the same, horizons map to turning points and f(r) to −˙a2. In black holes we usually restrict our analysis to the untrapped region f > 0 whereas in this case the physically relevant region is the opposite, ˙a2>0, so that the cosmological stories lie to the right of the ρpσ/G+]curve. The singularities are also the same as in the black hole case and occur either as a→0 (Big Bang) or at points Υ0[G] = 0 where the acceleration (or equivalently ˙ G) is not determined by the equations of motion. In this case however, contrary to the black hole case this singularity is transversable. There is no reason `a priori for the energy density to be monotonous with G. We just need to change the affine parameter describing the trajectory to the length in phase space, ds =√da2+d˙a2and verify that the motion is then regular and the time spent finite. An analogous phenomenon has also been observed in the context of braneworlds in LGB gravity [110]. The curvature singularity appears just because the potential is multivalued and has a degenerate point as Υ0[G] = 0. The prototypical example of this is a potential with two branches of the form V±(a) = ±rak ? ak−1 (2.6.17) that degenerates at a=a?and becomes imaginary beyond that point. The acceleration is not well defined at the degenerate point ¨a=k 2 ak ? ak+1V(a)(2.6.18) diverging in opposite directions depending on wether we approach a?trough the + or − branch. On the contrary, if we perform the change of variable mentioned above we get da ds =1 q1 + ¨a ˙a2 =V(a)pE0−V(a) qV2(a)(E0−V(a)) + k2 16 ak ? ak+1 (2.6.19) d˙a ds =1 q1 + da ds 2(2.6.20) well defined evolution equations for aand ˙a. As we approach the singular point, a=a?, da/ds vanishes and then changes sign whereas d˙a/ds goes to one. In other words we have a turning point with finite ˙a! Our cosmological particle just follows the potential through the singularity changing from one branch to the other. The same happens for maxima and minima of the Lovelock polynomial in this context. If we do not encounter a turning point 2.6. LOVELOCK COSMOLOGIES 53 -2 -1 1 2 3 4 g U@gD Σ=1 Σ=0 Σ=-1 c0=1 c0=0 c0=-1 Κ=2 Κ=0.2 ΚNariai Figure 2.19: Linear polynomial corresponding to the usual EH-branch for negative (c0= 1), zero (c0= 0) and positive (c0=−1) cosmological constants (L= 1). The dashed lines are just κ(g/σ)d−1 2for d= 5, corresponding to pressureless matter. The crossing of these lines with the polynomial gives the turning points. The solid black line corresponds to the critical value of the mass, κNariai = 1/4, that in this case corresponds to the critical mass for the existence of a potential barrier in the dS case. For σ= 1 and κ>κNariai (or r+> rNariai), starting from the Big Bang singularity at a= 0 (equivalently Υ = ∞) the asymptotically dS branch describes an always expanding spacetime whereas for the other cases and topologies the turning point always exist. For the dS branch for κ < κNariai there is also a second type of solution that describes a spacetime that collapses from vacuum and then reexpands. This solution can be connected to the Big Bang by tunneling. in the way we may start for a→ ∞ in a given dS vacuum, pass through the singularity and end up in a different dS space after enough time. This would be the case, for instance, for LGB gravity with two dS branches (i.e. c0<0 and c2<0). We can take the analogy with the one particle system a bit further. The existence of turning points will imply in some cases the existence of forbidden regions separating possible cosmological trajectories. Quantum-mechanically we may calculate the tunneling probability by performing a Wick rotation and computing the Euclidean action of the resulting trajectory with the same energy. In the gravitational context we may think of doing the analogous thing, this would amount to the computation of the Euclidean on-shell action, b I, of our cosmological solution while going through the barrier. The tunneling probability will be proportional to e− b I. As we will see in the next chapter, the Euclidean section has very important applications also in the context of black holes. We can now take any of the figures corresponding to black hole solutions from previous sections and reinterpret their trapped regions as possible cosmological solutions for a pressureless fluid. As an example we may analyze graphically the case of Einstein-Hilbert gravity with cosmological constant, our Lovelock cosmologies lie to the right of the black dashed lines. Chapter 3 Black hole thermodynamics “Pour atteindre la v´erit´e, il faut une fois dans la vie se d´efaire de toutes les opinions qu’on a re¸cues, et reconstruire de nouveau tout le syst`eme de ses connaissances.” Ren´e Descartes In the early 1970s, Bekenstein [111] argued that the second law of thermodynamics requires one to assign a finite entropy to a black hole, even though this seemed to contradict the fact that, as classical objects, they have zero temperature. Na¨ıvely they would absorb matter without ever emit anything. Bekenstein’s worry was that one could collapse any amount of highly entropic matter into a black hole – which is an extremely simple object – leaving no trace of the original disorder. This would in turn violate the second law of thermodynamics, which asserts that the entropy of any closed system can never decrease. However, adding mass to a black hole will increase its size, which led Bekenstein to suggest that the area of a black hole is a measure of its entropy. This conviction grew when, in 1972, Hawking proved that the surface area of a black hole, like the entropy of a closed system, can never decrease. The similarity between black holes and thermodynamic systems was considerably strengthened when Bardeen, Carter, and Hawking [112] showed that these enigmatic obey a complete set of rules that parallel exactly the laws of thermodynamics, •The surface gravity κis constant over the event horizon (zeroth law). •For any two stationary black holes differing only by small variations in the mass M, angular momentum J, and charge Q(first law), δM =κ 8πGN δA + ΩHδJ + ΦHδQ (3.0.1) 55 56 CHAPTER 3. BLACK HOLE THERMODYNAMICS where ΩHand ΦHare the angular velocity and the electromagnetic potential at the horizon respectively. •The area of the event horizon of a black hole never decreases (second law), δA ≥0 (3.0.2) •It is impossible by any procedure to reduce the surface gravity κto zero in a finite number of steps (third law). These laws were later shown to be much more general than the particular 4-dimensional setting where they were discovered. In particular, the first law holds for much more general gravitational actions, for which the entropy can be understood as a Noether charge [113]. In these laws the rˆole of the temperature is played by the surface gravity but Bardeen, Carter and Hawking, in their original paper, added: “It should however be emphasized that κ/8πGNand Aare distinct from the temperature and the entropy of the black hole. In fact the effective temperature of a black hole is absolute zero. In this sense a black hole can be said to transcend the second law of thermodynamics.” At the time, there seemed to be a fundamental contradiction between the hypothesis of black holes’ entropy being non-zero and them being perfectly absorptive objects. Classical black holes are, after all, black: when placed in contact with a heat bath they will absorb energy while emitting none, thus behaving as if they have a temperature of zero. Couple of years later, Hawking himself came up with the resolution of this apparent paradox, the problem being solved by quantum theory [114]. In the presence of a horizon quantum fields were shown to be emitted with a thermal spectrum of temperature proportional to the surface gravity of the hole, TH=κ/2π. This temperature has not yet been directly observed, that would pressumably grant Hawking the Nobel prize! For typical black hole masses, Hawking’s temperature is several orders of magnitude smaller than the cosmic microwave background, thus impossible to identify. To have a Hawking temperature larger than 2.7 K (and actually be able to evaporate), a black hole needs to have less mass than the Moon. Such a black hole would be smaller than a needle’s eye1. The above results establish that the parallel between the laws of black hole mechanics and the laws of thermodynamics is not a mere coincidence. Indeed they seem to be hints of some very deep physics, intertwining classical and quantum properties of these objects. The Hawking effect establishes that the surface gravity of a black hole can indeed be interpreted as a physical temperature. Further, mass in black hole mechanics is mirrored by energy in thermodynamics, and we know from relativity theory that mass and energy are actually equivalent. Connecting the two sets of laws also requires linking the surface area of a black hole with entropy, as Bekenstein had suggested. This black hole entropy is called its Bekenstein entropy, and is proportional to the area of the event horizon of the black hole. The Generalized Second Law of thermodynamics then conjectures that the sum of the entropy of the matter outside a black hole and its own never decreases. Black holes can be described as thermodynamic ensembles to which all the usual machinery of thermodynamics 1Data from Einstein online 57 can be applied, allowing the description of new and interesting phenomena, including phase transitions. The phase structure of General Relativity is quite well understood at present. As we increase the dimensionality however this phase structure gets increasingly intricate and diverse. In dimensions greater than four the metric has many more degrees of freedom and, as a result, the spectrum of the theory gets richer. We may have extended black objects, like strings or branes; but also more rotation planes and extra dimensions that may be compactified `a la Kaluza-Klein. The analysis of this phase structure is very interesting for a wide variety of reasons. First of all it may help to elucidate which features of general relativity are universal and which are on the contrary d-dependent. This particular question is exacerbated in the context of Lovelock, or more general gravity theories, as we also increase the number of tunnable couplings as we increase the dimension. In this context even the properties of static solutions are poorly understood and seemingly pathological features have appeared. The detailed analysis of the thermodynamics of these solutions is crucial for the understanding of the consistency of the theory. On the other hand the proposed study may uncover the existence of critical dimensions where properties of black holes change dramatically, it will yield information about the stability and phase transitions, and the possibility of thermodynamic and perturbative stability being correlated. Also, information about static and stationary solutions may provide some clues about the endpoint of instabilities and even about time dependent trajectories connecting different phases. Many interesting questions that deserve further investigation. The thermodynamics of Lovelock theories is not as well understood as for the general relativistic case, neither at the level of the basic principles. The zeroth law has been shown to hold also in this context for matter respecting the dominant energy condition [115], exactly as in the Einstein-Hilbert case. The first law is also verified, as expected in general grounds, and has been used to extract the expression for the entropy [116,117] of Lovelock black holes that no longer coincides with the area of the event horizon. In particular, it has been shown that the entropy may become negative in some cases, in conflict with any microscopic interpretation. We will comment more on this below. An extended version of the first law has also been proposed [76] where the Lovelock couplings play the rˆole of extra thermodynamic variables. The second law of thermodynamics has a crucial rˆole in the thermodynamic picture of black hole dynamics as it enforces the irreversibility intrinsic to thermodynamic processes. It has been shown that the second law also holds for general Lovelock theories in a number of cases [118–120], namely a physical process version has been proven on [121]. Some words on the third law can be also found in [122]. In the next sections we will take the usual thermodynamic interpretation inherited from General Relativity for granted, discussing its implications for the stability and phase transitions of static black holes in Lovelock gravity. Some seemingly pathological features will appear, its resolution, when available, being proposed and discussed. We will start with some clarifying description of the connection between gravity and thermodynamics and the presentation of the basic formulas for the thermodynamic variables. 58 CHAPTER 3. BLACK HOLE THERMODYNAMICS 3.1 The path-integral approach to Quantum Gravity The first attempts to the quantization of gravity were based naively on the usual techniques of Quantum Field Theory combined with general considerations about the nature of the gravitational interaction itself. Even though these theories were not completely consistent they yield some partial results compelling enough to be trusted as a good first approximation to the problem. These results are related to the Hawking effect, the connection between black holes and thermodynamics and other semi-classical features. The most useful approach to make this connection manifest is what is usually called Euclidean Quantum Gravity. The starting point for this approach is the idea that one can represent the amplitude to go from a state with metric g1and matter fields φ1, on a spacelike hypersurface S1, to a state with a metric g2and matter fields φ2, on a hypersurface S2, as a path integral over all field configurations gand φwhich take the given boundary values on S1and S2[123]. More precisely, hg2, φ2, S2|g1, φ1, S1i=ZD[g, φ]eiI[g,φ].(3.1.1) This is the usual way a path integral is defined in any quantum-mechanical system, where D[g, φ] is a measure in the space of all field configurations gand φand I[g, φ] is the action of the fields. Not all the components of the metrics g1and g2are physically relevant. We need to specify only the three-dimensional induced metric hon S1and S2, up to diffeomorphisms which map this two surfaces into themselves. Consider now an intermediate surface S0between the two boundary surfaces. One would expect that hh2, φ2, S2|g1, φ1, S1i=X h0,φ0hh2, φ2, S2|h0, φ0, S0ihh0, φ0, S0|g1, φ1, S1i.(3.1.2) This is a general property of amplitudes in quantum mechanics that one would want the theory to verify. The amplitude to go from the initial state to the final state should be obtained also by summing over all possible configurations on the intermediate surface. The usual Lovelock bulk action alone does not verify this property, it needs to be supplemented with the boundary terms discussed in section (1.2) [123]. This is an alternative motivation for its inclusion, now in the quantum framework. 3.1.1 Spacetime complexification and thermodynamics For real Lorentzian metrics gand real matter fields the action I[g, φ] will be real. The path integral (3.1.1) will then oscillate and it is not clear whether it converges or not. In quantum field theory one usually deals with this difficulty by a Wick rotation in the complex time variable, i.e. t=−iτ. This idea applied to our general case leads to a path integral of the form Z=ZD[g, φ]e− b I[g,φ],(3.1.3) 3.1. THE PATH-INTEGRAL APPROACH TO QUANTUM GRAVITY 59 where ˆ I=−iIis called the Euclidean action, b I[g, φ] = −1 16πGN(d−3)! K X k=0 ck d−2kZMLk−Z∂M Qk−Zd4x√gLm,(3.1.4) and gand hare now positive-definite. A very important use of the Euclidean section is to construct the canonical ensemble for a field theory. Consider a field φ. The amplitude to propagate from a configuration φ1at time t1to a configuration φ2at time t2is given by the path integral hφ2, t2|φ1, t1i=ZD[φ]eiI[φ],(3.1.5) with the given boundary values. Using the Schr¨odinger picture we can also write this amplitude as hφ2|e−iH(t2−t1)|φ1i=ZD[φ]eiI[φ],(3.1.6) where His the Hamiltonian driving the time evolution of the system. By going to periodic complex time via a Wick rotation with t2−t1=−iβ and φ2=φ1, and summing over a complete orthonormal basis of configurations, we get the partition function, Z=X En e−βEn=ZD[φ]e− b I[φ],(3.1.7) for the field φat temperature T=β−1, where Enis the energy of the n-th eigenstate. In this last expression the path integral is taken over all fields φwhich are real in the Euclidean section and periodic in imaginary time with period β. The same idea can be applied directly to any gravitational system making the connection with thermodynamics explicit. The canonical partition function associated with a gravitational system at temperature Tcan be defined as a path integral (3.1.3) extended to all configurations with given boundary values and identified in with period βin Euclidean time. Henceforth, we are going to consider only the gravitational part of the action setting all matter fields to zero. Once the partition function of our theory has been computed, it can be used to extract information about the system. In particular we can derive all the thermodynamic magnitudes of interest using the usual relations from statistical mechanics, namely the Helmholtz free energy, relevant thermodynamic potential for the canonical ensemble, F=M−TS =−Tlog Z,(3.1.8) which tends to be minimum. The average energy and the entropy can be calculated from it as 60 CHAPTER 3. BLACK HOLE THERMODYNAMICS hEi=1 ZX En Ene−βEn=−∂ ∂β log Z,(3.1.9) S=−∂F ∂T =βhEi+ log Z.(3.1.10) In the canonical ensemble, the temperature is an external constraint applied over the system and cannot be determined using this framework. In the gravitational context the temperature of a black hole solution is computed from the required regularity of the Euclidean section of the solution. Finally, if one is to recover the classical gravitational theory we started with in the limit of macroscopic objects, one expects that the dominant contribution to the partition function will come from metrics which are an extremum of the action, i.e. solutions of the classical equations of motion. This is known as the stationary-phase or saddle point approximation. As a extreme version of this we can estimate the partition function by the single contribution of the metric with least action, log Z ≈ −b I[gmin],(3.1.11) and the free energy will coincide essentially with it, ˆ I=βF =βM −S. As we will see in the next section, the free energy is in general divergent due to infinite volume of the spacetime. In order to regularize the Euclidean action we will then substract the contribution of some reference background that customarily is taken to be the maximally symmetric vacuum for the considered asymptotics. 3.1.2 AdS spacetime and the canonical ensemble The canonical ensemble describes a system in thermal equilibrium with an infinite heat reservoir so that the temperature of the system is fixed. This ensemble is ill-defined in asymptotically flat spacetimes [124] because of the attractive nature of gravity and the possibility of having a black hole. To illustrate this, let us consider four dimensional flat space at temperature T. In this hypothetical situation every region of spacetime would be filled with homogeneous thermal energy density, ρ∼T4. Due to the infinite volume of the spacetime, the total energy would be thus also infinite and the configuration, albeit simple, inconsistent. The backreaction of this infinite mass would lead the spacetime to warp, and it would be no longer flat. Another, argument from the classical point of view, is that thermal perturbations of wavelength larger than the Jeans length scale2grow exponentially and collapse to form a black hole [124]. Thermal flat space is thus unstable to the formation of black holes and these black holes are also unstable, in this case thermodynamically. They cannot be in equilibrium with their enviroment, either decaying into pure thermal radiation or engulfing the whole spacetime. 2Critical radius of a cloud (typically a cloud of interstellar dust) where thermal energy, which causes the cloud to expand, is counteracted by gravity, which causes the cloud to collapse. 3.2. LOVELOCK BLACK HOLES AND THERMODYNAMICS 67 1 2 3 4 5 6 g U@gD r+ rc ∆r+>0 ∆rc<0 È È È È È È È Κ+∆Κ Κ ∆Κ>0 Figure 3.1: Determination of the sign of the temperature for the cosmological and black hole horizon of a dS branch. The cosmological horizon has Tc∝dκ/drc<0 whereas the event horizon of the black hole has T+∝dκ/dr+>0. Recall that sign(δg+) = −sign(δr+) and the same holds for every horizon. If we take a derivative with respect to the mass of a product of two horizons from the same black hole, thus corresponding to the same mass, charge, etc. we get 1 S+S− ∂ ∂M S+S−=1 T+S+ +1 T−S− (3.2.20) The product of entropies will be independent of the mass when the above derivative vanishes, i.e. T+S++T−S−= 0, and this will happen in very specific cases within higher curvature theories. In general we will be just interested in the outermost horizon and we will keep for it the name of r+. 3.2.1 Vacuum horizons and Einstein-Hilbert gravity Let us start this subsection by discussing the horizon structure of the vacuum solutions. The general form of the metric function fis in this case f(r) = σ−Λr2,(3.2.21) so it can vanish at r=pσ/Λ, whenever σand Λ have the same sign, thus for hyperbolic AdS and spherical dS spacetimes. These horizons are observer dependent features since these spacetimes are maximally symmetric and, thus, any point can be considered as the origin. The dS case is widely known [141], this corresponding to the cosmological event horizon. The AdS case is, however, more obscure as long as the horizon is actually cloaking a finite size region in a similar way as a regular black hole horizon does. The black hole horizon is actually just a deformation of this ‘vacuum’ horizon. This has a problematic interpretation and has led to the proposal that the true ground state for hyperbolic spacetimes is not the massless one, but an extremal negative mass solution [133,134]. The cosmological horizon 68 CHAPTER 3. BLACK HOLE THERMODYNAMICS -2 -1 1 2 3 4 g U@gD Σ=1 Σ=0 Σ=-1 c0=1 c0=0 c0=-1 Κ=2 Κ=0.2 ΚNariai Figure 3.2: Linear polynomial corresponding to the usual EH-branch for negative (c0= 1), zero (c0= 0) and positive (c0=−1) cosmological constants (L= 1). The dashed lines are just κ(g/σ)d−1 2for d= 5. The solid black line corresponds to the critical value of the mass, κNariai = 1/4, for dS black holes with spherical horizon. The crossing of these lines with the polynomial gives the possible values for gat the horizon and then of r+(and rc). For σ= 1 and κ>κNariai (or r+> rNariai), the asymptotically dS branch describes a big crunch spacetime (f < 0, ∀r) without horizons. of pure dS spacetime has negative temperature, while for the AdS case the temperature is positive as for a regular black hole horizon. In order to analyze the horizon structure, let us focus on the asymptotically AdS, dS and flat black holes in Einstein-Hilbert gravity with cosmological constant [134]. We include this simple case here for completeness and as an illustration of our method. As clearly depicted in figure 3.2, the only case accepting all three distinct topologies without exhibiting naked singularities is the asymptotically AdS configuration, the other two cases being well-defined just for spherical topology. This AdS case, furthermore, has just one horizon for all three topologies. The asymptotically flat spherical black hole has one event horizon as well. The asymptotically dS spherical black hole has in general two horizons: One of them is just the deformation of the cosmological horizon already present in the maximally symmetric solution, while the other corresponds to the black hole. As the mass increases both horizons get closer to each other until, for some critical value of the mass, the so-called Nariai mass, κNariai =2Ld−3 d−1d−3 d−1d−3 2 ,(3.2.22) they actually meet (they disappear for masses above that value). The untrapped region, the spacetime as we usually consider it, is comprised between the two horizons and so for this extremal case it seems to disappear. A proper limiting procedure [142] shows that the geometry remains perfectly regular as κ→κNariai, and becomes the geometry of the Nariai solution. This space is the direct product of a dS2and a Sd−2, both with the same radius. Above this critical mass, though, it describes a big crunch spacetime. In the asymptotically AdS case, a negative mass extremal hyperbolic black hole has been 3.2. LOVELOCK BLACK HOLES AND THERMODYNAMICS 69 -4 -3 -2 -1 g U@gD Σ=0 Κ0 Κ=0.2 Figure 3.3: Negative mass hyperbolic black holes in Einstein-Hilbert gravity. The dashed line corresponds to a black hole with an outer and inner horizon (segment in red), while the solid line represents the extremal case, κ0=−κNariai. proposed as the ground state in Einstein-Hilbert gravity. The same would apply to any Lovelock theory. Black holes with larger (but negative) mass than this extremal one, κ0=−κNariai =−2Ld−3 d−1d−3 d−1d−3 d−1 ,(3.2.23) have two horizons, in a way reminiscent of the asymptotically dS spherical black hole with positive mass. The difference being that in the asymptotically dS case the two correspond respectively to the cosmological and black hole horizons, while in the AdS case they are the outer and inner horizons of a black hole (see figure 3.3). It is worth noticing that for negative masses we are exploring a completely different section of the polynomial than for positive values, the similarities being just due to the extremely simple form of the polynomial in the case under current analysis. In general, positive and negative mass black holes may have dramatically different features. As we can easily see in this simple example, the existence of just one black hole horizon in all positive mass cases implies that the singularity at the origin is always spacelike, while it is timelike in the negative mass hyperbolic case due to the presence of two black hole horizons that become degenerate in the extremal limit. In the case where the horizon and the singularity coincide, the nature of the latter is null. 3.2.2 Black hole entropy at extremality As we have seen in the last section, the vacuum (M= 0) state with hyperbolic topology has non-zero temperature as it displays a horizon. This is related to the fact that an accelerated observer in AdS would see a horizon with temperature related to its acceleration, in a similar manner as for the Unruh effect in Minkowski space. This makes it difficult to assume this 70 CHAPTER 3. BLACK HOLE THERMODYNAMICS state as groundstate, it cannot be considered at arbitrary temperature as it already has one. The alternative is to consider an extremal state instead. For that we have to make some comments on the special properties of extremal black holes. There has been some debate (see [143,144] for some recent discussion) about whether the entropy of extremal black holes corresponds to the usual Bekenstein-Hawking (or Wald in more general cases) or it’s simply zero as seems to indicate the semiclassical calculation [145–147]. Both approaches yield the same result in the non-extremal case, the reason for this discrepancy at extremality being the qualitative difference in the near horizon topology in both cases. In the Euclidean section, it has the topology of R2×Σd−2in for non-extremal black holes case whereas it is R×S1×Σd−2in the extremal case, effectively removing the horizon from the geometry. In the first case regularity of the flat factor forces the Euclidean time to be identified such that the period is the inverse of the temperature. This introduces a non-trivial temperature dependence on the solution and thus in the Euclidean on-shell action, b I, that it is at the origin of the entropy in this semiclassical picture. The entropy is defined as S=β∂ ∂β −1b I(3.2.24) In the extremal case the geometry is regular independently of the period βand the solution does not depend on this parameter either. The action depends on βjust because of the integration of the volume of the time circle, b Ibeing just proportional to β. The entropy thus vanishes as it can also be explicitly seen from the on-shell action computation. The na¨ıve result is b I=βM−˜ TS(3.2.25) where ˜ T=f0(r+)/4πis the usual expression for the black hole temperature. In the nonextremal case, the regularity of the solution at the horizon requires β=˜ T−1in such a way that the β-dependence of the secod term cancels and we get b I=βM −S(3.2.26) as expected. The extremal black hole is different as βand ˜ Tare unrelated in that case. Actually ˜ T= 0 yielding b Ie=βMe. The entropy would then vanish for the extremal black hole. Some other approaches yield the same form of the Bekenstein or Wald entropy but they usually rely on extremal limits of near-extremal solutions, whereas in our case we have asumed that the extremality condition holds `a priori. In case we want to consider any extremal state as ground state for a sector of the theory we will need it to be identified with arbitrary periodicity in Euclidean time. This cannot be done as we take the extremal limit of near-extremal solutions as in that case the temperature would be bound to vanish. We may however include the thermal extremal states at any temperature with zero entropy as, as we already explained, in that case the periodicity is not fixed. In what follows we will in general include extremal states in this way even though we will make some comments on the alterative situation in which they are not present. As we will see the latter case would have much more problematic interpretation. In particular, the introduction of extremal states as 3.3. TAXONOMY OF LOVELOCK BLACK HOLES 71 asympt. σ=−1σ= 0 σ= 1 EH g U@gD r®0 r* ¥ r+ Κ>0 HaL HbL g U@gD r®0 r* ¥ r+ Κ>0 HaL HbL g U@gD r+ r* ¥ r®0 Κ>0 HaL HbL Table 3.1: Taxonomy of the EH-branch black holes. groundstate will avoid the presence of negative entropy states, at least as globally preferred phases, as the free energy of any such state would be bigger than the extremal one F=M−TS > M > Me(3.2.27) This also happen for the other situations where negative entropies appear. In the case of type (b) spherical branches the analogous rˆole of the extremal state is played by the vacuum. 3.3 Taxonomy of Lovelock black holes We will study generic features of maximally symmetric Lovelock black holes in a case by case basis, considering the previously introduced classes of black hole branches (table 2.1). Some work in this direction has already been done considering just the LGB case [94,148]. Let us have a look on the different cases that we can encounter. 3.3.1 The Einstein-Hilbert branch The Einstein-Hilbert branch is just a deformation of the Schwarzschild-AdS black hole (c0= L−2) and can be identified as the branch crossing g= 0 with slope Υ0[0] = 1, exactly as in the Einstein-Hilbert case, and so the slope will be positive for the whole branch. This condition protects this branch from Boulware-Deser-like instabilities. When real, the cosmological constant associated with this branch is negative and so the spacetime is asymptotically AdS. We can proceed with this analysis in an analogous way for asymptotically dS spaces, just by changing the sign of the explicit cosmological constant in the action c0→ − 1 L2, or for asymptotically flat ones, just by setting c0= 0. We include the relevant part of table 2.1 below, for the reader convenience. Even though the EH-branch is just a deformation of the usual Schwarzschild-AdS black hole, it can be a quite dramatic one. For instance, it may happen that the polynomial has a minimum at gmin <0 (if there are several, gmin refers to the lowest one in absolute value), such that Υ[gmin]>0. A naked singularity would arise for large radius: the solution does not approach AdS asymptotically. This case was first discussed in [2,38] for third order Lovelock 72 CHAPTER 3. BLACK HOLE THERMODYNAMICS theory, but the same applies in the general case for a vast region of the space of parameters that will be named, following the aforementioned reference, the excluded region. In order to avoid the excluded region the value of Υ[gmin] at the biggest negative minimum, gmin, has to be negative. Notice that this does not depend on the topology of the solution, the excluded region being the same for all of them. The sector of the parameter space where this new kind of singularity appears has to be excluded in general, not only because of its nakedness but in reason of the perturbative instability of the corresponding solution [6], as we will see in section 4.4. For hyperbolic or planar topology, as this branch always crosses g= 0 with positive slope, Υ0[0] = 1, it has always a horizon hiding the singularity of the geometry that is located (see table 3.1) •either at r= 0 [(a) type], •or at the value corresponding to a maximum of Υ[g] [(b) type]. For hyperbolic horizons we have again the possibility of considering negative mass black holes, for masses above a critical value corresponding to the extremal case. This makes no difference with respect to the same situation taking place in an AdS branch and, thus, will be discussed at length below. The spherical case is quite more involved. For high enough mass, the existence of the horizon is ensured, but this is not the case in general. For the (a) type EH-branch the existence of the horizon can be elucidated by analyzing (2.3.6) in the limit of small mass g+→ ∞, Υ[g+]≈cKgK +≈κg d−1 2 +(3.3.1) For the horizon to exist in this limit, we need the right hand side of the equation to be bigger than the left hand side. This is ensured for d > 2K+ 1 as in this case the biggest power in the left hand side would be smaller than (d−1)/2. The existence of a horizon in the small mass limit ensures the existence of (at least) one for all values of the mass, simply due to the continuity of Υ[g]. The case d= 2K+ 1 is critical. There will be a minimal mass (κcrit ≤cK) below which a naked singularity appears. In principle, for high enough orders of the Lovelock polynomial, more than one horizon can exist but for the critical case, at some point, all of them disappear. The number of black hole horizons determines the type of singularity situated at r= 0, space or timelike. For d > 2K+ 1 we will always have an odd number of horizons (taking into account possible degeneracies), since the (spacelike) singularity is in the trapped region of the spacetime. For d= 2K+ 1, the number of horizons depends on the value of the mass. For masses above cKthe number is odd and at least one horizon will always exist, whereas for masses below this critical mass the number of horizons will change to an even quantity, and will actually disappear at some point. In any case the minimal mass horizonful solution always corresponds to a zero temperature state with a gap to the actual vacuum. This is similar to what happens for quasi-topological black holes with the difference of the number of horizons being always even. The zero tempearature state is reached as two of these horizons merge in this case. 3.3. TAXONOMY OF LOVELOCK BLACK HOLES 73 -4 -3 -2 -1 1 2 3 g 0.5 1.0 1.5 2.0 2.5 3.0 3.5 U@gD r®0 ¥ r+ Κ=2 Κ=0.19 HaL -2 -1 1 2 3 4 g 0.5 1.0 1.5 2.0 2.5 3.0 3.5 U@gD r* ¥ r+ Κ=2 Κ=0.2 Κ* HbL Figure 3.4: EH-branch in GB gravity in 5 dimensions for λ= 1/5 and λ=−1/4 respectively (L= 1). The dashed curves correspond to the κ(g/σ)d−1 2for spherical topology (σ= 1). The singularity becomes naked for κ≤λ= 0.2 in the first case and for κ≤κ?= 0.5 in the second case. For higher dimensions, the second figure would be qualitatively the same with just a different value of the critical mass. The first one, instead, changes as long as the horizon exists for all positive values of κin that case. For the (b) type branches the singularity is always located at r?=√−2λ L. The case where the EH-branch ends up at a maximum for some positive value of g=g? (r=r?), or (b) type branch, is even simpler. There is a critical value of the mass for which r+=r?. Below that mass a naked singularity appears. This is very similar to the situation described in the previous paragraph, the only difference being that in that case the radius of the singularity is zero, r?= 0. Also, for type (b) black holes, the temperature diverges as we approach the radius of the singularity. The simplest example of Lovelock theory is GB gravity [149], where we have just two branches, one of them suffering from BD instabilities. The remaining branch is thus an EHbranch. For λ > 0 this branch is of the (a) type, extending all the way up to a singularity situated at r= 0. For λ < 0, instead, this branch has a maximum at positive values of g (see figure 3.4). This is a singularity at a finite value of rthat may or may not be naked depending on the value of the mass. The mass for which the horizon coincides with the singularity is κ?=1 2(1 −4λ)(−2λL2)d−3 2.(3.3.2) For bigger masses we have a well defined horizon while below this bound the singularity becomes naked. Another intriguing possibility, that cannot be observed within the simple setting of GB gravity, is the would be appearance of several black hole horizons. For this to happen we need inflection points in Υ[g], and so the minimal example would be the cubic Lovelock theory. In the critical d= 7 case, we have the possibility of obtaining two black hole horizons for some regions of the space of parameters, while this number is three in higher dimensions. One remarkable thing worth noticing here is that, as couples of horizons appear or disappear 74 CHAPTER 3. BLACK HOLE THERMODYNAMICS asympt. σ=−1σ= 0 σ= 1 AdS g U@gD ¥ r+ r+ r* Κ<0HbL Κ>0HaL -4 -3 -2 -1 g U@gD ¥ r* Κ<0HbL Κ>0HaL -4 -3 -2 -1 g U@gD ¥ r* Κ<0HbL Κ>0HaL Table 3.2: Taxonomy of the AdS (other than EH) branches. when we vary the value of the mass, the value of (the biggest horizon) r+may change discontinuously. Thus, the temperature as a function of the mass also varies discontinuously when crossing the values of κfor which the outermost couple of horizons appear or disappear. One of the sides of the discontinuity has zero temperature (since the black hole is extremal for such critical mass) while the other has finite temperature. We must recall that the inner horizons are in general unstable [150–153] and this possibly means that we should not trust our solution behind the outermost inner horizon. We may interpret these extremal states as black hole ground states, each one for a given range of masses. These seem naively accessible by evaporation and, thus, point towards a violation of the third law of black hole dynamics [122]. They are in general unstable solutions [154,155], though (see also section 4.3). At low temperatures, this particular branch will have several possible black hole masses and transitions might occur among them and the thermal vacuum. As for temperatures close to zero the free energy essentially coincides with the mass, the globally preferred solution is the less massive one: the vacuum. We will comment more on this later on. 3.3.2 AdS (other than EH) branches The second class of branches that we describe in what follows are asymptotically AdS black holes different from the EH-branch. The latter will be included just for the discussion of negative mass solutions since the analysis is exactly the same. Consider first the positive mass solutions, for which the AdS branches always end at a maximum of the polynomial (see table 3.2). As before, the existence of a horizon cloaking the singularity fixes the topology of such branches: As the considered sections of the polynomial run over negative values of g, horizons exist just for σ=−1. On the other cases, the solutions describe a spacetime with a timelike naked singularity. The condition for the existence of a horizon sets an upper bound on the mass, κ < κmax,κmax corresponding to the critical value for which the radius of the horizon coincide with that of the singularity (r+=rmax), i.e., κmax =rd−1 max Υ[gmax].(3.3.3) When we encountered a naked singularity with positive mass in the Einstein-Hilbert theory, it corresponded to the low mass limit of a multi-horizon black hole (below a given 3.3. TAXONOMY OF LOVELOCK BLACK HOLES 75 -6 -4 -2 2 g U@gD Σ=1 Σ=0 Σ=-1 Κmax Κ1 Κ2 Figure 3.5: Qualitative representation of (positive mass) AdS branches. The blue and red branches will be referred to as type (b1) and (b2) respectively, when considered for negative masses. The green one is an excluded branch. There are asymptotically AdS massive black holes for κ < κmax (red branch), otherwise the geometry displays a naked singularity. mass, the two horizons merge and disappear altogether, leaving the singularity naked). The case here is somehow different as the black hole horizon cannot degenerate as we increase the mass approaching the critical value. Thereby, the solution infinitesimally close to the critical one has non-zero temperature, diverging as we approach the bound. This is more reminiscent of the low mass limit of Schwarzschild black holes (ultimately leading to a regular geometry) than of the usual naked singularities in Einstein-Hilbert gravity. It is also similar to what happens for type (b) spherical black holes. For negative mass solutions, the analysis of the existence of black hole horizons and its number is more involved. The main qualitative feature is the possibility of having a minimum of the polynomial associated to the branch under analysis (see blue and red branches in figure 3.5 for instance). We will refer to the case without such minimum (blue branch) as type (b1) solution and as type (b2) for the other one (red branch). The structure of horizons and the type of singularity will differ in both types of branches. When d= 2K+ 1 and the branch we are considering is of (b1) type, there is a minimal mass for which, instead of an extremal regular spacetime with a degenerate horizon, we have a naked singularity (see figure 3.6). The temperature also vanishes asymptotically as we decrease the mass. We will not comment further on these kind of solutions as they are gravitationally unstable against perturbations [9]. In case we have a well defined extremal negative mass black hole, we always have at least two horizons for (b1) type branches and d > 2K+ 1, as we depart from extremality. For (b2) type branches, however, the inner horizon disappears when its radius coincides with the radius of the singularity, changing its nature from timelike to spacelike, or viceversa. There is a critical mass for which this happens. This is irrelevant for an outside observer who cannot extract information from the inner horizon. Figure 3.7 shows both kinds of solutions in the simplest case of GB gravity, for (b1) type (λ < 0) and (b2) type (λ > 0), respectively. 76 CHAPTER 3. BLACK HOLE THERMODYNAMICS -6 -5 -4 -3 -2 -1 g -5 -4 -3 -2 -1 1 U@gD Κ1 Κ2 Κ0 Κ3 Hb1L Figure 3.6: EH-branch in the cubic theory in 7 dimensions for λ= 0.4 and µ= 0.2 (L= 1). The dashed curves correspond to κ(g+/σ)d−1 2with κ0=−µ/3 = −2/30, κ1=−0.022, κ2=λ=−0.015, and κ3=−0.008 (σ=−1). For masses above κ0we have one horizon with no distinction between positive and negative masses. For κ≤κ0there is a naked singularity at r= 0. No extremal state exists. -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 g -6 -4 -2 U@gD r* r+ Κ1 Κ2 Κcrit Κ0 Hb1L -4 -3 -2 -1 g -1.0 -0.5 0.5 1.0 1.5 U@gD Κ1 Κ2 Κ0 Κ* Hb2L Figure 3.7: EH-branch in GB gravity in 5 dimensions for λ=−1/2 and λ= 0.15 respectively (L= 1). The dashed curves correspond to κ(g+/σ)d−1 2with κ0=−0.75, κ1=−0.6, κcrit =λ=−0.5, and κ2=−0.3 (left) and κ0=−0.1, κ1=−0.08, κ?=−0.06 and κ2=−0.05 (right) (σ=−1). In both cases we have two horizons for κ0< κ < κ?and one for κ?< κ. For κ=κ0we have a degenerate horizon. The singularity becomes naked for κ≤λ−1/4 in both cases. In higher dimensions, the behavior is qualitatively the same. For the (b2) type branches the singularity is always located at r?=L√−2λ, while it is at the origin in the (b1) case. 3.4. HEAT CAPACITY AND LOCAL THERMODYNAMIC STABILITY 83 3.4.2 Hyperbolic black holes in the AdS-branches Most of the discussion on hyperbolic black holes in the EH-branch also applies, on general grounds, to the AdS-branches. The only difference being that, in general, there is a maximal mass for which the temperature diverges. Thus, close enough to that point the heat capacity has necessarily to be positive and the black hole thermodynamically stable. This can be seen directly from (3.4.1), as the heat capacity close to the maximum approaches dT dr+≈d−1 2π g+Υ[g+] Υ00[g+] Υ0[g+]2,(3.4.5) diverging as well when we reach the critical mass. Υ[g+] is positive due to the positivity of the mass. The second derivative Υ00[g+] is negative as we are close to a maximum, but g+is negative as well. The plot of temperature versus horizon radius will be in general qualitatively similar to that corresponding to the EH-branch, with the difference that the temperature diverges at some finite value of r+. 3.4.3 Spherical black holes in the dS-branches The low mass regime of the spherical solutions corresponding to dS branches is very similar to that of the EH-branch. The high mass regime, instead, is very different. These black holes may increase their mass until they reach a maximal (so-called Nariai) mass, which is set by the shape of the polynomial. This is an extremal state with zero temperature and, as we reach it from lower mass configurations, the system is thermodynamically unstable close to it. We may construct dS branches of (a) and (b) types using GB gravity with positive cosmological constant (setting c0=−L−2). In this case, the (b) type branch (figure 3.12, left) is unstable for all allowed values of the mass, since we are considering d= 5 (which is d= 2K+ 1 in this case), while a stable region of small black holes appears for (a) type branches (figure 3.12, right) [164]. This stable region disappears in higher dimensions. In general, spherical dS branches may have some stable intermediate region, but they are unstable (or even non-existent) for high enough temperature. Let us summarize the main results of this section. Hyperbolic black holes generically have two stable domains, at low and high temperatures. For intermediate temperatures, these solutions may have more than one possible mass, some of them unstable. The only difference of AdS branches with respect to the EH case is that the high temperature regime has a maximal finite mass in the former. The case of spherical black holes exhibit quite distinct features. From the thermodynamical point of view, we may distinguish those situations where there is, or there is not, a minimal temperature, Tmin, for the black holes to exist. In the former case, we do not reach any extremal black hole, neither in the low mass, nor in the high mass regimes. In the case of the EH-branch, for T < Tmin, only the thermal vacuum may exist whereas, for high enough temperatures, a black hole may exist with two very different masses, one close to κmin (which is unstable) and one very high (stable). For intermediate temperatures close to Tmin, we may in principle encounter several stable and/or unstable black holes. In the latter 84 CHAPTER 3. BLACK HOLE THERMODYNAMICS 0.0 0.2 0.4 0.6 0.8 1.0 r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 T 0.0 0.2 0.4 0.6 0.8 1.0 r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 T Figure 3.12: Temperature versus horizon radius (equivalently, mass) for spherical black holes in d= 5 GB gravity with positive cosmological constant, L2=−1 (in blue). The first figure, λ=−0.1, corresponds to a (b) type dS branch. We can identify the zero temperature state in the high mass regime with the Nariai solution. The temperature diverges as we approach the lower bound, κmin, indicated by the dashed blue line. For λ= 0.1, in the second figure, a spherical (a) type dS branch arises. We may identify again the extremal state with maximal mass with the Nariai solution, for which the temperature goes to zero, as well as the radius of the black hole horizon. The ‘zero size black hole’ has finite mass in this case. For higher dimensions the stable region of small black holes disappears (gray line corresponds to d= 6), the spherical black holes being unstable as their (b) type counterparts. The only difference is that the temperature diverges in the zero mass limit. case, instead, black hole solutions exist for the whole range of temperatures, except in the case where a dS-branch reaches an extremal state at low as well as at high masses. For the EH-branch, in this situation, we have two stable phases again, one in the low and one in the high temperature regimes. At intermediate temperatures, the black hole may have several possible masses, some of them unstable. 3.5 Hawking-Page-like phase transitions The existence of unstable phases, as well as the several possible black hole solutions at the same temperature, suggest the occurrence of Hawking-Page-like phase transitions, as already observed in the case of LGB gravity [129,137]. These phase transitions should be also relevant when studying the physics of the dual CFT plasma. In order to analyze this we need to discuss the global stability of the solutions. Any system in thermal equilibrium with an infinite heat reservoir (and thus at constant temperature) will be described by the canonical ensemble, whose relevant thermodynamic potential is the Helmholtz free energy, F. The preferred, and so globally stable, solution is the one that minimizes F. For instance, the free energy of the black hole solution calculated in (3.2.9), is the free energy with respect to the vacuum solution, except for the hyperbolic case where the finite ground state free energy (its mass) must be subtracted. Therefore, the sign of the free energy determines which solution is globally preferred at any given temperature, the appropriate black hole (if 3.5. HAWKING-PAGE-LIKE PHASE TRANSITIONS 85 several are possible) or a thermal bath for the groundstate (vacuum or extremal). The general analysis is, again, hard and not very enlightening. We will just concentrate in showing general features of these black hole solutions without entering into the details of the different cases. We will consider the same regimes analyzed for the local stability, as there we can easily find the expression for the free energy. In the planar case the analysis is very simple since the free energy reads F=−Vd−2 16πG rd−1 + L2.(3.5.1) The black hole is then always the preferred solution, as indicated by the negative sign of the free energy, and no phase transitions occur. This will be the situation in the large mass limit of the other topologies in the EH-branch. As for large enough r+the free energy may be as large as one wants, ambiguities on the reference background do not matter in this limit. 3.5.1 Spherical black holes For spherical black holes, we will restrict our discussion to the two most generic situations. For the EH-branch, we will consider separately the case of having a stable low temperature phase (as in d= 5 GB gravity with positive λ), and the case where a minimal temperature is needed for black hole solutions to exist. The second case is the analog of the usual situation in Einstein-Hilbert gravity. At low temperatures, the thermal vacuum can be considered as the globally stable solution whereas, for higher temperatures, two or more black hole solutions are possible. For high enough temperature just two of them remain. The small one has always positive free energy. For an (a) type branch, F=(d−2)Vd−2 16πG rd−2K−1 + d−2KcK,(3.5.2) whereas for the (b) type case the temperature diverges as we approach the maximum g+→ g?, and the free energy is F≈ −TS. Then, for positive entropy, the small black hole solution has negative free energy and is stable against the vacuum. However, it is not the minimum of the free energy since the big black hole has always a lower one. This is quite easy to see by realizing that the small black hole entropy goes to a constant as we approach g?whereas the entropy grows indefinitely for big black holes, since they approach the planar limit. Thus, the small black holes are not just locally but also globally unstable. The big black holes have, in general, negative free energy. We have then a Hawking-Pagelike phase transition, from the thermal vacuum at low temperatures to big black holes at high temperatures. The difference with respect to the Einstein-Hilbert case is that we may have several black holes at intermediate temperatures, with either sign of the free energy. For ranges of temperature where several black holes have negative free energy, transitions among them may happen, the globally preferred solution being the one with the lowest free energy. This would be an example of a new kind of phase transition, different from the Hawking-Page one, where one of the phases is always the thermal vacuum. If the EH-branch has stable low temperature black holes, i.e., for (a) type in d= 2K+1, these are globally unstable as indicated by their positive free energy that asymptotes a 86 CHAPTER 3. BLACK HOLE THERMODYNAMICS constant when r+→0, F=(d−2)Vd−2 16πG cK,(3.5.3) actually to the mass as both the entropy and temperature vanish in this limit. This is exactly equal to the earlier formula (3.5.2) for the given dimension. The same happens for the would be extremal black holes that one may encounter in the EH-branch. In the limit of low temperatures, the free energy coincides with the mass and, as such states have positive mass, they are globally unstable. The globally preferred phase is the thermal vacuum which is the minimal mass solution. Then, again, one has the same kind of transition described in the previous paragraph. Another situation we did not comment at length is the possibility of having negative entropy for the spherical black hole with critical mass, κmin. This happens already in the simplest possible case of GB gravity, for negative λ, where there is a maximum in the EHbranch situated at g?=−1/2λ. As pathological as it may seem, the consequence of this from the global stability point of view is clear. Again, the globally preferred solution is the big black hole as before, and the discussion goes through. This is quite general: a negative entropy state necessarily has bigger free energy than the vacuum (characterized by minimal mass and vanishing entropy). For dS branches the situation at low mass is exactly the same as for spherical solutions in the EH-branch. At low temperatures the free energy approaches the value of the mass and the globally stable phase is always thermal vacuum. For (a) type branches and d > 2K+ 1, no high temperature black hole exists, thereby the preferred phase in that regime would be trivially the thermal vacuum and no Hawking-Page-like phase transition seems to occur (see, for instance, [164], for the GB case). In any other situation (e.g., (b) type branches) with positive entropy, the globally stable solution would be the near-critical black hole approaching the maximum of the polynomial. Therefore, these branches seem to display phase transitions, even though high temperature black holes are locally thermodynamically unstable. The inclusion of the extremal Nariai space with arbitrary temperature and zero entropy does not change these conclusions. As it has higher mass than the vacuum it is always globally (and locally) unstable 3.5.2 Hyperbolic black holes For the hyperbolic black holes one may compute the free energy at the high and low temperature regimes as before. As the maximally symmetric space has temperature in this case it is not clear how to use it as a ground state. Instead, we will consider the extremal negative mass black hole as the reference state –with vanishing entropy– given that it can be identified with any temperature [147]3, as explained earlier. Otherwise, the analysis would become trivial with just one or more black hole solutions, no matter the value of the temperature. No Hawking-Page-like phase transitions would occur in that case, just the possibility of transitions among black holes of different masses at intermediate temperatures. 3This has been disputed by some authors (see [154] for instance) in reason of the semiclassical instability of these solutions. 3.6. DISCUSSION 87 For the EH-branch in the high temperature regime we just have one possible black hole solution. It has negative free energy as it approaches the planar limit and so it is globally preferred. The same happens for the AdS branches, that end up at a maximum of the polynomial. As we approach the critical mass, κmax, the free energy approaches F≈ −TS that is arbitrarily negative for positive entropy. For negative entropy at the maximum, which corresponds to the biggest possible black hole in the AdS branch, every black hole has negative entropy. In this case the globally preferred phase is the reference state, since it has zero entropy and minimal mass. For lower temperatures we have to consider black holes close to the extremal one. In the zero temperature limit only these extremal states matter and their free energy is simply given by their mass. Then, the globally preferred phase in that limit is the lowest mass state. For slightly higher temperatures one expects that the globally preferred solution is still described by the same minimal mass extremal state (identified with finite temperature) or the corresponding black hole solution that is just a smooth deformation of it. It is however hard to elucidate in general which of both solutions has the lowest free energy, and then the existence or not of Hawking-Page-like phase transitions. As we further increase the temperature, we might also encounter transitions among extremal or near extremal solutions associated to different extremal masses. 3.6 Discussion In this and previous chapters we presented a novel approach to deal with the full classification and description of black hole solutions with constant curvature horizons in Lovelock gravity. Our proposal allows to treat the generic case where the whole set of Lovelock coupling constants is arbitrary, contrary to most studies existing in the literature where the analysis is restricted to particular cases. Most of these cases, moreover, correspond to degenerate vacua of Lovelock gravity, while our approach is valid in general and is most useful in the non-degenerate case. We discussed the main features of all possible configurations, focusing in the neutral case. In particular, we have established a recipe to scrutinize the number of horizons and their evolution with the mass, something that we expect to be useful to visualize and gain intuition in physical processes involving black holes in such theories: evaporation, mass accretion and appearance of naked singularities [9]. We will comment more on this on the next chapter. The analysis of charged black holes and even cosmological solutions can be performed in a very similar manner. The same happens for some more general classes of higher curvature gravities described recently [69] that share the form of the black hole solutions (and consequently their thermodynamic properties) with Lovelock theories. Most of the results of this chapter are also of direct application there. In particular quasi-topological share some crucial properties with Lovelock’s in their critical dimension, d= 2K+ 1, despite their higher curvature order. We presented some general features of Lovelock black holes’ thermodynamics, analyzed their local and global stabilities and the possible existence of phase transitions. Even if these solutions show some seemingly pathological features, such as negative values for the entropy, these are avoided if we restrict ourselves to the globally preferred phase. 88 CHAPTER 3. BLACK HOLE THERMODYNAMICS For asymptotically AdS solutions (either in the EH or AdS-branches), global stability in the high temperature regime always selects the biggest black hole, being the one with biggest entropy. If we apply the same criterium to all possible solutions, regardless of the branch to which they belong, the selected solution is always the one approaching the planar limit, since it is the only one that has arbitrarily big entropy. The comparison of solutions belonging to different branches is not really allowed as they have different asymptotics. The usual Euclidean prescription says that we must compare all solutions with the same boundary conditions what certainly includes the asymptotics. However the existence of bubble solutions [55,56,165] separating regions corresponding to different vacua suggest the possible existence of mixed solutions and transitions between branches. In that context, the high temperature phase would na¨ıvely always correspond to the universal planar limit. This kind of branch transitions will be the focus of chapter 5where we will show that the situation is not as simple. The usual Einstein-Hilbert gravity admits in principle topological solutions displaying naked singularities. These may arise as a result of a bad choice of topology or as associated to negative mass, below the extremal one for hyperbolic horizons. The latter are just a special case of trans-extremal solutions, where the values of the parameters are chosen in such a way that, an otherwise well defined black hole with positive temperature, is beyond the extremal state. Example of this are the Reissner-N¨ordstrom or the negative mass hyperbolic black holes. In principle, all these situations are ruled out by the cosmic censorship conjecture that states that naked singularities do not form in the evolution of generic initial conditions. For instance, the evaporation process for black holes with several horizons should stop at the extremal state as it has zero temperature and this avoids the formation of trans-extremal solutions in that case. The situation in generic Lovelock theories of gravity is rather different. For this wide family, there are several situations that suggest a possible violation of the cosmic censorship conjecture, as we have seen analyzing the case of static uncharged black holes. In addition to the cases pointed out in the previous paragraph, new kinds of naked singularities appear, some of them which naively seem to be formed in the evolution of these black holes. This can happen for the otherwise well-behaved Einstein-Hilbert branch, and it certainly happens generically on the extra ‘higher order’ (A)dS-branches. Most of these naked singularities arise because the branch of interest ends up at a maximum or a positive (for positive mass) minimum of the Lovelock polynomial. The latter corresponds to a complex cosmological constant associated with this particular branch. The former, in turn, appear in a variety of cases. They constitute a maximal mass for hyperbolic black holes of AdS-branches as well as a minimal mass for some spherical black holes in the EH or dS-branches. The other possibility for naked singularities to appear is just the spherical case in the maximal d= 2K+ 1 Lovelock theory, for the EH or dS-branches when they extend all the way to r= 0 without encountering any singularity (maxima or minima of Υ[g]). In those cases we find a naked singularity for masses below a critical value. Any other possible naked singularity may be considered in the same class as those appearing in Einstein-Hilbert gravity. As we think of the evolution of the black holes studied in this thesis, we realize that naked singularities seem easy to form, at least naively. Consider for instance the evapora- 3.6. DISCUSSION 89 tion of spherical black holes. For (b) type EH or dS branches these black holes always reach a critical mass where the horizon coincides with the singularity. At that point the temperature diverges but a finite mass naked singularity remains. The naked singularity inevitably forms. We emphasized the word naively before since the present analysis just considers the thermodynamic stability of the solutions. These solutions are locally and globally unstable, however they are still valid solutions that may form under evolution of generic spherically symmetric initial conditions. In the next chapter we will perform a more detailed analysis in order to elucidate whether naked singularities may form or not in these theories [9]. We have fixed, throughout this paper, the values of the cosmological constant and the Newton constant appearing in the lagrangian to their customary values in AdS Lovelock gravities. It is worthwhile mentioning that a straightforward generalization of this work amounts to studying the case of dS Lovelock theories (note that there are AdS vacua also in this case), as well as theories where the Newton constant has negative sign. On the one hand, we shall mention that this sign flip was already considered in the context of three dimensional topologically massive gravity, where it was found that a negative Newton constant is useful to render otherwise negative energy modes harmless for the stability about flat space [166]. Furthermore, in higher dimensions, even though GN<0, the generic structure of branches discussed in this paper will remain, and there will always be solutions corresponding to well-defined gravities with positive Newton’s constant. Lovelock theories have the remarkable feature that lots of physically relevant information is encoded in the characteristic polynomial Υ[g]. Boulware-Deser-like instabilities, for instance, can be simply written as Υ0[Λ] <0, which has a beautiful CFT counterpart telling us that the central charge, CT, has to be positive. Now, Υ0[Λ] can be thought of as the asymptotic value of the quantity Υ0[g] that is meaningful in the interior of the geometry, and has to be positive all along the corresponding branch. Naked singularities taking place at extremal points of the polynomial are suggestive of the fact that Υ0[g] should be a meaningful entry of the holographic dictionary (see [40] for related ideas) that does not exist in the case of Einstein-Hilbert gravity. The relevance of Lovelock and more general higher curvature gravity theories in the context of the AdS/CFT correspondence will be the topic of the second part of the thesis. Chapter 4 Metric perturbations and stability “The most incomprehensible thing about the universe is that it is comprehensible” Albert Einstein Perturbation analysis is a very powerful tool to investigate the dynamical response of a system against small disturbances. This is particularly important for the case of gravity due to their strong non-linearity. This is already true for general relativity but even more for Lovelock or other higher curvature theories which are much more complicated systems. The equations of motion on any of these theories are very hard to solve analytically and exact solutions are known just on very special circumstances. Perturbation analysis of exact solutions plays a crucial rˆole in many physical situations and opens a window into the intricate dynamics of gravity in four and higher dimensions. For our purposes, the most important examples of application of perturbative analysis in the context of gravity are the studies of black holes. Such an investigation was first systematically done for the Schwarzschild black hole by Regge and Wheeler [167] in 1957 and completed some 13 years later by Zerilli [168] and Teukolsky [169]. Among the many applications of these methods it is of particular importance the analysis of the stability of black holes. This is a fundamental question at many levels as unstable solutions are less likeky to form through any physical process and even when they do they will certainly not remain on that state for very long. This has many implications, from the dynamics of black holes to the determination of the final fate of gravitational collapse. Apart from the stability issue, perturbation analysis also tells us a lot about basic properties of black hole solutions. For instance the study of stationary perturbations of a stationary black hole solution provides a criterion for uniqueness and the search for new solutions. The stability of higher dimensional black holes in EH theory has been intensively studied 91 92 CHAPTER 4. METRIC PERTURBATIONS AND STABILITY (for a review see for instance [170] and reference therein), higher dimensional Schwarzschild black holes being stable for any type of perturbations. For static spherically symmetric Lovelock black holes, the analysis is not as straightforward. This is due to the complicated form of the equations of motion, even on the linear case, but also to the form of the solutions themselves the existence of branches, etc. Stability analyses under all type perturbations have been performed however [127,156–162,171–175], also for the charged case [102,103], and instabilities have been found in some situations. Perturbative analysis as the one that follows rely greatly on the developement of master equations for gauge invariant gravitational perturbations, a much simpler and intuitive approach. In the context of Lovelock theories of gravity, generic master equations have been found in [174], however, except for some restricted efforts, most of the work has been performed however in the context of LGB gravity. We will make use of these master equations, or rather the effective potentials they define, in order to analyse stability of black hole solutions in a particular regime, that of high momentum gravitons. This restricted analysis will greatly simplify the computations, nonetheless it will still be general enough to uncover some very interesting features of Lovelock black hole solutions. All the computations will be performed analytically and will be useful in order to gain general intuition about gravitational instabilities in Lovelock gravities. In section 1we already used metric perturbations in order to show one of the pathologies associated with Lovelock gravities. We saw that the sign of the kinetic term of the perturbations about a given vacuum, Λiis proportional to Υ0[Λi] in such a way that a negative slope indicates the presence of ghosts. This in turn ruled out the black hole solutions associated with this vacuum for the same reason. In the case of static black holes, we will find for instance that for planar black holes we have to restrict the possible values of the Lovelock coefficients in order to avoid instabilities. For non planar black holes the situation is much more involved but we will still be able to describe some important qualitative features of these instabilities. We will uncover their close relation to the Cosmic Censorship conjecture, its relevance for the analysis of black hole evaporation and also for ruling out particular black hole solutions that otherwise lead to some troublesome behavior. All these effects, rather than being pathological, seem to provide the instabilities found with some very precise physical significance. 4.1 Graviton potentials Throughout this chapter we will be considering the same black hole solutions of the preceeding sections as described by the metric form (2.1.3) and the corresponding black hole polynomial (2.1.8) defining f. We add a generic perturbation of the metric, hab, with fixed frequency, ω, and momentum, qin a fixed direction. These fluctuations split into three channels according to their polarization relative to the momentum, namely the tensor, shear and sound channels [176] (or equivalently helicity/spin two, one and zero respectively). The equations of motion for these dynamical degrees of freedom, φh(r), the subindex h= 0,1,2 indicating the corresponding helicity, can be recast as Schr¨odinger type equations [174], that 4.1. GRAVITON POTENTIALS 93 in the large momentum limit they reduce to a very simple form1 −~2∂2 yΨh+Uh(y) Ψh=α2Ψh,~≡1 q→0,(4.1.1) where α=ω2/q2,yis a dimensionless tortoise coordinate defined as dy/dr =√−Λ/f(r), and Ψh(y) = Bh(y)φh(y), where Bh(y) are functions of the metric whose specific expression can be found in [174]. For a regular solution Ψh(y), the metric perturbation φh(y) blows up as Bh(y) approaches zero. In such case, it could not be considered any longer as a perturbation and, in that sense, the linearized analysis would be spoiled. We then need to make sure that the function Bh(y) is non-vanishing. In our regime of interest, the effective potentials Uh, can be determined as the speed of large momentum gravitons in constant yslices, Uh(y) = (c2 i(y)y < 0, +∞y= 0 ,(4.1.2) y= 0 being the boundary of the spacetime. The explicit computation for the helicity two graviton and details on the other helicities are given on appendix A For a generic Lovelock theory, in terms of the original radial variable, one finds for the tensor, shear and sound channels, respectively [2]: c2 2(r) = L2f(r) (d−4) r2C(2) d[g, r] C(1) d[g, r], c2 1(r) = L2f(r) (d−3) r2C(1) d[g, r] C(0) d[g, r],(4.1.3) c2 0(r) = L2f(r) (d−2) r2 2C(1) d[g, r] C(0) d[g, r]−C(2) d[g, r] C(1) d[g, r]!, where C(k) d[g, r] are functionals involving up to kth-order derivatives of gdefined in (A.1.13). Notice also that there is a quite simple relation between the three potentials, (d−2)c2 0(r)−2(d−3)c2 1(r)+(d−4)c2 2(r) = 0 (4.1.4) in such a way that any of the three can be written as a combination of the other two. These expressions are valid in general, also for charged black holes. In the uncharged case however the black hole equation (2.1.8) is simpler and this allows us to make a simplifying change of variable. Instead of rwe take Υ as independent variable (the relation is one-to-one in absence of charge) and we define x≡log L2Υ and F≡log L2Υ0so that r∂r=−(d−1)∂x 1In the notation of [174], we must identify γi≡γ=q2L2, and our potentials are related to theirs, Ui→ −Vi/γ Λ, as γ→ ∞. 100 CHAPTER 4. METRIC PERTURBATIONS AND STABILITY but also to having a negative coefficient for the graviton kinetic term in the black hole background and thus to unitarity. As we reduce the mass approaching the extremal value the black hole becomes unstable, earlier than any jump in the black hole radius. The solution is unstable also for any lower mass as the value of gfor the extremal point ge belongs to the untrapped region. The puzzle posed by these solutions is then solved, the pathological behavior being related to non-unitarity of the gravitons and instabilities of the black hole. Before any pathological behavior is encountered an instability sets in driving the system somewhere else. This also forbids the possibility of violations of the third law of thermodynamics. In the case where several extremal points (or masses) exist the relevant one is always the most massive. Notice that in general the instability is triggered for masses slightly above that extremal one. The previous analysis can be performed in almost the same manner for other extremal states that appeared in the classification of Lovelock black hole solutions. These examples are dS black holes at the Nariai mass(es) and extremal hyperbolic black holes. These are present also in general relativity and it is easy to show that they respect the stability constraint at the extremal point. The value of F0 eis not bigger but lower than the critical value of (d−3)/(d−1) in these situations as either we have negative mass or the curve κeg+is not below but above Υ[g+] close to the extremal point. We then do not in general expect instabilities for such backgrouds. This supports the consideration of some of these extremal solutions as groundstates, nonetheless they might still be unstable for the other channels. 4.4 The Cosmic Censorship Conjecture and stability The result of previous section is also relevant for the discussion of the Cosmic Censorship Conjecture in Lovelock theories. For the minimal dimension d= 2K+1 at any given order K it may happen that the two merging horizons are the only ones of the solution and thus the singularity behind them would then become naked. It is hard to imagine any physical process that would reduce the black hole mass below the extremal threshold, nevertheless, even if it existed, we have just shown that the instability sets in before the singularity becomes naked, in fact before the black hole becomes extremal. However, we also need to care about black holes of lower masses as they can `a priori be created directly by collapse. We have just shown that these solutions are also unstable, a strong indication that they cannot be the en point of gravitational collapse on generic circumstances. For matter collapsing to a regular black hole with a horizon, the formation of the latter contitutes a critical moment for the matter contained within it. Think of a spherically symmetric configuration. The causal properties of event horizons force all the matter to end up at the central singularity, no matter the details of the matter distribution, and it cannot scape from there as causally it would imply travelling backwards in time. This is radically different if the end point is a naked singularity. As this solution is not provided with an event horizon, matter may in principle scape from the singularity without violating any causal structure and the final configuration may be much more sensitive to the details of the configuration under collapse. The Penrose diagrams of both such processes are schematically depicted in figure 4.3. It makes then sense to analyze the stability of such hypothetical solutions in 4.4. THE COSMIC CENSOR 101 r= 0 r=rH r= 0 r= 0 Figure 4.3: Penrose diagrams for the collapse of a shell of radiation (thick line) to a black hole (left) and a naked singularity (right). In the case of the naked singularity the radiation has no obstacle to scape across (or bouncing on) the singularity, the hypothetical trajectory corresponding to the dashed line. order to assess whether or not they represent good candidates for endpoints of gravitational collapse. In the precedent case we have shown such singular solutions to be unstable and most probably any small departure from spherical symmetry would imply that such naked singularity would not be formed. Even in the spherically symmetric case the singularity might just be spurious, mathematically just a result of all the matter ending up at the same point at the same time due to the rigid symmetry imposed. Furthermore, once it reaches the singularity, matter may still bounce back, even in presence of other non-gravitational interactions. This is even clearer if we distort slightly the matter configuration in such a way that we avoid this coincidence problem. Different parts of the matter configuration will arrive at different times at slightly different points in such a way that the singularity is not formed. e.g. if we provide some angular momentum the centrifugal barrier would also do the job, at least in some cases. In the context of Lovelock theories of gravity solutions displaying naked singularities na¨ıvely appear in many different situations as seen in the classification of chapter 2, and some of them seem to be formed through a plausible physical process. This may occur for the otherwise well-behaved Einstein-Hilbert branch as well as for the higher curvature branches. Most of these naked singularities arise because the branch of interest ends up at a maximum of the Lovelock polynomials before a horizon is encountered. Another type of singularity is the one associated with positive minima of the same polynomial, that would correspond to a complex cosmological constant associated with this particular branch. The latter type of singularity can never be avoided by any choice of mass parameter and will not be considered here. The above argument makes clear that the stability of these solutions is an essential point to be analyzed in order to study the status of the Cosmic Censorship 102 CHAPTER 4. METRIC PERTURBATIONS AND STABILITY Conjecture in these theories. It can be shown that the instabilities mentioned in the previous paragraph show up whenever a violation of the cosmic censor may happen, as we will now see. Naked singularities may correspond either to r= 0 (or g=±∞) in d= 2K+ 1, for type (a) branches or to a finite value of the radius, r=r−1(g?), where Υ0[g?] = 0, this corresponding to type (b) or AdS branches. In the latter case it is enough to probe the solution close to the singularity in order to show its instability. We just need to analyse the behavior of the potentials for values of gclose to the critical one, g?. In this regime we can approximate the polynomial by Υ[g]≈Υ[g?] + 1 2Υ00[g?](g−g?)2(4.4.1) and use this to compute the leading contribution to the derivatives of F(x), that diverge in the vicinity of the singularity, F0[g] =≈Υ? Υ00 ?(g−g?)2→ −∞ ;F00[g]≈ −2F0[g]2(4.4.2) with negative sign as Υ00 ?<0 for a maximum. Consequently some of the potentials become negative, c2 2≈(d−1)L2f? (d−4) r2F0,(4.4.3) c2 1≈(d−1)L2f? (d−3) r2(−F0),(4.4.4) c2 0≈(d−1)L2f? (d−2) r2(−3F0).(4.4.5) The instability always appears as we have that the potentials corresponding to the different modes diverge with different sign, at least one of them being negative close to the naked singularity. It is important to note that not just the solution containing the naked singularity is unstable, any black hole solution with a horizon close enough to the singularity will also be so. Hence the black hole cannot be continuously connected to the singular solution by any physical process as the instability would inevitably show up before the singularity becomes actually naked. As the horizon approach the singularity, given that F0diverges in that limit, the instability is increasingly important and dramatic when we get there. Remark that in this cases the threshold between black holes and naked singularities is not extremal. The other possibility for naked singularities to show up was for spherical black holes in the limit of small mass when the highest Lovelock coupling is positive, cK>0, and d= 2K+ 1. In that case we can explore all the way down to r= 0 and it will actually be enough to show an instability in that limit. This corresponds just to g→ ∞ and the instability has been previously observed [173] without any reference to the cosmic censor. In that case F0≈(K−1)/K +AK/g2and F00 ≈ −2AK/(Kg2) to leading order, for some constant AK, 4.5. INSTABILITIES AND BLACK HOLE EVAPORATION 103 yielding for the potentials c2 2≈ − 5L2f (2K−3) r2,(4.4.6) c2 1≈0,(4.4.7) c2 0≈5L2f (2K−1) r2.(4.4.8) the constraint (4.1.4) being still verified. We found again an instability for the tensor mode, and again the instability is not just for the naked singularity but also for black holes close enough to it. Notice that once the shear potential becomes zero to leading order, according to (4.1.4) the two remaining potentials are bound to have opposite signs. The instabilities uncovered in the previous discussion, both for the naked singularities and black holes approaching that limit, clearly show that these naked singular solutions cannot be reached either by evolution of black holes by any physical process or via gravitational collapse3. 4.5 Instabilities and black hole evaporation In our quest to qualitatively understand in which situations Lovelock black holes become unstable we pass on to the case of evaporating black holes. This case is relevant as in previous analysis the instabilities under study seem to appear whenever the black hole horizon approaches too much the singularity. This is for instance what happens for type (b) spherical solutions. As the black hole evaporates it looses mass approaching the critical value (for which the temperature diverges) in finite time. This would leave behind a naked singularity but we already proved that neither the extremal solution nor the naked singularity can be reached, the solution becomes unstable as the horizon gets close to the singularity. The same happens for the other cases analyzed throughout this chapter and the same will also hold for other classes of evaporating spherical black holes. We have also seen that for type (a) branches in d= 2K+ 1 the black hole solution becomes unstable as well before reaching the r+= 0 state, that in this case is extremal. The black hole would spend an infinite amount of time to become extremal but just a finite amount to reach the instability. Hence, the instability seem to play a rˆole in the evaporation process of black holes in Lovelock gravities, at least for spherical topology. The remaining case to be understood is that of type (a) solutions, either on the EH or dS branches, for dimensions bigger than the critical, d > 2K+ 1. As the mass of the black hole shrinks to zero the horizon also shrinks approaching the central singularity. In this case the singularity can never become naked but still instabilities show up before the black hole shrinks to zero size. In [173,180] it has been observed that when all possible Lovelock couplings are turned on – the highest one being positive, cK>0 – the instability always shows up, in even and odd dimensions. Here we will generalize that analysis. 3Contrary to what has been proposed in [181,182]. 104 CHAPTER 4. METRIC PERTURBATIONS AND STABILITY Expanding the tensor and scalar potentials around g→ ∞ we simply get c2 2≈(d−3K−1)L2f K(d−4) r2,(4.5.1) c2 1≈(d−2K−1)L2f K(d−3) r2,(4.5.2) c2 0≈(d−K−1)L2f K(d−2) r2.(4.5.3) Then any spherical branch under the above conditions will present a tensor instability in the small mass limit for 2K+ 1 <d<3K+ 1, where Kis the order of the Lovelock theory. Einstein-Hilbert is a special case from this perspective. It would be unstable in d= 3 but in that dimension it escapes our analysis as it corresponds to a AdS-Chern-Simons theory, being topological. The d= 3K+ 1 is special and we need to go to the following order in the tensor potential. In that case wether the black hole is stable or not will depend on the actual values of (cK−1/cK)2and cK−2/cK. The case of Einstein-Hilbert gravity is stable in any case this being a consistency check of our computations. Summarizing, for generic Lovelock gravities we have encountered instabilities at the end of the evaporation process of any spherical black hole in dimensions lower than 3K+ 1, regardless of the type of singularity, (a) or (b), or the asymptotics. The latter dimensionality has to be considered in a more detailed way. Whether this instability is pointing towards an inconsistency of the theory, and as that we should constrain K≤d−1 3, or to some unknown but normal process taking place during the evaporation of a black hole has yet to be answered. 4.6 Discussion Perturbative analysis of exact solutions appears to be an extremely useful tool to get a deeper understanding of the dynamics of gravity in four and higher dimensions, specially for Lovelock theories. In this chapter we have presented and analyzed the graviton potentials for generic Lovelock gravities in some particularly simple regime. Still, despite the simplicity of the adopted approach, the results are general enough to provide some valuable information about the behavior of black holes in these higher curvature theories. Lovelock solutions display several seemingly pathological features, ranging from naked singularities to violations of the third law or discontinuous changes on the horizon radius – and consequently also on their associated thermodynamic variables. We have shown that all the puzzling properties of these solutions are ruled out once their stability is considered. Before any naked singularity may show up the corresponding solution becomes unstable, this applying as well for black holes undergoing any physical process as for the collapse of any kind of matter. Therefore, naked singularities cannot be formed under the evolution of generic initial conditions. The existence of such solutions for collapse with exact symmetry has no bearing on the truth of the conjecture as we can consider such initial conditions as fine-tuned, a zero density set on the space of initial conditions. Any perturbed set of initial conditions will not end up with the formation of the naked singularity due to the instability. 4.6. DISCUSSION 105 The Cosmic Censorship Hypothesis can be saved in Lovelock gravities, at least for now. The same happens also for the third law of thermodynamics, the solutions that would lead to any violation of this law being also unstable. As a consequence, the thermodynamic quantities for the remaining (stable) ranges of parameters are always continuous. These results provide a new and unsuspected insight into these instabilities, too often taken as pathological. Contrary to that belief they seem to have the opposite rˆole, preventing the appearance of real pathological behavior. The instabilities uncover through the last pages get in this way physical relevance, their analysis being crucial for the understanding of the dynamics of black holes in Lovelock and possibly other theories containing higher powers of the curvature. In particular, we have shown that Lovelock black holes with spherical symmetry generically become unstable as they evaporate. Then, the instabilities play also a rˆole in the way black holes decay. In most cases there is a mass gap between the lightest stable black hole and the corresponding maximally symmetric vacuum. The instability may lead the gravitational system to some new solutions that may fill this gap. The only possibilities for such hypothetical states are stars made either of regular matter or gravitational hair4. In some cases we might need to break spherical symmetry and provide some angular momentum, as the perturbations considered do. For hyperbolic AdS-branches the stability of the black hole solutions provides an upper bound on the gravitational mass. If we assume that the bound cannot be of fundamental nature we may still find more massive solutions in the form of hairy black holes. The dressing of the horizon provides the extra required energy. In both cases, the fact that the instability is restricted to some finite radius region of the spacetime seems to support these hypothesis, this is the region filled by our matter configuration or geon, the rest of the spacetime being untouched. We would need a more detailed analysis in order to confirm this intuition. On any of the cases, the instability is associated with a particular value of g, the minimal one for which some of the potentials become negative, gu< g?. The threshold of the instability is then the value of the mass for which the radius of the horizon corresponds to that critical value of g,gu=σ/r2 crit, thus κcrit =gu σd−1 2Υ[gu] (4.6.1) For values of the mass lower than this one we can translate the stability constraints into a bound on the amount of matter that can be contained in a sphere of radius r. For any quantity of matter κ(r) the radius filled by it has to be bigger than the would be naked singularity, r > ru, or equivalently g(r)< gu. Notice that this radius grows along with the mass. For a continuous distribution of matter this has to be verified for all the values of the radius up to r= 0 so that the configuration is stable. The equation for the metric function gis trivially modified in that case to Υ[g(r)] = κ(r) rd−1≤κ(r) rd−1 u = Υ[gu] (4.6.2) The polynomial plays the rˆole of an effective density and its value at the threshold of the instability can be interpreted as the maximal one so that the instability is avoided. We could 4See [183] for an example of the class of solutions we are referring to. 106 CHAPTER 4. METRIC PERTURBATIONS AND STABILITY -5 -4 -3 -2 -1 0 g 0.5 1.0 1.5 2.0 2.5 U 0 1 2 3 4 5 g 0.5 1.0 1.5 2.0 2.5 U Figure 4.4 have played the same game with the value of gfor the singularity g?, and we get another, less stringent, bound on Υ so that the naked singularity is not formed, Υ[g]<Υ[g?]. Notice that the bounds set by stability are always finite whereas the one that avoid the nakedness of the singularity is infinite in the case of r= 0 singularities. The polynomial Υ[g?] diverges in that case and any finite density matter configuration would be regular. We can schematically plot the matter density in terms of gin the same way as we did in order to find the horizon position, we will plot κ(pσ/g)(g/σ)d−1 2together with the polynomial. In figure (4.4) we have depicted some possible configurations, either black holes or stars. Even in the case these matter configurations are not possible, there is another possible endpoint for gravitational collapse. The collapsing matter may just disperse again as it not trapped by any event horizon. In case the symmetry is relaxed the matter will not collapse exactly to one point in such a way that the density may remain finite through the evolution. Then it may disperse back to infinity or form some type of bound state. Chapter 5 Bubbles and new phase transitions “Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life.” Marcus Aurelius Phase transitions between two competing vacua of a given theory are a quite common phenomenon in physics. They occur when some parameter of the system is varied so that the (free) energy of the actual vacuum become greater than the other. If the energy barrier between the two is big enough, the system may stay in the false vacuum for some time (metastability), and proceed to decay via quantum tunneling or, at finite temperature, jump over the wall due to a thermal kick. The decay of metastable systems usually proceed by nucleation of bubbles of true vacuum inside the false vacuum. In field theories at zero temperature, this was first studied by Coleman in his classic paper [184]. There, he introduced Euclidean methods for computing the probability of the quantum nucleation of a bubble, whose dynamics, after nucleation, may be followed classically. In the first (tree-level) semiclassical approximation, the probability of bubble nucleation is given by P∝e−IE,(5.0.1) where IEis the Euclidean action of the system evaluated at the appropriate solution; in this case, the instanton. It is a time-dependent solution which, in the simplest case of a particle in a potential, starts and ends its trajectory at the bottom of the false vacuum (given that the potential in the Euclidean section is the negative of its Lorentzian counterpart, it is a local maximum). This is the point x=xFin figure 5.1(b), where the particle starts its trajectory, then bounces at x=xB, and finally gets back to xFin infinite time. The work of Coleman generalizes this mechanism to a scalar field theory. The instant on corresponds to a scalar 107 108 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS V x xF xT E0 -V x xF xT xB (a) (b) Figure 5.1: In (a) we depict a particle in a potential with two local minima.At xF, the false vacuum, while the true vacuum is at x=xT. There is a potential barrier of height E0between the two vacua. In (b) the Euclidean counterpart of the same system. The potential is now −V. field configuration with SO(4) symmetry. The technique was then generalized by Linde, who considered a scalar field at finite temperature [185]. In this case, the probability of nucleation is still given by (5.0.1), but the Euclidean action has to be evaluated using a different classical configuration, which has SO(3) symmetry. In the mechanical example of figure 5.1, if the temperature is not high enough, this is the solution oscillating inside the Euclidean well, between two points in (xF, xB). The period βof that solution is identified with the inverse temperature, β= 1/T. This period has a minimum for small oscillations at the bottom of the well. For temperatures higher than that one may use the static solution with the particle at the bottom of the well, which, of course, has any periodicity. It is easy to see that for this case, (5.0.1) gives precisely the Boltzmann factor e−βE0one expects for the probability of the particle to jump over the barrier. This is the sphaleron or thermalon [186,187]. Gravitational instantons where first discussed by Coleman-de Luccia in [188], where a scalar field with a potential interacts with a dynamical metric. Now, the different vacua correspond to solutions with different cosmological constants. The false vacuum decays by nucleating an expanding bubble of true vacuum. Later, Brown and Teitelboim [189,190] found an analog instanton when gravity was coupled to an electromagnetic 3-form potential and its sources, electrically charged membranes. In that case, there are infinitely many false vacua, and the decay may proceed many times, changing the (positive) cosmological constant at each step. The authors showed that this mechanism could relax the cosmological constant, providing a possible mechanism for understanding the cosmological constant problem [191]. For finite temperatures, this same physical system may also decay. Now, a thermalon solution controls the decay rate, and, interestingly enough, the decay of a pure de Sitter geometry, turns out to leave a black hole behind [187]. In this chapter we show an analog process that occurs in higher-curvature theories of gravity. In general, this theories contain degenerate vacua even in the absence of matter. Furthermore, one vacuum may decay into the other by 5.1. HIGHER ORDER FREE PARTICLE 109 nucleating bubbles made of nothing but gravity itself [5,6,8]. Due to the non-linearity of the equations of motion, these theories generally admit more than one maximally symmetric solution, Rµναβ = Λi(gµαgνβ −gµβgνα); (A)dS vacua with effective cosmological constants Λi, whose values are determined by a polynomial equation [44], Υ[Λ] ≡ K X k=0 ckΛk=cK K Y i=1 (Λ −Λi) = 0 .(5.0.2) Kbeing the highest power of curvature (without derivatives) in the field equations. c0= 1/L2 and c1= 1 give canonically normalized cosmological and EH terms, ck≥2are the LGB and higher order couplings (see chapters 1and 2for more details). Any vacua is `a priori suitable in order to define boundary conditions for the gravity theory we are interested in; i.e. we can define sectors of the theory as classes of solutions that asymptote to a given vacuum [4]. In that way, each branch has associated static solutions, representing either black holes or naked singularities, ds2=−f(r)dt2+dr2 g(r)+r2dΩ2 d−2, f, g r→∞ −−−→ −Λir2,(5.0.3) and other solutions with the same asymptotics. The main motivation of the present work is that of studying transitions between different branches of solutions. This is important in order to investigate whether a new type of instability involving non-perturbative solutions occurs in the theory. This new kind of phase transitions have been recently investigated in the context of LGB [5] and Lovelock gravities [8]. 5.1 Higher order free particle The existence of branch transitions in higher curvature gravity theories is a concrete expression of the multivaluedness problem of these theories. In general the canonical momenta, πij, are not invertible functions of the velocities, ˙gij [57]. An analogous situation may be illustrated by means of a simple one-dimensional example [192]. Consider a free particle lagrangian containing higher powers of velocities, L( ˙x) = 1 2˙x2−1 3˙x3+1 17 ˙x4(5.1.1) In the hamiltonian formulation the equation of motion just implies the constancy of the conjugate momentum, d dtp= 0. However, being this multivalued (also the hamiltonian), the solution is not unique. Fixing boundary conditions x(t1,2) = x1,2, an obvious solution would be constant speed ˙x= (x2−x1)/(t2−t1)≡vbut we may also have jumping solutions with constant momentum and the same mean velocity. Obviously for that to happen at least one of the degenerate velocities has to be bigger than vand one smaller. In our example, for mean velocities in the range (v1, v2) that correspond to multivalued momentum, the solutions are infinitelly degenerate as the jumps may occur at any time and unboundedly in number, as long as the mean velocity is the same. This degeneracy is lifted however when the value of the action is taken into account. The minimal action path is the 116 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS of the solution. In fact, regularity of the Euclidean section (see figure 5.4) demands the inner solution is a black hole and fixes its temperature. Once the periodicity in Euclidean time is fixed to attain a regular horizon, the periodicity in the outer time will be determined through pf−(a?)β−=pf+(a?)β+≡β0,(5.3.15) where β−is the usual inverse Hawking temperature of the inner solution while β+is the one seen by an observer at infinity, that will be different, in general, from the Hawking temperature corresponding to the black hole of the given mass on either branch. β0corresponds to the periodicity in the new variable τ. Consider the induced vielbein basis eτ=dτ and ei=a(τ) ˜eϕi, which is intrinsic to Σ. If we call xµ ±(ζa) the coordinates of the embedding of Σ with intrinsic coordinates ζA= (τ, ϕi) on M±, the unit normal vectors read, in Lorentzian signature, nµ ±= ( ˙a f±(a),p˙a2+f±(a),0,··· ,0) .(5.3.16) The extrinsic curvature is then given by KAB =−n±µ ∂2xµ ± ∂ζA∂ζB+ Γµ αβ ∂xα ± ∂ζA ∂xβ ± ∂ζB!,(5.3.17) yielding K±τ τ=¨a+1 2f0 ±(a) p˙a2+f±(a), K±ϕi ϕj=p˙a2+f±(a) aδij.(5.3.18) The intrinsic curvature, in turn, is given by Riτ 0=¨a aei∧eτ, Rij 0=σ+ ˙a2 a2ei∧ej.(5.3.19) Notice that it is trivially the same as seen from either side as it is calculated from the induced metric and this is continuous. The vielbein basis is continuous but the spin connection is not. In that case it should be possible to write the surface term as the difference of (1.2.33) seen from each of the bulk regions. It is clear that all the components aligned along the normal direction of this intrinsic spin connection are zero in the same way as it happens for the corresponding intrinsic curvature. In the Euclidean signature we may just change the signs of the squared velocity and its acceleration as (˙a2,¨a)→(−˙a2 E,−¨aE). 5.3.2 Junction conditions and bubble dynamics The junction conditions (5.3.9) for the configurations of interest have just diagonal components related by a conservation equation (Bianchi identity) that constrains them in such a way that only the ττ component matters. The rest are related to that one as [54,55] d dτ ad−2π± ττ = (d−2) a2˙a π± ϕiϕi,∀i , (5.3.20) 5.3. JUNCTION CONDITIONS 117 in such a way that if π± ττ verifies (5.3.9), all the components automatically do. This is reminiscent of the analogous field equations for cosmological solutions studied in section 2.6 Thus, we just need to compute π± ττ . This junction condition involves just the angular components of both the intrinsic and extrinsic curvatures; the expression reduces to Π± (ττ)≡p˙a2+f±(a) aZ1 0 dξ Υ0σ−ξ2f±(a) + (1 −ξ2) ˙a2 a2,(5.3.21) where we avoid the inclusion of some (irrelevant for our discussion) factors, and the polynomial Υ is again seen to play a central rˆole. In the future we will also avoid the use of indices, Π±referring to the (ττ)-component. If we define e Π=Π+−Π−, because of (5.3.20) the junction equations can be written as e Π = ∂τe Π = 0. Notice that the dimensionality of spacetime is somehow irrelevant in this expression. In the case of LGB, this expression reduces to the one in [56]. If we conveniently introduce the functions1 g±≡g±(a) = σ−f±(a) a2, H ≡H(a, ˙a) = σ+ ˙a2 a2,(5.3.22) we can rewrite Π±[g±, H] = pH−g±Z1 0 dξ Υ0ξ2g±+ (1 −ξ2)H,(5.3.23) where it becomes clear that all the information about the branches is contained in g±, that are implicitly given by (2.1.8). Notice that g±≤Hin order to have a real value for Π± (which tantamount to f±(a)≥0 in order to have a real Euclidean boundary action). This implies that the static bubble corresponding to the thermalon necessarily forms at a?≥ max(rH+, rH−), outside the would be black hole horizons corresponding to both branches. Moreover, this reality condition is also necessary for the equation to yield real values of the velocity, at least for some region of the spacetime. The difference in canonical momenta e Π may be rewritten in a couple more useful ways as e Π = Z1 0 dξ pH−g+−pH−g−Υ0H−ξpH−g++ (1 −ξ)pH−g−2(5.3.24) that is the variation of (5.3.6), or changing the integration variable e Π = Z√H−g+ √H−g− dx Υ0[H−x2] (5.3.25) that is more compact and easier to manipulate. We can interpret e Π(˙a, a) = 0 as a conservation equation with a non-canonical kinetic term. We can make this statement more precise by noticing that Π2 += Π2 −⇐⇒ K−1 Y i=1 1 2˙a2+Vi(a)= 0 ,(5.3.26) 1Not to be confused with the previously used notation for the horizon value of g, 1/r2 + 118 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS where we have to take into account that some of the roots or potentials, Vi(a), correspond to the actual junction conditions (5.3.9), while some other may correspond to the reversal orientation, Π+=−Π−, that corresponds to gluing two interiors or two exteriors (wormhole). Besides some roots may be discarded in case they yield imaginary momenta, g±> H. This reality condition amounts to ˙a2≥ −f±; ˙a2 E≤f±(5.3.27) in the Lorentzian and Euclidean sections respectively. Remark that in the Euclidean version the brane may just propagate outside the horizon whereas in the Lorentzian case the bubble may cross it. This leads to some seemingly pathological situations, namely the distruction of the horizon by the bubble leading to a violation of the Cosmic Censorship Hypothesis. Roots of e Π = 0 and Π++ Π−= 0 may just join where Π+= Π−= 0. The same also happens for solutions yielding real and imaginary values of momenta, as this happens for instance for H=g−or equivalently ˙a2=−f−something that is possible just inside the horizon or for negative values of ˙a2, forbidden region of the potential. In these particular points the potential ceases to be a solution of Π+−Π−= 0 to become a solution of Π++Π−= 0. This is generic feature of any Lovelock theory as √H−g−does not change sign at H=g−, as it can be seen in the figure, whereas Π−changes sign there. When this happens inside the horizon it is unclear what happens to the bubble. It cannot go further but it cannot turn back as it would be travelling backwards in time. We will comment more on this later on. See figure 5.5 for specific examples of these facts. The potential becomes unphysical beyond the point where it meets f−/2. Notice that we can get to the origin if we reduce κ+ or, for fixed κ+if we increase κ−, actually for κ+≤4κ−. Once a particular potential is chosen, say Vj(a), we can use (5.3.20) to determine the corresponding acceleration equation that governs the dynamics of the bubble ¨a=−V0 j(a).(5.3.28) Notice that this dynamics may be difficult to determine, in general, since for generic Kit might be impossible to have an explicit expression for the potentials, and, on top of that, several of them may provide a suitable dynamics for the bubble. In all the expressions the rˆoles of the two branches can be exchanged yielding the same dynamics of the bubble. As there is no matter on the bubble, the corresponding equations are blind to which is the inner/outer solution, the bubble behaves in exactly the same way. There are two limiting cases of the junction conditions that are of special interest. On the one hand we are interested in the static configurations (thermalons) and their stability. For that it is enough to consider the slow limit of (5.3.9), in a double expansion about a? and ˙a= 0, e Π≈e Π?+∂e Π? ∂H ˙a2 a2 ? +∂e Π? ∂a (a−a?) + 1 2 ∂2e Π? ∂a2(a−a?)2,(5.3.29) where the upper star means that a quantity is being evaluated after the replacement H→ H?≡σ a2 ?and g? ±→g±(a?). Besides the two conditions e Π?=∂e Π? ∂a = 0 ,(5.3.30) 5.3. JUNCTION CONDITIONS 119 the junction condition (5.3.9) at a=a?adopts the canonical form of an energy constraint for an auxiliary harmonic system described by a(τ), 1 2˙a2+Veff(a)=0,(5.3.31) where Veff(a) = 1 2k(a−a?)2, k =a2 ? 2 ∂e Π? ∂H !−1∂2e Π? ∂a2(5.3.32) provided the denominator is non-vanishing, which is equivalent to the potential being a smooth function of the radius at a?. That factor vanishes when two potentials merge, thus becoming complex beyond that point. The bubble cannot go inside the region of complex potential, analogously to the case of branch singularities in the cosmological context (see section 2.6), it will turn back along the other root that merges with the one it was following. The other interesting limiting case corresponds to the speed of the brane becoming very large, since it will be relevant to discuss the asymptotical behavior of the running away bubble. As a→ ∞, the behavior of Hhas to be given by a power law, and so it can either diverge or asymptote to a constant. We can verify that one of the solutions diverges as H∼ad−1whereas the rest (K−2) are asymptotically constant. We can prove that the maximum number of solutions is K−1 in general just by squaring each side of the Π+= Π− equation in order to obtain a polynomic equation on H. Even if the na¨ıve degree of such equation is 2K−1 it is easy to show that the first non-vanishing coefficient of such equation is order K−1 in H. In the limit of large speeds (5.3.24) can be evaluated directly using H−ξpH−g++ (1 −ξ)pH−g−2≈ξg++ (1 −ξ)g−+1 4Hξ(1 −ξ) (g+−g−)2 (5.3.33) pH−g+−pH−g−≈1 2√H(g−−g+)1 + 1 4H(g−+g+)(5.3.34) so that we can verify that e Π≈ − 1 2√H(Υ[g+]−Υ[g−]) −1 2HZg+ g− dx Υ[x]−(g+Υ[g+]−g−Υ[g−]) (5.3.35) to next-to-leading order in 1/H, where we have used the change of variable x=ξg++(1−ξ)g− thus solving H≈1 2(κ+−κ−)ad−1Zg+ g− dx Υ[x]−(g+κ+−g−κ−)(5.3.36) In the lightlike limit (H→ ∞), we get a consistent equation for Heither if a→ ∞ and H∼ad−1or κ+→κ−which coincides with the result of [195]. In the former case the asymptotic behavior of the potential can be just read as H≈ad−1 2(κ+−κ−)ZΛ+ Λ− dx Υ[x] (5.3.37) 120 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS 0.5 1.0 1.5 2.0 2.5 a -25 -20 -15 -10 -5 5 V, f2 Figure 5.5: Potential (in blue) of a brane gluing spherically symmetric stable and unstable branches of LGB gravity with parameters λ= 0.001, L= 1, κ−= 1 and κ+= 2,2.4,7,102,103,104from left to right. In red f−/2, metric function corresponding to the stable branch. The blue curve slides up the red one as we increase κ+. The dashed curve corresponds to the asymptotic behavior (5.3.37) for the three lower values of κ+. and this solution is physical in the sense of being solution of e Π = 0 as opposed to Π++Π−= 0 roots. The asymptotic solution (5.3.37) is useful to analyze in which cases it is possible for the brane to reach infinity. For that, one has to also analyze the sign of the asymptotically constant solutions, when present, and also determine along which branch is propagating the bubble. The constant roots are the solutions of e Π[Λ±, H] = 0 (5.3.38) that is degree K−2 in Has the leading power is proportional to Υ[g+]−Υ[g−]∼1/ad−1and vanishes as we approach the asymptotic region. In that limit the topology of the horizon becomes irrelevant. In case the bubble may run away to infinity, it asymptotically approaches the speed of light. This can be seen for instance from (5.3.37), as ˙a2grows faster than the function f. the radial speed is then dr dt =˙a pf+ ˙a2f→f(5.3.39) exactly the same as for a null geodesic in that background. In that way, as AdS space has a timelike boundary, the bubble gets there in a finite time and as such it can be interpreted as a change of boundary conditions for the theory, i.e. as a jump from one branch of solutions to the other. The time from a position a0far away to the boundary is actually ∆τ=Z∞ a0 da ˙a∼Z∞ a0 da ad+1 2∼1 a d−1 2 0 <∞(5.3.40) and is also finite for the asymptotically constant values of H, although the asymptotic speed is a fraction of the speed of light in that case. In the case of LGB gravity there is just one possible potential that determines the dynamics of the brane separating two solutions belonging to the two different branches. We will follow the same notation as in (2.2.1), the (−) branch being the stable one. This effective potential can be simply written as V(a) = ad+1 ∆ [g(3 + 2λg)2] 24λ∆κ+σ 2(5.3.41) 5.3. JUNCTION CONDITIONS 121 Figure 5.6: Potential (in blue) of a brane gluing spherically symmetric stable and unstable branches of LGB gravity with parameters λ=−0.01, L= 1, κ−= 1 and κ+= 1.01,1.2,1.4,2,3,4,5 from left to right. In red f−/2, metric function corresponding to the stable branch and in orange the one corresponding to the unstable branch, f+/2, for κ+= 5. Singularities are located where the curves end, the potential ending when we find the outermost of them. Before that we encounter a point where V=f+/2 beyond which the potential is unphysical. The dashed curve corresponds to the asymptotic behavior (5.3.37) for κ+= 5. 0.2 0.4 0.6 0.8 1.0 1.2 a -20 -15 -10 -5 V, f2 It verifies all of the general properties described above. For instance, as the integral of Υ[g] between Λ−and Λ+is always positive the bubble can scape to infinity as long as κ+> κ−, i.e. the mass of the unstable branch has to be higher than the stable one. Remember that any of the two can be in principle on either side of the brane and the dynamics is exactly the same. The tension depends on the position and there is no pressure term as it would depend on the side that corresponds to each branch. The bubble has tension despite the fact that it contains no matter. Depending on the values of the masses the stable branch may have a horizon and the naked singularity of the unstable solution may be located at a higher or lower radial position. Moreover, depending on the choice of external and internal solutions and the position of the bubble, the (dynamical) metric may display the horizon in some situations. Quite generally (e.g. see figure 5.5 for low enough κ+) the horizon is accesible in the sense that the potential is finite and negative at that position. In case the stable branch is the inner one the metric may display a horizon but, as the bubble may cross that point, it can be destroyed. It will necessarily undergo susequent collapse until it reaches the naked singularity that becomes visible. For that we just need the radius of the horizon to be larger than the singularity, otherwise the latter would become naked even before the bubble reaches the horizon. In the opposite case, with the unstable branch inside, the contraction of the brane would instead create a horizon. In fact, even if these examples might seem fine-tuned we will provide specific examples. The case of a expanding looks a bit more natural, at least with the unstable branch outside. In that case one may think of a bubble poping out from a naked singularity and creating a horizon, leaving a regular black hole behind. This points towards a possible instability of the BD-unstable branch, namely their naked singularity metrics, decaying through the formation of this kind of bubbles. Remember that these solutions are also perturbatively unstable. We will analyze this issue, although in a different manner in the next sections. 122 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS 5.3.3 Bubbles, horizons and the cosmic censor The destruction of the horizon by the bubble might seem unnatural but seems to be what really happens. In order to verify that this is indeed the case one has to be careful and check that nothing goes wrong at the horizon or inside it. In particular, in order for the metric to be continuous, the existence of a commom induced metric on the junction surface is not enough. We also have to ensure that the change of variables between the coordinate frames on both sides is regular. In the vicinity of the junction (ρ= 0) we might write the metric in terms of a coordinates set adapted to the surface ds2≈ −dτ2+dρ2+a2(τ)dΣ2 σ,d−2(5.3.42) the same in both sides and ρbeing the normal. We then get constraints on the coordinate functions of the brane T±(τ, ρ) and a±(τ, ρ) (such that T±(τ, 0) ≡T±(τ) and a±(τ, 0) ≡a(τ)). One is the Lorentzian analog of the previously found (5.3.13) and additionally, −f±T02 ±+a02 ± f± = 1 ; −f±T0 ±˙ T±+a0 ±˙a± f± = 0 (5.3.43) where primes indicate derivatives with respect to the new variable ρ. Remark that the radial variables are not equal in this case even though they are at the junction. In this way we may write the change of variables as dt±=˙ T±dτ +˙a f± dρ dr±= ˙a dτ +f±˙ T±dρ (5.3.44) that corresponds to a boost in the (t±, r±)-plane. This can be easily verified using the orthonormal frame (2.1.5)  e0 ± e1 ± =U± dτ dρ  U±=  pf±˙ T±˙a √f± ˙a √f±pf±˙ T±  .(5.3.45) the boost matrix having unit determinant with an inverse obtained just by changing the sign of ˙a. Remember that ˙ T±can be written in terms of ˙ausing (5.3.13), being invariant under that change. The change of variables between inner and outer coordinates corresponds thus to a composition of boosts e+=U+U−1 −e−, a transformation that does not change the causal structure, i.e. null geodesics are continuous across the bubble. We can now address the question of the behavior of the bubble as we cross the horizon. In that case we have to change pf−=ipkf−k, as f−goes negative, in such a way that the timelike and spacelike vielbeins exchange their rˆoles2preserving the orientation (e0 −∧e1 −= ˆe0 −∧ˆe1 −) e0 −=iˆe1 −;e1 −=iˆe0 −.(5.3.47) 2Inside the horizon ˆe0=−dr pkfk; ˆe1=pkfkdt (5.3.46) 5.4. THERMALONS AND THERMODYNAMICS 123 Consequently the change of variables behind the horizon  ˆe0 − ˆe1 − =ˆ U− dτ dρ  ˆ U−=1 pkf−k −˙a−pf−+ ˙a2 −pf−+ ˙a2−˙a .(5.3.48) again corresponds to a boost and is completely well defined. The change is actually continuous across the horizon, the bubble being able to cross it. After all there is nothing special – locally – about that point. Besides, once it gets to the horizon the causal structure makes it impossible for the bubble to get back (see figure 4.3 for the Penrose diagram of the process) as it would be travelling backwards in time. This can also be seen from (5.3.48) as the diagonal components of the ˆ U−matrix have to be positive and bigger than one. This is also true for U±outside the horizon. It is remarkable that, even though the original horizon is destroyed by the bubble the causal structure inherited by it makes it impossible for the bubble to go back. In this case we cannot blame the symmetry of the solutions considered as we did in chapter 4. Once the horizon is crossed, its (previous) presence forces the bubble to actually reach the central singularity, regardless of the details of the collapse. The singularity unavoidably becomes naked, this being a clear violation of the Cosmic Censorship Hypothesis. Nonetheless, this violation might be a marginal one if, due to the instabilities suffered by those singular solutions (see chapter 4for details), the energy leaks the singularity, thus leaving regular pure AdS+behind. This process is allowed by the causal structure of the spacetime. Notice that the bubble itself may also emit part of its mass, M+−M−, to infinity before it actually reaches the singularity. However the naked singularity will still form as the mass contained in the inner solution, M−, cannot scape until the bubble reaches the center. 5.4 Thermalons and thermodynamics Bubble configurations when static are subject to the same discussion of chapter 3, concerning the thermodynamics aspects of black holes. In very much the same way, the new solutions will be characterized by the same thermodynamic variables. In this section we will compute the value of the Euclidean on-shell action for these thermalons. Such static solutions will exist in some cases as it can be seen in figure 5.5 for the case of LGB gravity with positive λcoupling. Being static, they trivially have a smooth Euclidean section whenever they display a horizon in the inner solution covering the singularity. We have to choose the periodicity in Euclidean time accordingly, so that the full configuration is smooth. Once the inner periodicity is fixed to attain a regular horizon, the periodicity in the outer time will be determined through (5.3.15), thus fixing the temperature. As for black holes, the Euclidean on-shell action is in general divergent and needs to be regularized, either by background substraction or by other means. We will measure the free energy with respect to the specific solution used as groundstate, usually the maximally symmetric one. In order to simplify the discussion we will calculate the on-shell action in terms of three parameters, a?and β±, the first being the equilibrium position of the 124 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS (−) (+) r=rH r= 0 r= 0 (+) (−) t= 0 Figure 5.7: Schematic Penrose diagram for a nearly static bubble that undergoes collapse from t= 0 onwards. The thick dashed line represents the trajectory of the bubble with inner black hole and outer naked singularity geometries. The spacelike singularity becomes timelike as the bubble reaches r= 0. The resulting spacetime is a (boosted) naked singularity, dashed lines representing constant t+slices. The lowest one corresponds to the Cauchy horizon introduced by the timelike singularity. The upper and right wedges represent the physical region of the spacetime corresponding to a black hole formed by collapse whereas the rest corresponds to the complementary white hole region. 5.4. THERMALONS AND THERMODYNAMICS 125 bubble and the others the periodicity in Euclidean time in the inner and outer regions. It is important to keep in mind that these two variables are not independent from each other. We will use + to denote the outer region and −for the inner one. This is more general but it will be consistent with the notation of the LGB branches when we undertake that analysis. Unlike the computation of the Hawking-Page effect described in section 3, where the fields are continuous, here we have to consider the contribution to the action arising on the bubble, when writing the Euclidean action in the form (5.3.8), b I=b I−+b IΣ+b I+.(5.4.1) The outer piece includes all the boundary terms at infinity necessary to both have a well defined variational principle. It regularizes its divergence by subtracting the background M+= 0 with the same periodicity at infinity, yielding b I+(a?, β+) = β+ (d−2)Vd−2 16πGN ∂rhrde Υ[g+]ia? ,(5.4.2) that has the same expression as for the usual black holes (3.2.2) evaluated at the position of the bubble instead of the horizon. The term b I−, in turn, is integrated from the horizon to the location of the bubble yielding two terms of the same form, one evaluated on the bubble and one at the horizon. b I−(a?, β−) = β− (d−2)Vd−2 16πGN∂rhrde Υ[g−]irH−∂rhrde Υ[g−]ia?.(5.4.3) where rHis the radius of the horizon, if any, or zero. Finally, b IΣis given by b IΣ=−b I− ∂(a?, β0) + b I+ ∂(a?, β0) (5.4.4) where the periodicity in Euclidean time is inherited from the bulk regions β0=pf±(a)β±. Then, we can collect all the contributions that depend upon the location of the bubble, b Ibub(a?, β0) = β0 (d−2)Vd−2 16πGN1 √f∂rhrd˜ Υ[g]ia? + 2 ∂rrd−2pfZ1 0 dt ˜ Υ0(1 −t2)gH+t2ga? + − (5.4.5) in the static case, otherwise we cannot perform the τintegration, and where F(g)|+ −= F(g+)− F(g−), indicates the difference between the terms evaluated on both sides of the brane. The rest can be consequently called b Ibh =b I − b Ibub =β−M−−S , (5.4.6) the usual contribution from the inner black hole. Trivially the surface term vanishes when the same solution is taken in both sides of the junction. 132 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS 1 2 3 4 x -0.5 0.5 1.0 1.5 UA-x2E x,U'@-x2D Figure 5.10: Υ[−x2]/x (solid blue line) and Υ0[−x2] (in gray) for cubic Lovelock theory with parameters L= 1, λ= 0.17 and µ= 0.02. The couples of points with the same color correspond to each of the three solutions of the static junction conditions (5.5.2) in that case. The area below the gray curve between each couple of points vanishes according to the second equation. We can also analyze the stability of the potential at the equilibrium point by means of (5.3.32), that up to irrelevant positive factors is Veff ∼ ∂e Π ∂H !−1 ∂2e Π ∂a2!∼1 √−g+Υ0[g+]−1 √−g−Υ0[g−] R√g+ √g−dx Υ0[−x2] x2 (5.5.3) This expression is positive for stable equilibria and negative in the opposite case. The above expression is negative for LGB gravity with λ > 0, as expected for unstable equilibria. In the cubic case we have two possible potentials and the equilibrium points can, in principle, be associated to any of them. This seems much more involved but one may still use (5.5.3) in order to study the stability of those static points. For our specific example we find that the local potential is positive for the red and green pairs whereas it is negative in the remaining case. Thus we have one possible stable bubble (green) and one unstable (blue) (see figure 5.11). We can get even more information from the asymptotic behavior of the potential. For the branch approximated by (5.3.37), we need to first realize that the outer mass is always bigger than the inner one due to the first condition in (5.5.2). We can compute the integral of the polynomial between the two stable vacua ZΛ+ ΛEH dx Υ[x]≈ −11 <0 (5.5.4) and from (5.3.37) we can then realize that His asymptotically negative. The bubble cannot reach infinity along this branch of the potential. It can however reach the boundary following the branch that asymptotes to a constant H≈0.76, the potential being asymptotically 5.5. GENERALIZED HP TRANSITIONS IN LOVELOCK GRAVITY 133 Figure 5.11: Bubble potentials for cubic Lovelock theory with λ= 0.17 and µ= 0.02. We just show the relevant branch of the potential connected to the physical equilibrium points, those with positive mass. The upper potential corresponds to the stable configuration (in green in figure 5.10) while the other is the unstable one (in blue also in figure 5.10). There is a minimum although it is hard to see in the plot. 1.1 1.2 1.3 1.4 a -0.004 -0.002 0.002 0.004 V negative along. This is actually the potential the two positive mass equilibrium points correspond to, as it can be seen in figure 5.11. One of the bubbles, the unstable one may reach infinity by expansion while the other is fixed on its position unless it can tunnel across the barrier to subsequently expand. The point to the left where the potentials end corresponds to a naked singularity of the outer solution that appears before we even reach the horizon, situated at the origin of the plot. We have analyzed a very particular example but the same analysis can be performed in general, in cubic or any higher order Lovelock theory. The possible situations one may encounter are extremely varied. As already seen in the cubic case, we may have stable or unstable equilibrium points whose number may change as we vary the values of the couplings, or a?for non-planar topology. These equilibria may in principle correspond to any of the K−1 bubble potentials of the theory, and each of these potentials may have any sign at infinity. This will determine whether the bubble may reach the boundary or not, thus the possibility of a change of branch. Also, depending on the couplings, the branches connected by the static configurations may change, all being connected to the EH-branch, some being connected or none being connected. Also for two given branches we may have several static configurations connecting them. All of them have to be compared in order to decide which is the globally stable phase. It might even happen that no static configuration exists. Depending on the characteristics of the globally preferred phase for given asymptotics, always a thermalon when it exists, the fate of the system may be very different. The junction conditions considered above determine not only the equilibrium configuration but also, in Lorentzian signature, the effective potential felt by the bubble and, consequently, its subsequent dynamics. If no thermalon connecting our choice of boundary conditions with the EH-branch exists the system will remain on the only possible solution, pure AdS+. This is the case for the unstable asymptotics of the cubic example depicted in figure 5.11. On the contrary, when that static configuration exists it will form but whether it changes the asymptotics or not will depend on the form of the potential. If the bubble is unstable and no potential barrier appears on its way to the boundary, we will have a change of branch. This is for instance what happens for LGB gravity with positive λ. For a stable bubble the situation is also quite interesting. In a sense this stable configuration provides a regular black hole to a branch of solutions that naively had none. The bubble being frozen at the equilibrium position, this situation is very similar to the Hawking-Page transitions studied in section 3.5, the system 134 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS 0.00 0.05 0.10 0.15 0.20 0.25 Λ 0 2 4 6 8 10 12 14 Tc 2 4 6 8 10 12 T 0.0 0.1 0.2 0.3 0.4 F Figure 5.12: Free energy versus temperature in 5d for λ= 0.04,0.06,0.09 (positivity bound), 0.219 (maximal F(T= 0)), and λ→1/4 (from right to left). The λdependence of the critical temperature is displayed in a separate box. remaining in the black hole phase. In this case however, when the asymptotic potential allows the bubble to reach the boundary, there might be a non-vanishing probability for the bubble to tunnel to that region. In that case the boundary conditions may eventually change and we again have a branch transition. This is the case of the unstable configuration found in our cubic example (see figure 5.11). In case the bubble collapses instead of expand, it will generally lead to the destruction of the horizon and the consequent formation of a naked singularity. We will not comment more on this here. By the arguments given in chapter 4and those outlined in the previous sections, we will just assume the system then comes back to the initial phase, pure AdS+. This is analogous to the formation of bubbles in a fluid. When these expand the phase transition to the gas phase proceeds while, when it collapses, the system remais liquid. The results of this section are more general than just the planar case as it also corresponds to the high mass limit of the other two topologies. Even though the equations are much more involved in case of spherical or hyperbolic symmetry, the analysis follows in the same way. In the next sections we concentrate in the case of LGB gravity with σ=±1. 5.5.1 Spherical LGB bubbles In the case of LGB gravity the thermalon configurations being described here are just relevant for unstable or ghosty boundary conditions, that of the unstable or (+) branch of solutions, and positive LGB coupling. Again, we have to compare the free energy of the thermalon configurations found in the preceeding sections with the corresponding thermal vacuum. From the Euclidean point of view this is the only other smooth metric with that unstable boundary conditions. The resulting phase diagram is similar to the usual Hawking-Page phase transition. For any value of the LGB coupling, the free energy, F, as a function of the temperature 1/β+ displays a critical temperature above which it becomes negative and, thus, the phase transition occurs (see figure 5.12). If the free energy is positive, however, the system is metastable. It decays by nucleating bubbles with a probability given, in the semiclassical approximation 5.5. GENERALIZED HP TRANSITIONS IN LOVELOCK GRAVITY 135 by e−β+F. Therefore, after enough time, the system will always end up in the stable, EH black hole solution. This is reminiscent of the HP transition, except for the fact that, here, the thermal AdS vacuum decays into a black hole belonging to a different branch.Tc(λ) is monotonically decreasing, the phase transition becoming increasingly unlikely the more we come closer to the EH – classical – limit. In this sense, it is a quantum mechanical phenomenon. 5.5.2 Hyperbolic LGB bubbles Asymptotically AdS spacetimes with hyperbolic topology are an interesting playground for checking various facts about the recently discovered type of phase transitions. Despite some unclear features of the thermodynamics of these spacetimes, they allow in particular for the discussion of transitions between two asymptotically AdS branches both having regular horizonful black hole solutions. In particular, for the simplest case of LGB theory with λ > 0, both, the EH and the ghosty branch black holes, may have horizons depending on the value of the mass. Even more, even though the range of masses for the ghosty branch is bounded from above (and below) this branch has black hole solutions for all possible temperatures. Besides, although one of the branches is unstable we would like to investigate if the discovered transition protects the theory against this instability in the sense that the unstable branch would never be the preferred phase. This is actually the case for LGB theory with planar or spherical symmetry. Despite the fact that the vacuum is preferred for some range of temperatures in the spherical case, the thermalon may be formed with small but finite probability in that case leading to a change of branch. This case is much richer, not only due to the possibility of transitions in both directions, but also because the spectrum of configurations gets richer. The existence of extremal black holes in both branches for specific values of the mass, leads to the corresponding extremal thermalons. These extremal configurations can be considered as qualitatively different from their non-extremal counterparts as regularity of the horizon in the Euclidean section does not fix their temperature. As a limit of non-extremal black holes the extremal solutions necessarily have zero temperature and the entropy corresponding to the Wald formula. Quite the opposite, ab initio extremal configurations may be identified with any temperature and zero entropy (see chapter 3for details). The same happens for the bubble configurations even though the limiting temperature is not zero. In addition to the non-extremal bubbles seen so far there will be extremal solutions where the horizon of the inner black hole is degenerate. For the direct transition, the one leading to the EH branch we will fix the BD-unstable asymptotics and compare the free energy of all possible configurations at the same temperature. These are in principle four families of solutions, extremal and non-extremal black holes and the corresponding bubble counterparts. The same will happen for the reverse transition. Let me first comment briefly on the thermodynamics of the unstable branch black holes. These are unstable not only `a la Boulware-Deser but also thermodynamically. Their specific heat is negative and they also have negative entropy. Due to this fact their free energy will always be bigger than the one corresponding to the extremal black hole that coincides with its mass. This extremal solution corresponds to the black hole with mass saturating the upper bound, κ=λin five dimensions or the one corresponding to a degenerate horizon in 136 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS higher dimensions. In the 5d case the degenerate horizon coincides with the singularity at r= 0 but this does not represent a problem as that point is at the end of an infinite throat and should be removed from the geometry. The near horizon geometry can be written as ds2∼2λdρ2+e2ρ−dt2 2λ+dΣ2 −1,3(5.5.5) where we have explicitly removed the point r= 0 considering a change of variables as ρ=Logr. In higher dimensions the horizon is at finite radius and the geometry completely regular. We will concentrate in the five dimensional case in what folows. Thus, this particular solution is a smooth geometry interpolating between AdS5at spatial infinity and the previous geometry as we approach r= 0. Also, it has zero specific heat and entropy and can be considered with any periodicity in Euclidean time and thus it is a well motivated candidate as groundstate of the theory for the sector we are considering. Also as the black holes are unstable all of them have higher free energy than the extremal one and thus the chosen groundstate is always preferred in a semiclassical basis. The only relevant configuration for the next step of the analysis will then be the extremal black hole. We will have to compare the free energy of our thermalon configurations to that of this groundstate. The free energy of the non-extremal configuration has the usual expression F=M−TS and the one corresponding to the extremal once again corresponds to a constant, thus implying a vanishing entropy. However the extremal free energy does not coincide with the mass in this case. In the extremal case the bubble is situated exactly at the inner horizon, f−(a?) = 0, in such a way that the rescaling of the temperature just cancels the zero of f0at the horizon, pf+ pf− f0 − 4π→˜ Te +.(5.5.6) This is the temperature corresponding to the black dot in the figure where the non-extremal bubble curve ends. The resulting on-shell action is then, b Ie=β+(Me +−˜ Te +Se) (5.5.7) Again attending at the usual semiclassical picture the entropy of the solution vanishes and the free energy corresponds to an effective mass that picks some contribution proportional to the Wald entropy. Fe,b =Me +−˜ Te +Se(5.5.8) Here the word extremal does not necessarily mean zero temperature but it is rather a question about the topology of the near horizon region. The curves for the free energy as a function of the temperature for the different configurations is shown in figure (5.13) for several values of the coupling λ. For the discussion of the phase diagram we will assume that the extremal configurations are present at any temperature with zero entropy. At the end of this section we will comment on the alternative approach not considering them but as limiting cases of their non-extremal counterparts. 5.5. GENERALIZED HP TRANSITIONS IN LOVELOCK GRAVITY 137 20 40 60 80 T -1.5 -1.0 -0.5 0.0 0.5 1.0 F 10 20 30 40 50 60 T -1.0 -0.5 0.0 0.5 1.0 1.5 F 2 4 6 8 10 T -0.5 0.0 0.5 1.0 1.5 2.0 F Figure 5.13: Free energy versus temperature for λ= 0.035,0.05,0.12,0.249. The dashed black curve corresponds to the unstable branch of black holes while the thick black one represents its extremal free energy. The thick red curve corresponds to the bubble with the same asymptotics and its extremal limit is indicated by the black dot. The thin red curve corresponds to the free energy of such extremal state when vanishing entropy is assumed. The dots indicate the locus of different kinds of possible phase transitions even though just some of them may take place in each case. The black dot when relevant represents a phase transition from a extremal to a non-extremal bubble and the red one a transition from the extremal black hole to the non-extremal bubble. The green dot indicate the minimal temperature for the existence of non-extremal bubbles when these are divided in two branches one thermodynamically stable and one unstable. 138 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS 0.00 0.05 0.10 0.15 0.20 0.25 Λ 20 40 60 80 Tcrit Figure 5.14: In red the bubble states, light for the extremal, and in gray the extremal (BD-unstable) black hole. The red dot indicate the value of λfor which the transition between the extremal unstable black hole and the non-extremal bubble (red line) begins to exist. The red line corresponds to the bubble extremal limit and the red one to the minimal temperature of these configurations when this does not correspond to the extremal one. We have different behaviours for different ranges of the LGB coupling. For small values, λ<λcrit ≈0.0404, the bubble is always the stable phase with a transition between the extremal configuration at low temperatures and the non-extremal one preferred as we increase the temperature. For values of λabove that critical value the low temperature phase is the extremal black hole instead, our hypothetical vacuum, the non-extremal bubble being again the stable phase at high temperature. Another interesting feature is that for values of λbigger 1/12 negative entropies appear for the bubble, notice the positive slope of the non-extremal curve in the third and fourth graphics. This negative entropy states however are never the preferred phase of the system and can safely be discarded as unphysical. This however would not happen if we consider the alternative approach to the extremal states. In that case these are just limiting cases of the non-extremal configurations already included as endpoints of the corresponding curves (black dot for the extremal bubble). We have to consider the same graphics of figure (5.13) but discarding the thick black and thin red curves corresponding to the extremal states. The stable phase at low temperature would then always be the black hole with bubble formation at high temperature. This situation is puzzling as the low temperature phase would in that case have negative specific heat and entropy. The negative entropy bubble states would also be the preferred phase for some range of the temperature in that case. We cannot just discard them as unphysical in this case as we would not have any available metric at low temperatures. The inclussion of the extremal states seems to solve the problem of negative entropy configurations, and reduces it to a thermodynamic instability. 5.5. GENERALIZED HP TRANSITIONS IN LOVELOCK GRAVITY 139 In the region of the λ−Tphase diagram of figure (5.14) colored in gray, the bubble configurations have both lower free energy than the vacuum, this being the thermodynamically preferred phase. Still the probability of bubble formation is non-zero being proportional to the exponential of the difference of the actions of both solutions. Thus after enough time a bubble will form leading again the system to the EH branch. For the inverse transition we proceed in the same way but setting the opposite asymptotics. The results are depicted in figure (5.15). Strikingly, due to several physical constraints, for low values of the LGB coupling bubbles, either extremal or non-extremal, do not exist, the critical value being λ=1 24(7 −√13) ≈0.14 where the two extremal states degenerate. They are shown in the figure (dashed red and thin red respectively for nonextremal and extremal bubbles) but they would be formed inside the outer horizon. Viable bubble geometries appear above the critical λvalue represented by the thick red line and they also have a well defined extremal extension (thin red line). For low enough values of the LGB coupling the black hole states are the only available configurations. Below λ= 1/12 the non-extremal solutions are stable and have positive entropy and as a consequence have lower free energy than the extremal vacuum. The nonextremal black hole is the stable phase of the system for all temperatures. For higher values of λ, negative entropy non-extremal states appear at low temperatures but the extremal configuration has lower free energy in that case. We can again safely remove the unphysical states. The low temperature phase corresponds to the extremal black holes with a transition to the non-extremal ones at high temperature. The stable low and high temperature phases are still the same above the critical λ= 1 24(7 −√13) where the bubble configurations appear. These can be divided in two branches, one stable that merges with the black hole curve at zero temperature and one unstable close to the extremal state that is irrelevant as it has higher free energy than the former. Any of these bubbles have higher free energy than the extremal black holes for all temperatures. Still and as for the transition in the other sense, the probability of a bubble being formed is non-zero and thus after enough time a bubble the BD-unstable phase would form. This would in principle drive the system to that pathological branch but it is also unstable to the formation of bubbles of the EH branch. The final fate of the system seems to be a chaotic situation with bubbles of both phases poping up everywhere. This seemingly problematic situation is avoided for λ < 1 24(7 −√13). In case we discard the free energy curves corresponding to the extremal states, thin red and black lines, we would encounter some problematic behavior for even lower values of λ. For λ > 1/12 negative entropy black holes appear and they would be the preferred phase for low temperatures. Above λ=1 24(7 −√13) the bubbles appear with also negative entropy and lower values of the free energy for some range of temperatures, between the lower thin red and green curves of figure (5.16). We can summarize saying that for values of the LGB coupling below λ=1 24(7 −√13) the BD-unstable branch of hyperbolic black holes is always driven to the EH one, this being stable against the formation of bubbles. For this scheme the EH-branch is protected and the only transition that takes place is a change from the extremal black hole at low temperatures to the non-extremal at higher ones. This transition only occurs when negative entropy states are in the spectrum, namely close to the extremal configuration. For higher values of λthe 140 CHAPTER 5. BUBBLES AND NEW PHASE TRANSITIONS situation is chaotic with bubbles of the opposite branch poping for any choice of the boundary conditions, the interpretation of this being unclear. Moreover, the transition mechanism presented here provides a possible resolution of the instabilities found for the ghosty branch of LGB gravity. For spherical and planar topology it is always driven to the EH-branch via bubble nucleation. We have just verified that the same happens for hyperbolic spacetimes as long as the LGB coupling is below λ=1 24(7 −√13). This will presumably be also valid in the cubic case for moderate values of µ. 5.6 Discussion During this chapter we have broaden the scope of our analysis of Lovelock theories and their solutions. In particular, we have included the possibility of distributional solutions for which the spin connection is discontinuous at some given junction (hyper)surface. In the absence of matter in the bubble, it glues two solutions that correspond to different branches of the same theory, i.e. to different asymptotics. From the hamiltonian point of view the existence of such configurations is allowed by the multivaluedness of the canonical momenta. We have also proven that it is possible to generalize the thermodynamic notions usually applied to black holes to these new solutions. Under certain regularity assumptions, static bubble configurations can be assigned temperature, mass, entropy and free energy, all verifying the expected thermodynamic relations. Having proven the consistency of the thermodynamic picture, we have then analyzed local and global stability of our system in this generalized context and the occurrence of phase transitions. We have restricted our attention to the LGB case even though the same kind of transitions occur also in the general case, as it can be explicitly shown in the planar case. This is a novel mechanism for phase transitions that is a distinctive feature of higher curvature theories of gravity. Specifically, phase transitions among the different branches of the theory are driven by this mechanism. Mimicking the thermalon configuration [187], a bubble separating two regions of different cosmological constants pops out, generically hosting a black hole. In the context of LGB gravity, this configuration is thermodynamically preferred above some critical temperature. The corresponding phase transition can be interpreted as a generalized HP transition for the high-curvature branches, driving the system towards the EH branch. This happens even for the hyperbolic case in which the reverse transition is also possible. For the EH-asymptotics the usual black hole is always the preferred phase. Below some critical λthermalon solutions do not even exist in this sector of the theory. The junction conditions do not just determine the existence of the static configurations but also their dynamics. In the LGB case, the bubble configuration, being unstable, dynamically changes the asymptotic cosmological constant, transitioning towards the stable horizonful branch of solutions, the only one usually considered as relevant. This is then a natural mechanism for the system to select the general relativistic vacuum among all possible ones. We are aware of the fact that the vacuum Λ+in the LGB theory exhibits ghosts. The phenomenon presented in this chapter, however, takes place in the Lovelock theory as well, where there are further healthy vacua than the one connected to the EH action [3]. The selection of the EH-branch among all the stable ones is not as universal as one may naively think, however, as not all branches of solutions are connected to the EH one by a 5.6. DISCUSSION 141 0.2 0.4 0.6 0.8 1.0 T -0.3 -0.2 -0.1 0.0 0.1 0.2 F 0.5 1.0 1.5 2.0 2.5 3.0 T -0.3 -0.2 -0.1 0.0 0.1 0.2 F 1 2 3 4 5 6 7 T -0.2 0.0 0.2 0.4 0.6 F 0.1 0.2 0.3 0.4 0.5 0.6 T -0.10 -0.05 0.00 0.05 0.10 F 0.2 0.4 0.6 0.8 T -0.10 -0.05 0.00 0.05 0.10 F 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T -0.1 0.0 0.1 0.2 0.3 F Figure 5.15: Free energy versus temperature for λ= 0.05,0.11,0.2(and zoom),0.21,0.23. In red the bubble states and in black the black holes in the stable branch. The thin black line correspods to the extremal geometry once assumed the vanishing of the entropy whereas the green line corresponds to the usual Wald entropy for the same geometry. The dashed red lines correspond to the extension for a < 1/√2 of M+−T−S. These are not viable bubble solutions in the same way as its extremal extension represented by the thin red line. Acceptable bubble geometries appear for λ≥1 24 7−√13represented by the thick red line and they also have a well defined extremal extension. The red dots indicate the locus of extremal bubble to black hole transitions and viceversa whereas the blue and the green ones indicate non-extremal extremal bubble transitions and bubble black hole transitions respectively. Below the green dot when there the black line is below the red one and so we have a low temperature black hole phase as well as a high temperature one. 251 Ideally, we would like to evaluate (D.0.13) for QCD close to confinement/deconfinement transition. While the recent lattice results provide a reliable equation of state3[263], rather than doing it from first principles, one has to rely on various models to evaluate transport coefficients of gauge theory plasma at strong coupling [216,246,264–266]. In what follows we present the first self-consistent estimate of (D.0.13) for a strongly coupled gauge theory plasma. Cascading gauge theory.— Consider [213]N= 1 four-dimensional supersymmetric SU(K+ P)×SU(K) gauge theory with two chiral superfields A1, A2in the (K+P, K) representation, and two fields B1, B2in the (K+P, K). This gauge theory has two gauge couplings g1, g2 associated with two gauge group factors, and a quartic superpotential W∼tr (AiBjAkB`)ikj` .(D.0.14) The theory is not conformal, and develops a strong coupling scale Λ through dimensional transmutation of the gauge couplings. In the UV/IR it undergoes the cascade of Seiberg [267] dualities with K→K±P. The net result of the duality cascade is that the rank Kof the theory becomes dependent on the scale Eat which the theory is probed [268]: K→Keff(E)≈2P2ln E Λ, E Λ.(D.0.15) While not QCD, the theory shares some of the IR features of the latter: when Kis an integer multiple of P, the cascade ends in the IR with SU(P) supersymmetric Yang-Mills theory which confines with spontaneous breaking of the chiral symmetry. Cascading gauge theory is always strongly coupled in the UV. In the planar limit and for large ’t Hooft coupling of the IR SU(p) factor, the theory is strongly coupled along its full RG flow, and thus can be studies using its holographic dual [213]. We focus on the cascading gauge theory in the regime where the holographic description is reliable. Thermodynamics of the cascading gauge theory plasma has been studied extensively in the past [253,269,270]: cascading gauge theory plasma simultaneously undergoes (firstorder) confinement and the chiral symmetry breaking at Tc= 0.6141111(3)Λ; at a slightly lower temperature Tχsb = 0.882503(0)Tcthe deconfined phase becomes unstable towards spontaneous development of a chiral condensate, finally, at Tu= 0.8749(0)Tc, the deconfined phase of the theory approaches a critical point with a divergent specific heat [266]. The shear viscosity of the plasma is universal for all phases and at all temperatures [271], η S=1 4π.(D.0.16) The bulk viscosity of the theory is technically difficult to compute — so far it is known only to the fourth order in the high temperature expansion, ln T Λ−1[253], which is not enough to determine its value at the critical point Tc. We use Eling-Oz formula [272,273] to compute bulk viscosity of the deconfined phase of the cascading gauge theory over all 3At least at vanishing baryon chemical potential. 252 APPENDIX D. CAVITATION EFFECTS 0.5 0.6 0.7 0.8 0.9 1.0 TL 0.03 0.04 0.05 0.06 0.07 ΖS Figure D.1: The ratio of the bulk viscosity ζto the entropy density Sin cascading gauge theory plasma (solid curve) and the bulk viscosity bound [265] (dashed). The dashed vertical line denote the critical temperature Tcof the confinement/deconfinement phase transition. temperature range. The results are presented in figure D.1. We find ζ ST=Tc = 0.04(8) .(D.0.17) Besides, the bulk viscosity bound [265] is respected all across the phase transition. We can now address the question whether or not cavitation is expected to affect the temperature of the deconfinement transition in cascading plasma. Here, the phase Aof a fluid is the deconfined phase of the plasma, and Bis the confined phase. Since in the planar limit both the transport coefficients and the entropy density are suppressed, we obtain combining (D.0.13) and (D.0.17) |δTc| Tc .ζA SA = 0.04(8) .(D.0.18) Discussion.— In this Letter we asked to which extent cavitation in confining gauge theories affects the critical temperature of the confinement/deconfinement transition. We used the specific example of a cascading gauge theory to argue that in the planar limit and at strong coupling the effect is small. 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