Geometric approaches to the thermodynamics of black holes
aa r X i v : . [ g r- q c ] M a y August 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 1 Geometric approaches to the thermodynamics of black holes
Christine Gruber ∗ Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,AP 70543, M´exico, DF 04510, M´exico ∗ E-mail: [email protected]
Orlando Luongo † Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch7701, Cape Town, South Africa.Astrophysics, Cosmology and Gravity Centre (ACGC), University of Cape Town, Rondebosch7701, Cape Town, South Africa.Dipartimento di Fisica, Universit`a di Napoli ”Federico II”, Via Cinthia, Napoli, Italy.Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli, Via Cinthia, Napoli, Italy. † E-mail: [email protected], [email protected]
Hernando Quevedo ‡ Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,AP 70543, M´exico, DF 04510, M´exico ‡ E-mail: [email protected]
In this summary, we present the main topics of the talks presented in the parallelsession “Black holes - 5” of the 14th Marcel Grossmann Meeting held in Rome, Italy inJuly 2015. We first present a short review of the main approaches used to understandthermodynamics by using differential geometry. Then, we present a brief summary ofeach presentation, including some general remarks and comments.
Keywords : Black hole thermodynamics; black hole entropy; phase transitions; equilib-rium space; phase space
1. Introduction
Differential geometry is a very important tool of mathematical physics with diverseapplications in physics, chemistry, engineering and even economics. As one of themost important applications from the point of view of theoretical physics, we canmention the case of the four interactions of Nature for which a well-establisheddescription in terms of geometrical concepts is known. Indeed, Einstein proposedin 1915 the astonishing principle ”field strength = curvature“ to understand thephysics of the gravitational field . In an attempt to associate a geometric structureto the electromagnetic field, Yang and Mills used in 1953 the concept of a principalfiber bundle with the Minkowski spacetime, as the base manifold, and the symmetrygroup U (1), as the standard fiber, to demonstrate that the Faraday tensor canbe interpreted as the curvature of this particular fiber bundle. Today, it is wellknown that the weak interaction and the strong interaction can be represented asthe curvature of a principal fiber bundle with a Minkowski base manifold and thestandard fiber SU (2) and SU (3), respectively.In very broad terms, one can say that all the known forces of Nature act among ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 2 the particles that constitute a thermodynamic system. Due to the large number ofparticles involved in the system, only a statistical approach is possible, from whichaverage thermodynamic values for the physical quantities of interest are derived.Although the thermodynamic laws are based entirely upon empirical results whichare satisfied under certain conditions in almost any macroscopic system, the geomet-ric approach to thermodynamics has proved to be very useful. In our opinion, thefollowing three branches of geometry have found sound applications in equilibriumthermodynamics: analytic geometry, Riemannian geometry, and contact geometry.One of the most important contributions of analytic geometry to the under-standing of thermodynamics is the identification of points of phase transitions withextremal points of the surface determined by the state equation of the correspond-ing thermodynamic system. For a more detailed description of these contributionssee, for instance, the books by H. B. Callen or K. Huang .Riemannian geometry was first introduced in statistical physics and thermo-dynamics by Rao , in 1945. Rao introduced a metric whose components in localcoordinates coincide with Fisher’s information matrix. Rao’s pioneering work hasbeen followed up and extended by a number of authors (for a review, see, e.g. thebook by S. Amari ). On the other hand, Riemannian geometry in the space ofequilibrium states was introduced by Weinhold and Ruppeiner , who definedmetric structures as the Hessian of the internal energy and the entropy, respectively.Both metrics have been widely used to study the geometry of the equilibrium spaceof ordinary systems and black holes. The approach based upon the use of Hessianmetrics is commonly known as thermodynamic geometry.Contact geometry was introduced by Hermann into the thermodynamic phasespace in order to formulate in a consistent manner the geometric version of the lawsof thermodynamics. One important property of classical thermodynamics is that itdoes not depend on the choice of thermodynamic potential which is equivalent tosaying that it is invariant with respect to Legendre transformations . Contact geom-etry allows us to consider a Legendre transformation as a coordinate transformationin the phase space . This fact was used by Quevedo to propose the formalismof geometrothermodynamics which takes into account the Legendre invariance ofclassical thermodynamics, and is considerably different from the approach of ther-modynamic geometry.One of the goals of this parallel session of MG14 was to allow researchers inter-ested in the relationship between geometry and thermodynamics to present theirresults and applications, especially in the context of black hole thermodynamics.This goal was reached to some extent. All the talks discussed problems relatedto black hole thermodynamics in different theories, its physical implications, phasetransition structures, equilibrium spaces, thermodynamic phase spaces and others.This work is split into several sections, each of which corresponds to one pre-sentation in the parallel session BH5. In this way, we hope to take into accountthe interests of all the speakers, and reflect the spirit of the session held at a highly ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 3 scientific level and carried out in full cordiality.
2. Possible Observation Sequences Randall-Sundrum Black Holes
In the first talk of the session, Erin Nikita presented his work on sperically symmetricRandall-Sundrum (RS) black holes, done in collaboration with Kristina A. Rannuand Stanislav O. Alexeyev. They investigated the ACPY-solution , which is anasymptotically Schwarzschild solution of 5-dimensional RS models, given by themetric ds = " − − Λ r r − M − . M F − r dFdr ! − Mr ! − dr − − Mr ! dt + " r + F − Λ d Ω . (1)Here, Λ is a negative cosmological constant in 5 dimensions, and F is a numericallyfound polynomial of 11 th order, F ( r ) = 1 − . Mr ! + 1 . Mr ! (2) − . Mr ! + . . . + 2 . Mr ! . (3)Even in the post-Newtonian regime RS models can hardly be distinguished fromEinstein gravity. A possibility to derive observational properties which could bechecked against observations is to consider accretion disks around and the thermo-dynamic properties of the black hole solution. Concretely, the lifetime of primordialblack holes in the RS frameworks, nucleated in the early universe at high temper-atures and density fluctuations, have been considered. Initial masses of these pri-mordial black holes which reach the end of their lifetime around the present timeare assumed to be of the order of M ≈ . × g . (4)Their radiation contributes to the cosmic microwave background , and in the finalstages of their evaporation they are expected to produce bursts of energy , pre-dicted to be gamma radiation in the range of MeV to TeV, at redshifts of z ≤ . .Present time telescopes are capable of detecting the evaporation of PBHs atmaximal distances d of d ≃ . (cid:18) Ωsr (cid:19) − . (cid:18) E GeV (cid:19) . (cid:18) T TeV (cid:19) . pc , (5)where Ω is the angular telescope resolution, T is the black hole temperature and E represents the energy range of the telescope. ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 4 For detection of differences between GR and RS models in the evaporation ofprimordial black holes, the difference in masses is supposed to be M RS M GR > . (6)The calculations of black hole evaporation, following the standard expression byHawking , for the ACPY solution provides a precise expression for the evaporationrate in terms of the mass up to 10 th order, − dMdt ≃ k B π M + 1512 k B π M + O (cid:0) M − (cid:1) . (7)This result shows that the present conditions and gamma ray bursts of currentlyevaporating primordial black holes are not enough to distinguish standard Einsteingravity black holes from RS black holes.
3. Constructing black hole entropy from gravitational collapse
A new perspective to describe thermodynamic properties of black hole formationhas been discussed by Aymen Hamid, developed in collaboration with GiovanniAcquaviva, George F. R. Ellis and Rituparno Goswami , through the use of gravi-tational collapse in semitetrad 1+1+2 covariant formalism. This approach extendsthe 1+3 formalism to any spacetime with a preferred spatial direction. Particularattention is here devoted to the simplest case of spherically symmetric metrics.Under the simplest hypothesis of Oppenheimer-Snyder-Datt collapse, a sphericaldust like star living on Schwarzschild geometry was assumed, the star’s exteriorclassified as Petrov type D. The entropy of the free gravitational field can be inferredfor a static observer, even even if no event horizons exist. Considering a notion ofentropy during the collapse process allows for an answer to the question whetherentropy in this context is a property of the horizon only, which emerges after thecreation of a black hole, or whether it can in principle be defined at all times andsmoothly changes from an initial configuration to its black hole result S BH = A/ S grav and its definition is basedon the Bel-Robinson tensor . In order to be compatible with standard features ofphysical entropy, it is necessary to demand that the entropy should be a measureof the local anisotropy of the free gravitational field, non-negative, vanishing forzero Weyl tensor, and in the limiting case should result in the Bekenstein-Hawkingentropy of black holes.From those requirements, assuming that the second law of thermodynamics stillcontinues to hold, the relation T grav dS grav = dU grav + p grav dV > T grav , U grav and p grav the effective temperature, internalenergy and isotropic pressure of the free gravitational field, and V the spatial vol-ume. With the gravitational pressure vanishing in a Coulomb-like field, p grav = 0, ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 5 and employing the equation of energy conservation, the second law and also thetemperature of the gravitational field can be expressed in terms of the kinematicaland Weyl quantities of the spacetime. Calculating the change of the entropy of thegravitational field during a time interval ( τ − τ ) then yields the result δS grav | ( τ − τ ) = α (cid:16) A ( τ ) − A ( τ ) (cid:17) , (9)where A ( τ ) is the surface area of the star at an arbitrary time τ > τ , and α is aconstant introduced in the definition of the gravitational energy-momentum tensor,which can be constrained to the value α = 1.The increase in the instantaneous gravitational entropy outside a collapsing starduring a given interval of time is thus proportional to the change in the surfacearea of the star during that interval, and thus the black hole form of the entropy isrecovered in the process of gravitational collapse.
4. A Heuristic Energy Quantization Of Equilibrium Black HoleHorizons
In his contribution, Abhishek Majhi aimed at deriving the energy quantization ofblack holes by employing heuristic arguments from thermodynamics and statisticalmechanics .The black holes are hereby considered as isolated horizons, i.e., 3-dimensionalnull inner boundaries of a 4-dimensional spacetime, and possess topology S (2) × R .The symplectic structure of the 4-d bulk spacetime induces a symplectic structureof a SU (2) Chern-Simons (CS) theory on the boundary, i.e., the isolated horizon.The quantization of the isolated horizon in a straightforward manner is not possible,since Chern-Simons theories are topological field theories in which the Hamiltonianvanishes identically, and thus no well-defined notion of energy exists. However,naturally the bulk spacetime enclosed by the boundary entails a notion of energy,and thus there is a first law of thermodynamics associated with the boundary,which in turn assigns a meaning of energy to the boundary field theory. Moreover,a correspondence of bulk and boundary states can be achieved in Loop Quantumgravity (LQG), where states are represented by sets of spins | j , ..., j N i . A generalHamiltonian should act on states asˆ H S | j , . . . , j N i = l p N X l =1 ǫ jl | j , ..., j N i , (10)with the energy eigenvalues ǫ jl .It is possible to use these states and the associated Hilbert space to count thepossible microstates on the isolated horizon, i.e., the entropy, under the assumptionof constant area, which corresponds to the constant entropy ensemble. Using anapproach from LQG, the spectrum of the spins representing the state of the systemis calculated, and the distribution function for the most probable configuration of ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 6 spins is obtained from extremizing the entropy. The entropy of the isolated horizoncan then be written as S = λ A IH πγl p , (11)where A IH is the classical area of the isolated horizon, γ is the Barbero-Immirziparameter and l p is the Planck length. A Lagrange multiplier λ has been introducedin the extremization procedure, taking on the value of λ for the most probable spinconfiguration.The constant entropy approach is however only one possible ensemble in whichthe system can be considered. The physical properties of the system should notdepend on the choice of ensemble, and therefore the treatment in another ensemble,e.g. one with constant energy, must be completely equivalent and yield the samephysical predictions. Extremizing the entropy under the constraint of constantenergy, using a Lagrange multiplier β , the energy and entropy of the isolated horizonare then related as S = βE IH l p . (12)With this expression, β can then be interpreted as the inverse temperature of thehorizon.Comparing the obtained relation of entropy/area and energy, δE IH = λ πγβl p δA IH (13)with the first law of thermodynamics, δE IH = κ IH π δA IH , (14)where κ IH is the surface gravity of the horizon, it is possible to identify1 β = γl p λ κ IH . (15)From the general formulation of the Hamiltonian of the system, and using thisrelation between the constants, a Hamiltonian is then defined asˆ H S | j , . . . , j N i = κ IH π | j , ..., j N i . (16)The Hamiltonian captures the physics associated with both near horizon Rindlerobservers as well as asymptotic observers, and allows for the formulation of blackhole thermodynamics in the usual energy ensemble. This work thus bypasses theproblem of the vanishing Hamiltonian in the boundary Chern-Simons theory witharguments from thermodynamics and statistical mechanics. ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 7
5. Thermodynamic Volume And Phase Transitions Of AdS BlackHoles
In his talk, David Kubiznak presented a new point of view on black hole ther-modynamics, in particular on black holes with a cosmological constant. If thecosmological constant is identified with pressure, and its conjugate quantity withthermodynamic volume, black holes can be understood from a chemical perspectivein terms of concepts such as van der Waals fluids, reentrant phase transitions, andtriple points. In a series of articles in collaboration with Robert Mann, NatachaAltamirano and Zeinab Sherkatghanad, , various types of AdS black holes, in-cluding rotating (around one or more axes), charged and higher dimensional ones,have been investigated.The principal idea is to treat the cosmological constant as a thermodynamicvariable, i.e., the pressure, including it into the first law of thermodynamics. Iden-tifying the pressure through comparison of the Smarr relation with Euler’s theoremfor homogeneous functions as P = − Λ8 π = ( d − d − πl , (17)where Λ is the cosmological constant, l the corresponding curvature radius, and d the number of spacetime dimensions, its conjugate is then defined as V = − π ∂M∂ Λ . (18)The modified first law of thermodynamics then reads dM = κ π dA + V dP , (19)plus possible black hole work terms as Φ dQ + P i Ω i dJ i , which indicates that themass of the black hole actually corresponds better to the enthalpy of an ordinarythermodynamic system, than to its internal energy. The mass of the black hole isthus equivalent to the amount of energy necessary to create the black hole and putit into its cosmological environment.The above mentioned articles have investigated various types of black holes witha cosmological constant, by analyzing the Gibbs free energy and its dependence onvarious thermodynamic quantities. For singly spinning black holes in d dimensions,a transition between small and large black holes has been found, similar to theliquid-gas phase transition in fluids, and the corresponding critical exponents arethe same as for a van der Waals fluid . Analogous investigations have been doneon singly and doubly spinning higher-dimensional AdS black holes in the canonicalensemble, i.e., fixed angular momentum (momenta). For the singly spinning ones,besides the usual transition between small and large black holes, there is a anotherreentrant phase transition from large back to small black holes, in analogy to thephase structure of multicomponent fluids . Ultimately, in multiply spinning blackholes in d = 6 dimensions, the thermodynamics of the system depends on the ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 8 ratio q = j /J of the two angular momenta. For q = 0, the system recovers thereentrant large/small/large phase transition structure of the singly spinning blackhole. With nonzero, but small q , the phase transition changes to an analogy of asolid/liquid phase transition, whereas for q ∈ [0 . , . q > . .With the introduction of the cosmological constant as an additional thermody-namic variable, it is thus possible to discover a very rich structure of thermody-namical properties and transitions in black holes, and by the analogy with fluidsreinterpret some aspects of black hole thermodynamics .
6. Thermodynamic Structure Of The Space Of Equilibrium StatesOf The Kerr-Newman Black Hole Family
In his talk, Miguel Angel Garcia-Ariza talked about a formulation of thermody-namics within a framework of differential geometry, taken in investigations in acollaboration with Merced Montesinos and Gerardo F. Torres del Castillo . Intheir work, they followed an approach to geometric thermodynamics defined byRuppeiner , employing a metric formalism in the abstract n -dimensional manifoldspanned by the extensive thermodynamic state variables, interpreted as coordinateson that manifold. Ruppeiner’s metric is defined as the Hessian of the entropy func-tion with respect to the extensive variables. The curvature defined by this metric issupposed to mirror the thermodynamic interactions of the system, i.e., a flat metricwith zero curvature would correspond to a system without thermodynamic interac-tions, whereas some curvature implies that interactions are present, and curvaturesingilarities mark points of major changes in the system’s properties.As an example of an simple hydrostatic thermodynamic system, the ideal gaswas considered, with its fundamental equation given by dU = T dS − pdV + µdN , (20)with U , T , S , p , V , µ , and N denoting the standard thermodynamic state variables,i.e., the system’s internal energy, temperature, entropy, pressure, volume, chemicalpotential, and number of particles, respectively. For systems following this generalfundamental relation, the thermodynamic metric in equilibrium phase space can bewritten as g R = c V T dT + 1 κ T T V dV , (21)where c V is the heat capacity at constant volume, and κ T is the compressibility atconstant temperature.Since the ideal gas, having no interactions, is described to a flat metric in themanifold of equilibrium states, the question arises whether the ideal gas is the onlyand uniquely determined system with a flat thermodynamic metric. Assuming that ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 9 for such systems, c V = const, and employing some coordinate transformations, itcan be shown that any thermodynamic system with a compressibility of the form k − T = e t [ tf ( V ) + f ( V )] , (22)where t = log T , and f , f are only functions of the volume, has a flat metric andvanishing curvature. As a consequence, the ideal gas is shown to represent onlya particular case of a closed hydrostatic system with C V = const. Such systemsin general may have spaces of equilibrium states with a flat metric, despite thepresence of interactions. This leads to some tension with the usual conjecture on thecorrespondence between curvature of the thermodynamic metric and interactions.An example was constructed, in which Ruppeiner’s metric was flat, and yet thesystem featured thermdynamic interactions.A geometric formalism of thermodynamics can be applied also in the field ofblack hole thermodynamics, concretely for the example of the Kerr-Newman familyof black holes. Some inconsistencies with the choice of the thermodynamic potentialwere pointed out, i.e., the dependence on the predicted critical points of the systemon the thermodynamic potential used.
7. Black Hole Thermodynamics In Finite Time
A slightly ’engineering’ point of view to black hole thermodynamics was provided inthe contribution of Christine Gruber, who in collaboration with Alessandro Bravettiand Cesar Lopez-Monsalvo investigated the impact of finite-time effects on blackhole engines .Finite-time thermodynamics is an approach to thermodynamics from a morerealistic point of view, dropping the assumption of perfectly reversible processesand instead trying to estimate dissipative losses that occur along the evolution ofa system in finite times – so to speak, calculating the energetic ’price of haste’.Thermodynamic processes carried out in finite times suffer from dissipative lossesbecause it is not possible to go through the path reversibly, i.e., as a sequence ofperfect local equilibria. During a process which is not perfectly reversible, in eachinfinitesimal step along the way small amounts of energy are dissipated. Intuitively,this can be understood from a so-called horse-carrot process. To drive a systemalong a path in thermodynamic phase space, it is brought into contact with amuch larger reservoir having slightly different values of the intensive thermodynamicquantities. Equilibration with a sequence of reservoirs causes the system to changethe values of its intensive quantities in infinitesimal steps, eventually reaching thefinal point of the path. If there is however not an infinite time at hand to establishthe perfect equilibrium in each step along the way, dissipative losses will occur andsum up over the length of the path. Quantitatively, these losses can be computedusing a geometric formalism of thermodynamics , as was already introduced inthe previous talk/chapter. A thermodynamic metric on the abstract manifold ofthermodynamic phase space can be defined as the Hessian of the thermodynamic ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 10 potential such as e.g. the entropy S , with respect to the extensive thermodynamicvariables x i of the system, ds = S ij dx i dx j (23)From this definition, the length of a process from the initial to the final point canbe defined in the thermodynamic phase space as L S = Z γ q − S ij dx i dx j . (24)It has been shown that this length can be related to the sum (or integral) ofthe infinitesimal energy dissipations in each step along the process,(∆ S ) diss ≥ L S ǫτ , (25)where ǫ is the infinitesimal amount of dissipation generated in each step along thepath, and τ is the duration of the process.The aim of finite-time thermodynamics is to obtain realistic limitations on ideal-ized scenarios, and it is thus a useful tool to reassess the efficiency of thermodynamicprocesses, with wide applications in industrial contexts. In the work by Gruber et.al, it has been applied to black holes in the context of Penrose-like processes, whichconsider the extraction of energy from a Kerr black hole. They investigated ther-modynamic processes decreasing the angular momentum of the black hole from theextremal to the Schwarzschild limit, or the extraction of mass, and calculated thethermodynamic length of these processes. The results showed that the dissipativelosses during the extraction of energy grow stronger close to the extremal limit, andthus in order to minimize dissipation, the extremal limit should be avoided.
8. The Volume of Black Holes
The contribution of Maulik Parikh was concerned with the definition of volume forblack holes, or stationary spacetimes in general . The need for such a definitionarises from the quantum gravitational argument that holography, or the encodingof information on the surface of a black hole instead of the bulk volume, leads to aradical decrease in entropy, since it is proportional to area /l p , instead of volume /l p .The problem here is however that this notion of volume is not well-defined, anddepends on the interpretation of time and space across horizons, or on the choiceof time slicing in a spacetime. Therefore, an invariant definition of volume wasproposed as V spatial = dV D dt , (26)where dV D ( t ) = Z t + dtt dt ′ Z d D − x √− g D , (27) ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 11 which is the differential spacetime volume of a given spacetime, and contains thedeterminant g ( D ) of the complete spacetime, instead of only the spatial part. Thisdefinition for the volume holds for spacetimes in which a timelike Killing vectorexists (even if that Killing vector is not a global one), which is equivalent to therequirement of thermal equilibrium, and is thus valid for static or stationary space-times. The proposed expression for the volume does not change with time, andmoreover does not depend on the choice of the stationary time-slicing. For a four-dimensional spherically symmetric spacetime, the volume takes on the form V d = 4 π r . (28)With these definitions at hand, the possibility of having a black hole with a finitesurface area, but an infinite volume was addressed. However, considering the sym-metry groups of 3- and 4-dimensional spherical, flat and hyperbolic spacetimes, itcan be argued that there is no class of solutions which admits an infinite volumethat is bounded by a finite horizon area. The generalization of these arguments tohigher dimensions remain to be done.In a second part of his talk, investigations on black hole nucleation have beenpresented . The rate Γ of nucleation of a certain type of instantons in a chosenbackground is given by Γ ≃ exp ( − I E [instanton])exp ( − I E [background]) , (29)where I E are the Euclidean action. The nucleation rate of black holes in Einstein andEinstein-Gauss-Bonnet gravity for 4-dimensional de Sitter spacetimes is calculated,resulting in Γ ≃ exp (cid:18) − πL G + 4 παG (cid:19) , (30)where G is Newton’s constant, L is the curvature radius of de Sitter, and α is theGauss-Bonnet coupling term. Thus, by adding a topological Gauss-Bonnet termto the gravitational action of four-dimensional de Sitter spacetime, the nucleationrate of black holes is greatly enhanced, which renders the theory very sensitive toinstabilities. The Gauss-Bonnet coefficient thereby serves as a stability bound onthe maximal curvature of spacetime.
9. Entropy in locally-de Sitter spacetimes
In her presentation, Adriana Victoria Araujo introduced work on the thermo-dynamic properties of de Sitter spacetimes, done in collaboration with J. G.Pereira . When constructing general relativity on a de Sitter background insteadof a Minkowskian one, the local Riemannian geometry is modified, in particularspacetime is endowed with a de Sitter-Cartan connection, changing the local dy-namics. As a consequence, the notion of entropy changes, which is directly related ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 12 to Noether charges derived from the system’s spacetime diffeomorphisms, whichnow are determined by the de Sitter group. The effects of the de Sitter backgroundon the thermodynamics of black holes has been investigated.Due to the presence of a cosmological constant, or a curvature radius l , aSchwarzschild-de Sitter spacetime possesses two horizons. The usual Schwarzschildhorizon is modified by terms including l , and given by r SdS = 2 M (cid:16) M l + · · · (cid:17) , (31)whereas the cosmological de Sitter horizon is r ′ SdS = l (cid:16) − Ml − M l + · · · (cid:17) , (32)expanded in powers of M/l , assuming that the de Sitter curvature radius ismuch larger than the Schwarzschild radius. The corresponding thermodynamics ofSchwarzschild-de Sitter spacetime is determined by both horizons and their prop-erties.Considering the Schwarzschild horizon with de Sitter modifications, the entropy S = A/ S = 4 πM (cid:16) M l + · · · (cid:17) . (33)Differentiating the entropy with respect to the mass, the temperature of the blackhole horizon can be obtained as T = 18 πM (cid:16) − M l + · · · (cid:17) = κ π . (34)which as well gives the surface gravity κ . Ultimately, the energy of the horizon ismodified by the de Sitter curvature radius as well, and can be determined as E = M + 8 M l + · · · . (35)In the case of de Sitter horizon, the entropy is calculated as S ′ = πl (cid:16) − Ml − M l + 3 M l + · · · (cid:17) , (36)with a corresponding horizon temperature of T ′ ≡ πr ′ SdS = 12 πl (cid:16) Ml + 3 M l + · · · (cid:17) . (37)The modified energy of the de Sitter horizon then is E ′ = l − M − M l + 13 M l · · · . (38)In summary, the thermodynamic properties of the horizons in Schwarzschild-deSitter spacetime have been investigated, and it was shown that the presence ofa cosmological constant, or a finite de Sitter curvature radius, leads to a mutualdependence of the horizon properties. ugust 27, 2018 7:19 WSPC Proceedings - 9.75in x 6.5in main-session˙final page 13
10. Conclusions
The main conclusion derived from the talks presented in this parallel session isthat nowadays black hole thermodynamics is a vast area of active research. Thedifferent geometric approaches are just one method which allows us to investigatethe structure of the equilibrium and phase spaces, the thermodynamic volume ofblack holes, the phase transition structure, and the novel idea of finite-time blackhole thermodynamics. Future works in this direction include the investigation of thecosmological constant and other physical parameters as thermodynamic variableswhich completely change the structure of the equilibrium and phase spaces. Also,the search for microscopic models for black holes is an open issue that can behandled by using geometric methods.However, other methods are necessary in order to attack the problem of under-standing the physical meaning of black hole thermodynamics. Entropy, which isperhaps the most important thermodynamic property of black holes, is far from be-ing completely understood. A classical origin of entropy is a possibility, although aquantum origin is certainly a very challenging idea. Both approaches were discussedin this parallel session. In addition, the idea of using black hole thermodynamicsto detect deviations of generalized models from Einstein gravity is a very promisingapproach.This parallel session was held at a highly scientific level, and carried out infull cordiality. We thank all the speakers and attendees for their contributions,discussions and suggestions that made possible this session.
Acknowledgments
CG was supported by funding from the DFG Research Training Group 1620 ‘Modelsof Gravity’, and by an UNAM postdoctoral fellowship program. O.L. wants tothank the National Research Foundation (NRF) for financial support and P. K. S.Dunsby for fruitful discussions. This work was partially supported by DGAPA-UNAM, Grant No. 113514, and CONACyT, Grant No. 166391.
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