Showing papers on "Black hole thermodynamics published in 1994"
TL;DR: It is proved that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole, and a local, geometrical prescription is proposed for the entropy, $S_{dyn}$, of a dynamical black hole.
Abstract: We consider a general, classical theory of gravity with arbitrary matter fields in n dimensions, arising from a diffeomorphism-invariant Lagrangian L. We first show that L alwasy can be written in a ``manifestly covariant'' form. We then show that the symplectic potential current (n-1)-form FTHETA and the symplectic current (n-1)-form \ensuremath{\omega} for the theory always can be globally defined in a covariant manner. Associated with any infinitesimal diffeomorphism is a Noether current (n-1)-form J and corresponding Noether charge (n-2)-form Q. We derive a general ``decomposition formula'' for Q. Using this formula for the Noether charge, we prove that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole. (For higher derivative theories, previous arguments had established this law only for stationary perturbations.) Finally, we propose a local, geometrical prescription for the entropy ${\mathit{S}}_{\mathrm{dyn}}$ of a dynamical black hole. This prescription agrees with the Noether charge formula for stationary black holes and their perturbations, and is independent of all ambiguities associated with the choices of L, FTHETA, and Q. However, the issue of whether this dynamical entropy in general obeys a ``second law'' of black hole mechanics remains open. In an appendix, we apply some of our results to theories with a nondynamical metric and also briefly develop the theory of stress-energy pseudotensors.
2,321 citations
TL;DR: The entropy per unit area is shown to be finite to all orders in superstring perturbation theory, and the importance of these conclusions to the resolution of the problem of black hole information loss is reiterated.
Abstract: In this paper the entropy of an eternal Schwarzschild black hole is studied in the limit of an infinite black hole mass. The problem is addressed from the point of view of both canonical quantum gravity and superstring theory. The entropy per unit area of a free scalar field propagating in a fixed black hole background is shown to be quadratically divergent near the horizon. It is shown that such quantum corrections to the entropy per unit area are equivalent to the quantum corrections to the gravitational coupling. Unlike field theory, superstring theory provides a set of identifiable configurations which give rise to the classical contribution to the entropy per unit area. These configurations can be understood as open superstrings with both ends attached to the horizon. The entropy per unit area is shown to be finite to all orders in superstring perturbation theory. The importance of these conclusions to the resolution of the problem of black hole information loss is reiterated.
734 citations
TL;DR: The thermodynamic properties of black holes in (3+1)- and (2+1-dimensional Einstein gravity with a negative cosmological constant are investigated and it is shown that the chemical potential conjugate to angular momentum is equal to the proper angular velocity of the black hole with respect to observers who are at rest in the stationary time slices.
Abstract: We investigate the thermodynamical properties of black holes in (3+1)- and (2+1)-dimensional Einstein gravity with a negative cosmological constant. In each case, ther thermodynamic internal energy is computed for a finite spatial region that contains the black hole. The temperature at the boundary of this regoin is defined by differentiating the energy with respect to entropy, and is equal to the product of the surface gravity (divided by 2\ensuremath{\pi}) and the Tolman redshift factor for temperature in a stationary gravitational field. We also compute the thermodynamic surface pressure and, in the case of the 2+1 black hole, show that the chemical potential conjugate to angular momentum is equal to the proper angular velocity of the black hole with respect to observers who are at rest in the stationary time slices. In 3+1 dimensions, a calculation of the heat capacity reveals the existence of a thermodynamically stable black hole solution and a negative heat capacity instanton. This result holds in the limit that the spatial boundary tends to infinity only if the cosmological constant is negative; if the cosmological constant vanishes, the stable black hole solution is lost. In 2+1 dimensions, a calculation of the heat capacity reveals the existence of a thermodynamically stable black hole solution, but no negative heat capacity instanton.
343 citations
TL;DR: The universal bound on entropy is rederive with the help of black holes while allowing for Unruh-Wald buoyancy and it is demonstrated by explicit calculation that, for an arbitrarily large number of particle species, the bound is indeed satisfied by cavity thermal radiation in the thermodynamic regime, provided vacuum energies are included.
Abstract: We rederive the universal bound on entropy with the help of black holes while allowing for Unruh-Wald buoyancy. We consider a box full of entropy lowered toward and then dropped into a Reissner-Nordstr\"om black hole in equilibrium with thermal radiation. We avoid the approximation that the buoyant pressure varies slowly across the box, and compute the buoyant force exactly. We find, in agreement with independent investigations, that the neutral point generically lies very near the horizon. A consequence is that in the generic case the Unruh-Wald entropy restriction is neither necessary nor sufficient for enforcement of the generalized second law. Another consequence is that generically the buoyancy makes only a negligible contribution to the energy bookkeeping, so that the original entropy bound is recovered if the generalized second law is assumed to hold. The number of particle species does not figure in the entropy bound, a point that has caused some perplexity. We demonstrate by explicit calculation that, for an arbitrarily large number of particle species, the bound is indeed satisfied by cavity thermal radiation in the thermodynamic regime, provided vacuum energies are included. We also show directly that thermal radiation in a cavity in D-dimensional space also respects the bound regardless of the value of D. As an application of the bound we show that it strongly restricts the information capacity of the posited black hole remnants, so that they cannot serve to resolve the information paradox.
272 citations
TL;DR: A generalized second law of thermodynamics is formulated, and shown to be valid under suitable conditions, and it is shown that, in this model, a black hole can consume an arbitrarily large amount of information.
Abstract: Black hole evaporation is investigated in a (1+1)-dimensional model of quantum gravity. Quantum corrections to the black hole entropy are computed, and the fine-grained entropy of the Hawking radiation is studied. A generalized second law of thermodynamics is formulated, and shown to be valid under suitable conditions. It is also shown that, in this model, a black hole can consume an arbitrarily large amount of information.
242 citations
TL;DR: The extremal limit of classical solutions describing charged dilaton black holes accelerating in a background magnetic field is studied, showing that near the event horizon the extremal solutions reduce precisely to the static extremal black hole solutions.
Abstract: Classical solutions describing charged dilaton black holes accelerating in a background magnetic field have recently been found. They include the Ernst metric of the Einstein-Maxwell theory as a special case. We study the extremal limit of these solutions in detail, both at the classical and quantum levels. It is shown that near the event horizon the extremal solutions reduce precisely to the static extremal black hole solutions. For a particular value of the dilaton coupling, these extremal black holes are five-dimensional Kaluza-Klein monopoles. The Euclidean sections of these solutions can be interpreted as instantons describing the pair creation of extremal black holes and/or Kaluza-Klein monopoles in a magnetic field. The action of these instantons is calculated and found to agree with the Schwinger result in the weak-field limit. For the Euclidean Ernst solution, the action for the extremal solution differs from that of the previously discussed wormhole instanton by the Bekenstein-Hawking entropy. However, in many cases quantum corrections become large in the vicinity of the black hole, and the precise description of the creation process is unknown.
183 citations
TL;DR: A one parameter family of static charged black hole solutions in $(2+1)$-dimensional general relativity minimally coupled to a dilaton $\phi\propto ln({r\over\beta})$ with a potential term e^{b\phi}\Lambda$ is obtained.
Abstract: A one parameter family of static charged black hole solutions in (2+1)-dimensional general relativity minimally coupled to a dilaton \ensuremath{\varphi}\ensuremath{\propto}ln(r/\ensuremath{\beta}) with a potential term ${\mathit{e}}^{\mathit{b}\mathrm{\ensuremath{\varphi}}}$\ensuremath{\Lambda} is obtained. Their causal structures are investigated, and the thermodynamical temperature and entropy are computed. One particular black hole in the family has the same thermodynamical properties as the Schwarzschild black hole in 3+1 dimensions. Solutions with cosmological horizons are also discussed. Finally, a class of black holes arising from the dilaton with a negative kinetic term (tachyon) is briefly discussed.
117 citations
TL;DR: The entropy of a scalar field is calculated semiclassically in the background of a dilatonic black hole to derive the area and cutoff dependences.
Abstract: The entropy of a scalar field is calculated semiclassically in the background of a dilatonic black hole. The area and cutoff dependences are normal except in the extremal case, where the area is zero but the entropy nonzero.
110 citations
108 citations
08 Sep 1994
TL;DR: In this paper, the authors discuss the existence of the universal bound on entropy regardless of acceleration buoyancy, and discuss the question of why macroscopic objects cannot emerge in the Hawking radiance.
Abstract: I review various proposals for the nature of black hole entropy and for the mechanism behind the operation of the generalized second law. I stress the merits of entanglement entropy {\tenit qua\/} black hole entropy, and point out that, from an operational viewpoint, entanglement entropy is perfectly finite. Problems with this identification such as the multispecies problem and the trivialization of the information puzzle are mentioned. This last leads me to associate black hole entropy rather with the multiplicity of density operators which describe a black hole according to exterior observers. I relate this identification to Sorkin's proof of the generalized second law. I discuss in some depth Frolov and Page's proof of the same law, finding it relevant only for scattering of microsystems by a black hole. Assuming that the law is generally valid I make evident the existence of the universal bound on entropy regardless of issues of acceleration buoyancy, and discuss the question of why macroscopic objects cannot emerge in the Hawking radiance.
101 citations
TL;DR: In this paper, a general definition of a black hole in terms of a future outer trapping horizon, a hypersurface foliated by marginal surfaces of a certain type, has been found.
Abstract: General laws of black hole dynamics, some of which are analogous to the laws of thermodynamics, have recently been found for a general definition of a black hole in terms of a future outer trapping horizon, a hypersurface foliated by marginal surfaces of a certain type. This theory is translated here into spin coefficient language. Most of the following results assume an energy condition. Second law: the area form of a future outer trapping horizon is generically increasing, otherwise constant. First law: the rate of change of the area form is given by an energy flux and the trapping gravity. Zeroth law: the total trapping gravity of a compact outer marginal surface has an upper bound, attained if and only if the trapping gravity is constant. Topology law: a compact future outer marginal surface has spherical topology. Signature law: an outer trapping horizon is generically spatial, otherwise null. Trapping law: spatial surfaces sufficiently close to a compact future outer marginal surface are trapped if they lie inside the trapping horizon. Confinement law: if the interior and exterior of a future outer trapping horizon are disjoint, an observer inside the horizon cannot get outside.
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TL;DR: In this paper, a pedagogical review of dilaton gravity, Hawking radiation, the black hole information problem, and black hole pair creation is given, along with a discussion of the relationship between the two problems.
Abstract: These lectures give a pedagogical review of dilaton gravity, Hawking radiation, the black hole information problem, and black hole pair creation (Lectures presented at the 1994 Trieste Summer School in High Energy Physics and Cosmology)
TL;DR: In this article, it was shown that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary, and that the edge states can contribute to black hole entropy.
Abstract: We show in the context of Einstein gravity that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary. Such a boundary occurs in certain approaches to the physics of black holes like the one based on the membrane paradigm. The edge states can contribute to black hole entropy in these models. A ``complementarity principle" is also shown to emerge whereby certain ``edge" observables are accessible only to certain observers. The physical significance of edge observables and their states is discussed using their similarities to the corresponding quantities in the quantum Hall effect. The coupling of the edge states to the bulk gravitational field is demonstrated in the context of (2+1) dimensional gravity.
01 Jan 1994
TL;DR: In this paper, the information puzzle in four dimensions: Can the Information Come Out Before the Endpoint?; Low-Energy Effective Descriptions of the Planckian Endpoint; Remnants?; Information Destruction?; The Superposition Principle; Energy Conservation The New Rules; Superselection Sectors, $\alpha$-parameters, and the Restoration of Unitarity.
Abstract: Contents:
1. Introduction
2. Causal Structure and Penrose Diagrams: Minkowski Space; 1+1 Dimensional Minkowski Space; Schwarzchild Black Holes; Gravitational Collapse and the Vaidya Spacetimes; Event Horizons, Apparent Horizons, and Trapped Surfaces
3. Black Holes in Two Dimensions: General Relativity in the $S$-Wave Sector; Classical Dilaton Gravity; Eternal Black Holes; Coupling to Conformal Matter; Hawking Radiation and the Trace Anomaly; The Quantum State; Including the Back-Reaction; The Large $N$ Approximation; Conformal Invariance and Generalizations of Dilaton Gravity; The Soluble $RST$ Model
4. The Information Puzzle in Four Dimensions: Can the Information Come Out Before the Endpoint?; Low-Energy Effective Descriptions of the Planckian Endpoint; Remnants?; Information Destruction?; The Superposition Principle; Energy Conservation The New Rules; Superselection Sectors, $\alpha$-parameters, and the Restoration of Unitarity
5. Conclusions and Outlook
TL;DR: This entropy is shown to satisfy an increase theorem on either the global or apparent horizon of a two-dimensional black hole.
Abstract: Black hole entropy is studied for an exactly solvable model of two-dimensional gravity\cite{rst1}, using recently developed Noether charge techniques\cite{wald1}. This latter approach is extended to accomodate the non-local form of the semiclassical effective action. In the two-dimensional model, the final black hole entropy can be expressed as a local quantity evaluated on the horizon. This entropy is shown to satisfy an increase theorem on either the global or apparent horizon of a two-dimensional black hole.
TL;DR: In this paper, a catastrophe-theoretic analysis of the potential function defined by black hole entropy is presented, where two types of self-gravitating particle solutions found in several theories with nonAbelian fields are smoothly connected by a family of non-trivial black holes.
Abstract: Two types of self-gravitating particle solutions found in several theories with nonAbelian fields are smoothly connected by a family of non-trivial black holes. There exists a maximum point of the black hole entropy, where the stability of solutions changes. This criterion is universal, and the changes in stability follow from a catastrophe-theoretic analysis of the potential function defined by black hole entropy.
TL;DR: In this article, the authors consider low energy string theory with a horizon and a spacelike symmetry and show that the dual solution has exactly the same Hawking temperature (surface gravity) and entropy (area) as the original solution.
Abstract: We consider solutions to low energy string theory which have a horizon and a spacelike symmetry. Each of these solutions has a geometrically different dual description. We show that the dual solution has a horizon with exactly the same Hawking temperature (surface gravity) and entropy (area) as the original solution.
TL;DR: In this paper, a quantum description of black holes is proposed, where the degrees of freedom to be quantized are identified with the microscopic degrees offreedom of the horizon, and their dynamics is governed by the action of the relativistic bosonic membrane in D = 4.
TL;DR: It is argued that in the context of Hawking radiation a limiting temperature for string theory leads to a limiting acceleration, which for a black hole implies a minimum distance from the horizon for an observer to remain stationary and effectively introduces a cutoff in Rindler space or the Schwarzschild geometry inside of which accelerations would exceed this maximum value.
Abstract: An accelerating observer sees a thermal bath of radiation at the Hawking temperature which is proportional to the acceleration. Also, in string theory there is a Hagedorn temperature beyond which one cannot go without an infinite amount of energy. Several authors have shown that in the context of Hawking radiation a limiting temperature for string theory leads to a limiting acceleration, which for a black hole implies a minimum distance from the horizon for an observer to remain stationary. We argue that this effectively introduces a cutoff in Rindler space or the Schwarzschild geometry inside of which accelerations would exceed this maximum value. Furthermore, this natural cutoff in turn allows one to define a finite entropy for Rindler space or a black hole as all divergences were occurring on the horizon. In all cases if a particular relationship exists between Newton's constant and the string tension then the entropy of the string modes agrees with the Bekenstein-Hawking formula.
TL;DR: In this article, a Liouville field is coupled to gravity in two space-time dimensions in a novel way, and a large class of asymptotically flat solutions to the field equations is obtained, one of which bears an interesting resemblance to the two-dimensional string-theoretic black hole.
TL;DR: In this article, a cornucopion model of the Bekenstein-Hawking entropy is presented, where information is lost to the original asymptotic observer, but the information encoded in the BH entropy remains in causal contact with him and is re-emitted with the Hawking radiation.
Abstract: Trieste Spring School Lectures describing the author's opinions about black hole evaporation and information loss. The remnant, or cornucopion scenario for the endpoint of Hawking evaporation is described in detail. In this picture information can be lost to the original asymptotic observer without violating the rules of quantum mechanics, because a black hole remnant is viewed as a large space connected onto our own by an almost pointlike opening. It does not behave like an elementary particle. Objections to remnants are refuted and the (remote) possibility of testing this scenario experimentally is discussed. Also included is a brief description of Susskind's picture of the stringy origin of Bekenstein-Hawking entropy. An attempt is made to argue that the cornucopion picture and Susskind's model of the states responsible for black hole entropy are compatible with each other. Information is lost to the asymptotic observer in Hawking evaporation, but the information encoded in the BH entropy remains in causal contact with him and is re-emitted with the Hawking radiation.
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TL;DR: In this paper, the authors show that the black hole partition function can be placed on a firm logical foundation by enclosing the black holes in a spatially finite "box" or boundary.
Abstract: The first objective of this article is to show that the black hole partition function can be placed on a firm logical foundation by enclosing the black hole in a spatially finite "box" or boundary. The presence of the box has the effect of stabilizing the black hole and yields a system with a positive heat capacity. The second objective of this article is to explore the origin of black hole entropy. This is accomplished through the construction of a path integral expression for the density matrix for the gravitational field, and through an analysis of the connection between the density matrix and the black hole density of states. Our results suggest that black hole entropy can be associated with an absence of certain "inner boundary information" for the system. (Based on the talk presented by J.D. Brown at the conference "The Black Hole 25 Years After", Santiago, Chile, January 1994.)
TL;DR: In this article, the authors investigated the thermodynamics of a static spherically symmetric black hole-global monopole system by two methods: surface gravity and the euclidean path integral.
TL;DR: The quantum stress tensor near a three-dimensional black hole is studied for a conformally coupled scalar field and the back reaction to the metric is investigated.
Abstract: The quantum stress tensor near a three-dimensional black hole is studied for a conformally coupled scalar field. The back reaction to the metric is also investigated.
TL;DR: In this article, the one-loop contribution to the entropy of a black hole from field modes near the horizon is computed in string theory and it is modular invariant and ultraviolet finite.
Abstract: The one-loop contribution to the entropy of a black hole from field modes near the horizon is computed in string theory. It is modular invariant and ultraviolet finite. There is an infrared divergence that signifies an instability near the event horizon of the black hole. It is due to the exponential growth of the density of states and the associated Hagedorn transition characteristic of string theory. It is argued that this divergence is indicative of a tree level contribution, and the Bekenstein-Hawking-Gibbons formula for the entropy should be understood in terms of string states stuck near the horizon.
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TL;DR: In this paper, a geometrical notion of entropy, definable in flat space, governs the first quantum correction to the Bekenstein-Hawking black hole entropy, and the entropy diverges in quantum field theory in the absence of an ultraviolet cutoff.
Abstract: We show that a geometrical notion of entropy, definable in flat space, governs the first quantum correction to the Bekenstein-Hawking black hole entropy. We describe two methods for calculating this entropy -- a straightforward Hamiltonian approach, and a less direct but more powerful Euclidean (heat kernel) method. The entropy diverges in quantum field theory in the absence of an ultraviolet cutoff. Various related finite quantities can be extracted with further work. We briefly discuss the corresponding question in string theory.
TL;DR: The string-theoretic calculation of black hole entropy is extended, first performed by Susskind and Uglum, and it is shown that the result agrees with that obtained from the classical action of string theory, using the Noether charge method developed by Wald.
Abstract: We extend the string-theoretic calculation of black hole entropy, first performed by Susskind and Uglum, away from the infinite mass limit. It is shown that the result agrees with that obtained from the classical action of string theory, using the Noether charge method developed by Wald. Also shown in the process is the equivalence of two general techniques for finding black hole entropies---the Noether charge method and the method of conical singularities.
TL;DR: It is shown that within this approach quantum hair arises naturally within the approach to the generalized uncertainty principle of quantum gravity, and implications for the no-hair theorem are discussed.
Abstract: We discuss the idea of black hole complementarity, recently suggested by Susskind and co-workers, and the notion of the stretched horizon, in the light of the generalized uncertainty principle of quantum gravity. We discuss implications for the no-hair theorem and we show that within this approach quantum hair arises naturally.
TL;DR: The apparent horizon is shown to be a powerful tool in the study of black hole spacetimes because the wavelength and damping time of the quasinormal modes and the rotation parameter in the rotating cases can be read off directly from oscillations in the geometry of the black hole horizons.
Abstract: Dynamic black hole spacetimes are studied by examining the evolution of apparent horizons surrounding the holes. We performed numerical evolutions of three different initial data sets: nonrotating black holes distorted by time symmetric (Brill) gravitational waves, distorted rotating black holes, and the time symmetric two black hole Misner data. Although the initial data sets represent different physical problems, the results for these systems are strikingly similar. At early times in the evolution, the apparent horizons may be very distorted and nonspherical (or disjoint in the case of two black holes, but the systems quickly settle down to a nearly spherical or oblate (in the case of rotating holes) configuration and the horizons are then seen to oscillate at the quasinormal frequency of the final black hole. In the case of two black holes with disjoint horizons, we see the appearance of a larger horizon surrounding both holes as they collide. From this point the horizon dynamics is very similar to the single distorted black hole systems. The wavelength and damping time of the quasinormal modes and the rotation parameter in the rotating cases can be read off directly from oscillations in the geometry of the black hole horizons. The apparent horizon is thus shown to be a powerful tool in the study of black hole spacetimes.
TL;DR: This work test the consistency of these principles by attempting to exceed the black hole extremality condition in various processes in which a U(1) charge is added to a nearly extreme Reissner-Nordstroem black hole charged with a [ital different] type of U( 1) charge.
Abstract: The stability of the black hole horizon is demanded by both cosmic censorship and the generalized second law of thermodynamics. We test the consistency of these principles by attempting to exceed the black hole extremality condition in various processes in which a U(1) charge is added to a nearly extreme Reissner-Nordstroem black hole charged with a [ital different] type of U(1) charge. For an infalling spherical charged shell the attempt is foiled by the self-Coulomb repulsion of the shell. For an infalling classical charge it fails because the required classical charge radius exceeds the size of the black hole. For a quantum charge the horizon is saved because, in order to aviod the Landau ghost, the effective coupling constant cannot be large enough to accomplish the removal of the horizon.