Non-extremal Black Holes Research Articles

Counting black hole microstates using string dualities

We discuss a previous attempt at a microscopic counting of the entropy of asymptotically flat nonextremal black holes. This method used string dualities to relate four and five dimensional black holes to the BTZ black hole. We show how the dualities can be justified in a certain limit, equivalent to a near horizon limit, but the resulting spacetime is no longer asymptotically flat.

Dirty black holes: Symmetries at stationary nonstatic horizons

We establish that the Einstein tensor takes on a highly symmetric form near the Killing horizon of any stationary but non-static (and non-extremal) black hole spacetime. [This follows up on a recent article by the current authors, gr-qc/0402069, which considered static black holes.] Specifically, at any such Killing horizon -- irrespective of the horizon geometry -- the Einstein tensor block-diagonalizes into ``transverse'' and ``parallel'' blocks, and its transverse components are proportional to the transverse metric. Our findings are supported by two independent procedures; one based on the regularity of the on-horizon geometry and another that directly utilizes the elegant nature of a bifurcate Killing horizon. It is then argued that geometrical symmetries will severely constrain the matter near any Killing horizon. We also speculate on how this may be relevant to certain calculations of the black hole entropy.

Pair creation of anti–de Sitter black holes on a cosmic string background

We analyze the quantum process in which a cosmic string breaks in an anti-de Sitter (AdS) background, and a pair of charged or neutral black holes is produced at the ends of the strings. The energy to materialize and accelerate the pair comes from the strings tension. In an AdS background this is the only study done in the process of production of a pair of correlated black holes with spherical topology. The acceleration $A$ of the produced black holes is necessarily greater than (|L|/3)^(1/2), where L<0 is the cosmological constant. Only in this case the virtual pair of black holes can overcome the attractive background AdS potential well and become real. The instantons that describe this process are constructed through the analytical continuation of the AdS C-metric. Then, we explicitly compute the pair creation rate of the process, and we verify that (as occurs with pair creation in other backgrounds) the pair production of nonextreme black holes is enhanced relative to the pair creation of extreme black holes by a factor of exp(Area/4), where Area is the black hole horizon area. We also conclude that the general behavior of the pair creation rate with the mass and acceleration of the black holes is similar in the AdS, flat and de Sitter cases, and our AdS results reduce to the ones of the flat case when L=0.

Quasinormal modes of the extremal BTZ black hole

Motivated by several pieces of evidence, in order to show that extremal black holes cannot be obtained as limits of non-extremal black holes, in this paper we calculate explicitly quasinormal modes for the Bañados, Teitelboim and Zanelli (BTZ) extremal black hole and show that the imaginary part of the frequency is zero. We obtain exact result for the scalar and fermionic perturbations. We also showed that the frequency is bounded from below for the existence of the normal modes (non-dissipative modes).

Type 0A 2D Black Hole Thermodynamics and the Deformed Matrix Model

Recently, it has been proposed that the deformed matrix model describes a two-dimensional type 0A extremal black hole. In this paper, the thermodynamics of 0A charged non-extremal black holes is investigated. We observe that the free energy of the deformed matrix model to leading order in 1/q can be seen to agree to that of the extremal black hole. We also speculate on how the deformed matrix model is able to describe the thermodynamics of non-extremal black holes.

Supersymmetric black holes in 2D dilaton supergravity: baldness and extremality

We present a systematic discussion of supersymmetric solutions of 2D dilaton supergravity. In particular those solutions which retain at least half of the supersymmetries are ground states with respect to the bosonic Casimir function (essentially the ADM mass). Nevertheless, by tuning the prepotential appropriately, black hole solutions may emerge with an arbitrary number of Killing horizons. The absence of dilatino and gravitino hair is proven. Moreover, the impossibility of supersymmetric dS ground states and of nonextremal black holes is confirmed, even in the presence of a dilaton. In these derivations the knowledge of the general analytic solution of 2D dilaton supergravity plays an important role. The latter result is addressed in the more general context of gPSMs which have no supergravity interpretation. Finally it is demonstrated that the inclusion of non-minimally coupled matter, a step which is already nontrivial by itself, does not change these features in an essential way.

On Black Hole Thermodynamics of 2-D Type 0A

We present a detailed analysis of the thermodynamics of two dimensional black hole solutions to type 0A with q units of electric and magnetic flux. We compute the free energy and derived quantities such as entropy and mass for an arbitrary non-extremal black hole. The free energy is non-vanishing, in contrast to the case of dilatonic 2-d black holes without electric and magnetic fluxes. The entropy of the extremal black holes is obtained, and we find it to be proportional to q^2, the square of the RR flux. We compare these thermodynamics quantities with those from candidate matrix model duals.

Extremal limits of theCmetric: Nariai, Bertotti-Robinson, and anti-NariaiCmetrics

In two previous papers we have analyzed the C-metric in a background with a cosmological constant, namely the de Sitter (dS) C-metric, and the anti-de Sitter (AdS) C-metric, following the work of Kinnersley and Walker for the flat C-metric. These exact solutions describe a pair of accelerated black holes in the flat or cosmological constant background, with the acceleration A being provided by a strut in-between that pushes away the two black holes. In this paper we analyze the extremal limits of the C-metric in a background with generic cosmological constant. We follow a procedure first introduced by Ginsparg and Perry in which the Nariai solution, a spacetime which is the direct topological product of the 2-dimensional dS and a 2-sphere, is generated from the four-dimensional dS-Schwarzschild solution by taking an appropriate limit, where the black hole event horizon approaches the cosmological horizon. Similarly, one can generate the Bertotti-Robinson metric from the Reissner-Nordstrom metric by taking the limit of the Cauchy horizon going into the event horizon of the black hole, as well as the anti-Nariai by taking an appropriate solution and limit. Using these methods we generate the C-metric counterparts of the Nariai, Bertotti-Robinson and anti-Nariai solutions, among others. One expects that the solutions found in this paper are unstable and decay into a slightly non-extreme black hole pair accelerated by a strut or by strings. Moreover, the Euclidean version of these solutions mediate the quantum process of black hole pair creation, that accompanies the decay of the dS and AdS spaces.

Challenges and obstacles for a bouncing universe in brane models

A brane evolving in the background of a charged AdS black hole displays in general a bouncing behavior with a smooth transition from a contracting to an expanding phase. We examine in detail the conditions and consequences of this behavior in various cases. For a cosmological-constant-dominated brane, we obtain a singularity-free, inflationary era which is shown to be compatible only with an intermediate-scale fundamental Planck mass. For a radiation-dominated brane, the bouncing behavior can occur only for background-charge values exceeding those allowed for non-extremal black holes. For a matter-dominated brane, the black-hole mass affects the proper volume or the expansion rate of the brane. We also consider the brane evolving in an asymmetric background of two distinct charged AdS black hole spacetimes being bounded by the brane and find that, in the case of an empty critical brane, bouncing behavior occurs only if the black-hole mass difference is smaller than a certain value. The effects of a brane curvature term on the bounce at early and late times are also investigated.

Thermal fluctuations and black-hole entropy

In this paper, we consider the effect of thermal fluctuations on the entropy of both neutral and charged black holes. We emphasize the distinction between fixed and fluctuating charge systems, using a canonical ensemble to describe the former and a grand canonical ensemble to study the latter. Our novel approach is based on the philosophy that the black-hole quantum spectrum is an essential component in any such calculation. For definiteness, we employ a uniformly spaced area spectrum, which has been advocated by Bekenstein and others in the literature. The generic results are applied to some specific models, in particular, various limiting cases of an (arbitrary-dimensional) AdS–Reissner–Nordstrom black hole. We find that the leading-order quantum correction to the entropy can consistently be expressed as the logarithm of the classical quantity. For a small AdS curvature parameter and zero net charge, it is shown that, independent of the dimension, the logarithmic prefactor is +1/2 when the charge is fixed but +1 when the charge is fluctuating. We also demonstrate that, in the grand canonical framework, the fluctuations in the charge are large, ΔQ ∼ ΔA ∼ S1/2BH, even when ⟨Q⟩ = 0. A further implication of this framework is that an asymptotically flat, non-extremal black hole can never achieve a state of thermal equilibrium.

Pair of accelerated black holes in a de Sitter background: The dSCmetric

Following the work of Kinnersley and Walker for flat spacetimes, we have analyzed the anti-de Sitter C-metric in a previous paper. In the de Sitter case, Podolsky and Griffiths have established that the de Sitter C-metric (dS C-metric) found by Plebanski and Demianski describes a pair of accelerated black holes in the dS background with the acceleration being provided (in addition to the cosmological constant) by a strut that pushes away the two black holes or, alternatively, by a string that pulls them. We extend their analysis mainly in four directions. First, we draw the Carter-Penrose diagrams of the massless uncharged dS C-metric, of the massive uncharged dS C-metric and of the massive charged dS C-metric. These diagrams allow us to clearly identify the presence of two dS black holes and to conclude that they cannot interact gravitationally. Second, we revisit the embedding of the dS C-metric in the 5D Minkowski spacetime and we represent the motion of the dS C-metric origin in the dS 4-hyperboloid as well as the localization of the strut. Third, we comment on the physical properties of the strut that connects the two black holes. Finally, we find the range of parameters that correspond to non-extreme black holes, extreme black holes, and naked particles.

Exactly solvable models in 2D semiclassical dilaton gravity and extremal black holes

Previously known exactly solvable models of 2D semiclassical dilaton gravity admit, in the general case, only non-extreme black holes. It is shown that there exist exceptional degenerate cases that can be obtained by some limiting transitions from the general exact solution, which include, in particular, extremal and ultraextremal black holes. We also analyse properties of extreme black holes without demanding exact solvability, and show that for such solutions quantum backreaction forbids the existence of ultraextreme black holes. The conditions under which divergencies of quantum stresses in a free-falling frame can disappear are found. We derive the closed equation with respect to the metric as a function of the dilaton field that enables one, choosing the form of the metric, to restore corresponding Lagrangian. It is demonstrated that exactly solvable models, found earlier, can be extended to include an electric charge only in two cases: either the dilaton–gravitation coupling is proportional to the potential term, or the latter vanishes. The second case leads to the effective potential with a negative amplitude and we analyse how this fact affects the structure of spacetime. We also discuss the role of quantum backreaction in the relationship between extremal horizons and the branch of solutions with a constant dilaton.

Relation between black hole entropy and quantum field spin

Starting from the master equations of governing arbitrary spin fields, we calculate the free energy of spherically symmetric space-time due to arbitrary spin fields. Then we study the entropy of seven kinds of static space-times. The result shows that there is a linear relation between nonextreme black hole entropy and quantum field spin. Taking into account the degeneracy due to spin, we have a direct proportion relation between nonextreme black hole entropy and quantum field degeneracy. As for the extreme black hole, we find that its entropy is zero.

Black hole entropy from nonperturbative gauge theory

We present the details of a mean-field approximation scheme for the quantum mechanics of N D0-branes at finite temperature. The approximation can be applied at strong 't Hooft coupling. We find that the resulting entropy is in good agreement with the Bekenstein-Hawking entropy of a ten-dimensional nonextremal black hole with a 0-brane charge. This result is in accord with the duality conjectured by Itzhaki, Maldacena, Sonnenschein and Yankielowicz. We study the spectrum of single-string excitations within quantum mechanics, and find evidence for a clear separation between light and heavy degrees of freedom. We also present a way of identifying the black hole horizon.

Late-time dynamics of rapidly rotating black holes

We study the late-time behavior of a dynamically perturbed rapidly rotating black hole. Considering an extreme Kerr black hole, we show that the large number of virtually undamped quasinormal modes (that exist for nonzero values of the azimuthal eigenvalue $m)$ combine in such a way that the field (as observed at infinity) oscillates with an amplitude that decays as $1/t$ at late times. This is in clear contrast to the standard late time power-law falloff familiar from studies of nonrotating black holes. This long-lived oscillating ``tail'' will, however, not be present for nonextreme (presumably more astrophysically relevant) black holes, for which we find that many quasinormal modes (individually excited to a very small amplitude) combine to give rise to an exponentially decaying field. At very late times this slowly damped quasinormal-mode signal gives way to the standard power-law tail (corresponding to a mixture of multipoles depending on the initial data). These results could have implications for the detection of gravitational-wave signals from rapidly spinning black holes, since the required theoretical templates need to be constructed from linear combinations of many modes. Our main results are obtained analytically, but we support the conclusions with numerical time evolutions of the Teukolsky equation. These time evolutions provide an interesting insight into the notion that the quasinormal modes can be viewed as waves trapped in the spacetime region outside the horizon. They also suggest that a plausible mechanism for the behavior we observe for extreme black holes is the presence of a ``superradiance resonance cavity'' immediately outside the black hole.

Macroscopic and microscopic description of black diholes

We study configurations consisting of a pair of non-extremal black holes in four dimensions, both with the same mass, and with charges of the same magnitude but opposite sign — diholes, for short. We present such exact solutions for Einstein–Maxwell theory with arbitrary dilaton coupling, and also solutions to the U(1) 4 theories that arise from compactified string/M-theory. Despite the fact that the solutions are very complicated, physical properties of these black holes, such as their area, charge, and interaction energy, admit simple expressions. We also succeed in providing a microscopic description of the entropy of these black holes using the ‘effective string’ model, and taking into account the interaction between the effective string and antistring.

One-loop corrected thermodynamics of the extremal and nonextremal spinning Banados-Teitelboim-Zanelli black hole

We consider the one-loop corrected geometry and thermodynamics of a rotating BTZ black hole by way of a dimensionally reduced dilaton model. The analysis begins with a comprehensive study of the non-extremal solution after which two different methods are invoked to study the extremal case. The first approach considers the extremal limit of the non-extremal calculations, whereas the second treatment is based on the following conjecture: extremal and non-extremal black holes ae qualitatively distinct entities. We show that only the latter method yields regularity and consistency at the one-loop level. This is suggestive of a generalized third law of thermodynamics that forbids continuous evolution from non-extremal to extremal black hole geometries.

BLACK HOLE THERMODYNAMICS FROM CALCULATIONS IN STRONGLY COUPLED GAUGE THEORY

We develop an approximation scheme for the quantum mechanics of N D0-branes at finite temperature in the 't Hooft large-N limit. The entropy of the quantum mechanics calculated using this approximation agrees well with the Bekenstein-Hawking entropy of a ten-dimensional non-extremal black hole with 0-brane charge. This result is in accord with the duality conjectured by Itzhaki, Maldacena, Sonnenschein and Yankielowicz. Our approximation scheme provides a model for the density matrix which describes a black hole in the strongly-coupled quantum mechanics.

Black hole thermodynamics from calculations in strongly coupled gauge theory.

We develop an approximation scheme for the quantum mechanics of N D0-branes at finite temperature in the 't Hooft large- N limit. The entropy of the quantum mechanics calculated using this approximation agrees well with the Bekenstein-Hawking entropy of a ten-dimensional nonextremal black hole with 0-brane charge. This result is in accordance with the duality conjectured by Itzhaki, Maldacena, Sonnenschein, and Yankielowicz [Phys. Rev. D 58, 046004 (1998)]. Our approximation scheme provides a model for the density matrix which describes a black hole in the strongly coupled quantum mechanics.

Does the third law of black holethermodynamics really have a serious failure?

The almost perfect correspondence between certain laws of classicalblack hole mechanics and the ordinary laws of thermodynamics isspoiled by the failure of the conventional back hole analogue of thethird law. Our aim in this addendum to a recent paper (2000 Class. Quantum Grav. 17 153-78) is to contribute to the associated discussionby shedding light on some simple facts of black hole physics.However, no attempt is made to lay to rest the correspondinglong-lasting debate. Instead, some evidence ismerely providedto make it clear that although the borderline between extremaland non-extremal black holes is very thin they are essentiallydifferent. Hopefully, a careful investigation of the relatedissues will end up with an appropriate form of the third lawand hence with an unblemished setting for black holethermodynamics.