Computation Of Black Hole Entropy Research Articles

Gravity’s Rainbow and Black Hole Entropy

We consider the effects of Gravity’s Rainbow on the computation of black hole entropy using a dynamical brick wall model. An explicit dependence of the radial coordinate approaching the horizon is proposed to analyze the behavior of the divergence. We find that, due to the modification of the density of states, the brick wall can be eliminated. The calculation is extended to include rotations and in particular to a Kerr black hole in a comoving frame.

Thermodynamics of higher spin black holes in AdS3

We discuss the thermodynamics of recently constructed three-dimensional higher spin black holes in SL(N, $ \mathbb{R} $ ) × SL(N, $ \mathbb{R} $ ) Chern-Simons theory with generalized asymptotically-anti-de Sitter boundary conditions. From a holographic perspective, these bulk theories are dual to two-dimensional CFTs with WN symmetry algebras, and the black hole solutions are dual to thermal states with higher spin chemical potentials and charges turned on. Because the notion of horizon area is not gauge-invariant in the higher spin theory, the traditional approaches to the computation of black hole entropy must be reconsidered. One possibility, explored in the recent literature, involves demanding the existence of a partition function in the CFT, and consistency with the first law of thermodynamics. This approach is not free from ambiguities, however, and in particular different definitions of energy result in different expressions for the entropy. In the present work we show that there are natural definitions of the thermodynamically conjugate variables that follow from careful examination of the variational principle, and moreover agree with those obtained via canonical methods. Building on this intuition, we derive general expressions for the higher spin black hole entropy and free energy which are written entirely in terms of the Chern-Simons connections, and are valid for both static and rotating solutions. We compare our results to other proposals in the literature, and provide a new and efficient way to determine the generalization of the Cardy formula to a situation with higher spin charges.

Loop quantum gravity without the Hamiltonian constraint

We show that under certain technical assumptions, including the existence of a constant mean curvature (CMC) slice and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantized on a partially reduced phase space, meaning reduced only with respect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce, in close analogy to shape dynamics, the generator of a local conformal transformation acting on both, the metric and the scalar field, which coincides with the CMC gauge condition. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantized by standard loop quantum gravity methods. Since this connection is invariant under the local conformal transformation, the generator of which is shown to be a good gauge fixing for the Hamiltonian constraint, the Dirac bracket associated with implementing these constraints coincides with the Poisson bracket for the connection. Thus, the well developed kinematical quantization techniques for loop quantum gravity are available, while the Hamiltonian constraint has been solved (more precisely, gauge fixed) classically. The physical interpretation of this system is that of general relativity on a fixed spatial CMC slice, the associated ‘time’ of which is given by the CMC. While it is hard to address dynamical problems in this framework (due to the complicated ‘time’ function), it seems, due to good accessibility properties of the CMC gauge, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. The corresponding calculation yields the surprising result that the usual prescription of fixing the Barbero–Immirzi parameter β to a constant value in order to obtain the well-known formula S = a(Φ)A/(4G) does not work for the black holes under consideration, while a recently proposed prescription involving an analytic continuation of β to the case of a self-dual space-time connection yields the correct result. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function and has consequences for loop quantum cosmology.

Black-hole horizons in modified spacetime structures arising from canonical quantum gravity

Several properties of canonical quantum gravity modify spacetime structures, sometimes to the degree that no effective line elements exist to describe the geometry. An analysis of solutions, for instance in the context of black holes, then requires new insights. In this paper, standard definitions of horizons in spherical symmetry are first reformulated canonically, and then evaluated for solutions of equations and constraints modified by inverse-triad corrections of loop quantum gravity. When possible, a spacetime analysis is performed which reveals a mass threshold for black holes and small changes to Hawking radiation. For more general conclusions, canonical perturbation theory is developed to second order to include back-reaction from matter. The results shed light on the questions of whether renormalization of Newton's constant or other modifications of horizon conditions should be taken into account in computations of black-hole entropy in loop quantum gravity.

Detailed black hole state counting in loop quantum gravity

We give a complete and detailed description of the computation of black hole entropy in loop quantum gravity by employing the most recently introduced number-theoretic and combinatorial methods. The use of these techniques allows us to perform a detailed analysis of the precise structure of the entropy spectrum for small black holes, showing some relevant features that were not discernible in previous computations. The ability to manipulate and understand the spectrum up to the level of detail that we describe in the paper is a crucial step towards obtaining the behavior of entropy in the asymptotic (large horizon area) regime.

Black hole entropy in loop quantum gravity: The role of internal symmetries

I will discuss here the role of the internal symmetry group in the computations of black hole entropy in loop quantum gravity according to the standard prescription given by Domagala and Lewandowski [1]. In particular I will show how it is possible to take into account the possible choice of either SO(3) or SU(2) as the internal symmetry groups of general relativity in Loop Quantum Gravity and how this choice changes the combinatorial problem of counting the black hole degrees of freedom.

On the computation of black hole entropy in loop quantum gravity

We discuss some issues related to the computation of black hole entropy in loop quantum gravity from the novel point of view provided by the recent number-theoretical methods introduced by the authors and their collaborators. In particular we give exact expressions, in the form of integral transforms, for the black hole entropy in terms of the area. We do this by following several approaches based both on our combinatorial techniques and on functional equations similar to those employed by Meissner in his pioneering work on this subject. To put our results in perspective, we compare them with those of Meissner. We will show how our methods confirm some of his findings, extend the validity of others and correct some mistakes. At the end of the paper, we will discuss the delicate issue of the asymptotics of black hole entropy.

Walls of marginal stability and dyon spectrum in Script N = 4 supersymmetric string theories

The spectrum of quarter BPS dyons in N=4 supersymmetric string theories can change as the asymptotic moduli cross walls of marginal stability on which the dyon can break apart into a pair of half BPS states. In this paper we classify these marginal stability walls and examine this phenomenon in the context of exact dyon spectrum found in a class of N=4 supersymmetric string theories. We argue that the dyon partition functions in different domains separated by marginal stability walls are the same, but the choice of integration contour needed for extracting the degeneracies from the partition function differ in these different regions. We also find that in the limit of large charges the change in the degeneracy is exponentially suppressed compared to the leading contribution. This is consistent with the fact that in the computation of black hole entropy we do not encounter any change as the asymptotic moduli fields move across the walls of marginal stability. Finally we carry out some tests of S-duality invariance in the theory.

Description of the Riemannian geometry in the presence of conical defects.

A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to the proposed prescription ${\cal M}_{\alpha}$ are considered as limits of the converging sequences of smooth spaces. This enables one to give a strict mathematical meaning to a number of invariant integral quantities on ${\cal M}_{\alpha}$ and make use of them in applications. In particular, an explicit representation for the Euler numbers and Hirtzebruch signature in the presence of conical singularities is found. Also, higher dimensional Lovelock gravity on ${\cal M}_{\alpha}$ is shown to be well-defined and the gravitational action in this theory is evaluated. Other series of applications is related to computation of black hole entropy in the higher derivative gravity and in quantum 2-dimensional models. This is based on its direct statistical-mechanical derivation in the Gibbons-Hawking approach, generalized to the singular manifolds ${\cal M}_{\alpha}$, and gives the same results as in the other methods.