Quantum geometry of isolated horizons and black hole entropy

Abstract

Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum Chern-Simons theory of a U(1) connection on a punctured 2-sphere, the horizon. Subtle mathematical features of the quantum Chern-Simons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (non-gravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all non-rotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated.An effort has been made to make the paper self-contained by including short reviews of the background material.

Noncommutative Spectral Invariants and Black Hole Entropy

“…This point of view has emerged in recent years in different works as in [49,11,2] where conformal symmetries on the horizon are used to compute black hole entropy.…”

Black hole entropy from loop quantum gravity in higher dimensions

“…Up to this, there are two main proposals for tracing: the first one is to sum over all spin networks with fixed total horizon area A H , thus having a variable number of punctures. This approach has been employed in the original calculation [24] and extended to the SU(2) approach [19] in [4,37]. We can readily apply it and obtain an entropy proportional to A H /β at leading order.…”

“…In order to be able to quantize them, we will select a maximally commuting subset adapted to the puncturing spin network in the next section, that is, we perform gauge unfixing [28]. This subset does however not form a closing algebra with the Gauß constraints, since the smearing functions Λ IJ are not constant on H. We thus further restrict to constant Λ IJ on H and note that also in the U(1) framework in 3 + 1 dimensions [15,24], the restriction to constant Λ IJ on H becomes necessary, however for different reasons [7].…”

“…Quantization of the second term cancels the first one. Still, comparison with the 3 + 1 dimensional treatment [24] suggests that the horizon Hilbert space should be further restricted by a global gauge invariance condition, which is derived there by "shrinking a loop around the back of the sphere". This procedure is not readily available in higher dimensions.…”

“…As usual, see e.g [24], we assume that there exists at least one solution of the Hamiltonian constraint in the bulk which is compatible with a given configuration of punctures. Otherwise, a gauge fixing as in [25] could be employed.…”