On Black Hole Entropy

TL;DR

This paper compares two methods for calculating black hole entropy in complex gravity theories, extending Wald's approach and introducing a field redefinition technique, with implications for understanding the statistical origins of black hole entropy.

Contribution

It develops a field redefinition method for black hole entropy calculation and extends Wald's geometric approach to arbitrary cross-sections of Killing horizons.

Findings

Wald's entropy expression is invariant under field redefinitions.

The field redefinition method yields exact results at first order.

The techniques are applicable to a wide class of covariant gravity theories.

Abstract

Two techniques for computing black hole entropy in generally covariant gravity theories including arbitrary higher derivative interactions are studied. The techniques are Wald's Noether charge approach introduced recently, and a field redefinition method developed in this paper. Wald's results are extended by establishing that his local geometric expression for the black hole entropy gives the same result when evaluated on an arbitrary cross-section of a Killing horizon (rather than just the bifurcation surface). Further, we show that his expression for the entropy is not affected by ambiguities which arise in the Noether construction. Using the Noether charge expression, the entropy is evaluated explicitly for black holes in a wide class of generally covariant theories. Further, it is shown that the Killing horizon and surface gravity of a stationary black hole metric are invariant under field redefinitions of the metric of the form $\bar{g}_{ab}\equiv g_{ab} + \Delta_{ab}$, where $\Delta_{ab}$ is a tensor field constructed out of stationary fields. Using this result, a technique is developed for evaluating the black hole entropy in a given theory in terms of that of another theory related by field redefinitions. Remarkably, it is established that certain perturbative, first order, results obtained with this method are in fact {\it exact}. The possible significance of these results for the problem of finding the statistical origin of black hole entropy is discussed.}