On Black Hole Entropy

TL;DR

Jacobson, Kang, and Myers analyze black hole entropy in generally covariant gravity with higher-derivative terms by comparing Wald's Noether-charge method to a field-redefinition approach. They prove that entropy is a local horizon functional defined on arbitrary cross-sections of a Killing horizon and is robust to Noether ambiguities for stationary, bifurcate horizons, while developing a field-redefinition technique to relate entropies across related theories. They provide explicit entropy expressions for broad classes of Lagrangians, including extensions with ∇R terms, and show that certain perturbative results can be exact through this method. The work advances understanding of the microscopic interpretation of black hole entropy and offers practical tools for evaluating entropy in complex gravitational theories, though questions remain for nonstationary horizons and the general second-law behavior.

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 $\bar{g}_{ab}\equiv g_{ab} + Δ_{ab}$, where $Δ_{ab}$ 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.}