A comparison of two constructions for dynamical corrections to Wald entropy

Abstract

In this work, we analyze the differences and similarities between two recent constructions, which are distinct in their methodologies for extending the Wald entropy of stationary black holes to non-stationary situations in general higher-derivative gravity. One of them, denoted by $S_\text{Wall}$, is constructed by exploiting the boost symmetry of the near-horizon geometry, whereas the other, denoted by $S_\text{dyn}$, is obtained from a covariant phase-space analysis based on the Wald-Iyer Noether charge formalism. While $S_\text{dyn}$ is, by construction, defined only for linearized fluctuations around a stationary black hole solution, $S_\text{Wall}$ does not require such a linearization for its construction. Although the linearization is necessary to interpret $S_\text{Wall}$ as a well-defined notion of entropy, the construction itself naturally contains terms that are higher order in the dynamical fluctuations. By comparing the technical structures underlying the two constructions, we clarify the fundamental differences between the methods on which they are based. We demonstrate that while the construction of $S_\text{dyn}$ given the $S_\text{Wall}$ is straightforward, the converse is more subtle. We develop an algorithm to obtain a local expression for $S_\text{Wall}$ from a known expression for $S_\text{dyn}$ in a generic diffeomorphism-invariant theory of gravity, provided certain technical conditions are satisfied. We justify our analytical findings with explicit demonstrations in a particular case: the Riemann-squared example of the higher-derivative theory of gravity.