Entropy-current for dynamical black holes in Chern-Simons theories of gravity

TL;DR

This paper develops an entropy current framework for dynamical black holes in Chern-Simons gravity theories, establishing a local second law of thermodynamics through explicit examples and a general proof in odd dimensions.

Contribution

It introduces a novel construction of entropy current for dynamical black holes in Chern-Simons theories and proves the linearized second law in arbitrary odd dimensions.

Findings

Entropy current constructed for specific CS theories in 2+1 and 4+1 dimensions.

Second law holds to quadratic order in perturbations for 2+1 dimensional CS theory.

The formalism is invariant under null horizon reparameterizations.

Abstract

We construct an entropy current and establish a local version of the classical second law of thermodynamics for dynamical black holes in Chern-Simons (CS) theories of gravity. We work in a chosen set of Gaussian null coordinates and assume the dynamics to be small perturbations around the Killing horizon. In explicit examples of both purely gravitational and mixed gauge gravity CS theories in $(2+1)$ and $(4+1)$-dimensions, the entropy current is obtained by studying the off-shell structure of the equations of motion evaluated on the horizon. For the CS theory in $(2+1)$ dimensions, we argue that the second law holds to quadratic order in perturbations by considering it as a low energy effective field theory with the leading piece given by Einstein gravity. In all such examples, we show that the construction of entropy current is invariant under the reparameterization of the null horizon coordinates. Finally, extending an existing formalism for diffeomorphism invariant theories, we construct an abstract proof for the linearised second law in arbitrary Chern-Simons theories in any given odd dimensions by studying the off-shell equations of motion. As a check of consistency, we verify that the outcome of this algorithmic proof matches precisely with the results obtained in explicit examples.