Horizon entropy and higher curvature equations of state

TL;DR

This paper explores extending the thermodynamic derivation of gravitational field equations to higher curvature theories, proposing a refined approach that relates horizon entropy to local Killing vectors and examining the conditions needed for such derivations.

Contribution

It introduces a refined method for deriving higher curvature gravitational equations from horizon thermodynamics, addressing previous limitations and proposing specific conditions for the derivation.

Findings

Horizon entropy can be linked to local Killing vectors in higher curvature theories.

Field equations can be derived from an algebraic Lagrangian involving the metric and Riemann tensor.

Derivation likely excludes derivatives of curvature in the Lagrangian.

Abstract

The Clausius relation between entropy change and heat flux has previously been used to derive Einstein's field equations as an equation of state. In that derivation the entropy is proportional to the area of a local causal horizon, and the heat is the energy flux across the horizon, defined relative to an approximate boost Killing vector. We examine here whether a similar derivation can be given for extensions beyond Einstein gravity to include higher derivative and higher curvature terms. We review previous proposals which, in our opinion, are problematic or incomplete. Refining one of these, we assume that the horizon entropy depends on an approximate local Killing vector in a way that mimics the diffeomorphism Noether charge that yields the entropy of a stationary black hole. We show how this can be made to work if various restrictions are imposed on the nature of the horizon slices and the approximate Killing vector. Also, an integrability condition on the assumed horizon entropy density must hold. This can yield field equations of a Lagrangian constructed algebraically from the metric and Riemann tensor, but appears unlikely to allow for derivatives of curvature in the Lagrangian.