Horizon thermodynamics in $f(R,R^{\mu\nu}R_{\mu\nu})$ theory

TL;DR

This paper explores the validity of the horizon first law in a complex modified gravity theory, deriving black hole entropy and energy, and applying the results to quadratic-curvature gravity for specific black hole solutions.

Contribution

It introduces a method to derive black hole energy in $f(R,R^{ u}R_{ u})$ gravity using the horizon first law and applies it to quadratic-curvature gravity.

Findings

The horizon first law holds in the studied theory.

Derived black hole entropy using Wald method.

Calculated energy for static spherically symmetric black holes.

Abstract

We investigate whether the new horizon first law still holds in $f(R,R^{\mu\nu}R_{\mu\nu})$ theory. For this complicated theory, we first determine the entropy of black hole via Wald method, then we derive the energy by using the new horizon first law, the degenerate Legendre transformation, and the gravitational field equations. For application, we consider the quadratic-curvature gravity and firstly calculate the entropy and the energy for a static spherically symmetric black hole, which reduces to the results obtained in literatures for a Schwarzschild-(A)dS black hole.