Horizon Thermodynamics and Gravitational Field Equations in Horava-Lifshitz Gravity

TL;DR

This paper demonstrates that in Hořava-Lifshitz gravity, the gravitational field equations at a black hole horizon can be reformulated as the first law of thermodynamics, leading to consistent expressions for entropy and mass.

Contribution

It establishes a thermodynamic interpretation of gravitational field equations in Hořava-Lifshitz gravity, including a generalized Misner-Sharp energy and its conservation properties.

Findings

Field equations can be expressed as the first law of thermodynamics at the horizon.

Derived consistent formulas for entropy and mass of black holes.

Defined a conserved generalized Misner-Sharp energy in this gravity theory.

Abstract

We explore the relationship between the first law of thermodynamics and gravitational field equation at a static, spherically symmetric black hole horizon in Ho\v{r}ava-Lifshtiz theory with/without detailed balance. It turns out that as in the cases of Einstein gravity and Lovelock gravity, the gravitational field equation can be cast to a form of the first law of thermodynamics at the black hole horizon. This way we obtain the expressions for entropy and mass in terms of black hole horizon, consistent with those from other approaches. We also define a generalized Misner-Sharp energy for static, spherically symmetric spacetimes in Ho\v{r}ava-Lifshtiz theory. The generalized Misner-Sharp energy is conserved in the case without matter field, and its variation gives the first law of black hole thermodynamics at black hole horizon.