First Law of Black Hole in the Gravitational Electromagnetic System

TL;DR

This paper derives the first law of black hole thermodynamics in a quantum-corrected Einstein-Maxwell theory, showing its potential validity even with higher-curvature and nonminimally coupled electromagnetic fields.

Contribution

It provides a generic derivation of the first law for black holes in a quantum-corrected gravitational electromagnetic system without requiring smooth electromagnetic fields on the bifurcation surface.

Findings

First law holds for asymptotically flat stationary black holes with quantum corrections.

No need for electromagnetic field smoothness on the bifurcation surface.

Supports the validity of black hole thermodynamics in quantum-corrected theories.

Abstract

After considering the quantum corrections of Einstein-Maxwell theory, the effective theory will contain some higher-curvature terms and nonminimally coupled electromagnetic fields. In this paper, we study the first law of black holes in the gravitational electromagnetic system with the Lagrangian $\math{L}(g_{ab}, R_{abcd}, F_{ab})$. Firstly, we calculate the Noether charge and the variational identity in this theory, and then generically derive the first law of thermodynamics for an asymptotically flat stationary axisymmetrical symmetric black hole without the requirement that the electromagnetic field is smooth on the bifurcation surface. Our results indicate that the first law of black hole thermodynamics might be valid for the Einstein-Maxwell theory with some quantum corrections in the effective region.