Corrected form of the first law of thermodynamics for regular black holes

TL;DR

This paper identifies an inconsistency in the traditional first law of black hole thermodynamics for regular black holes and proposes a corrected formulation based on the energy-momentum tensor structure.

Contribution

It introduces a modified first law for regular black holes that accounts for the matter fields' energy-momentum tensor, challenging the conventional interpretation of black hole mass as internal energy.

Findings

The conventional first law is inconsistent with the Bekenstein-Hawking area law for regular black holes.

A corrected first law includes an extra factor when the mass parameter is in the energy-momentum tensor.

The black hole mass M cannot be regarded as the internal energy in the corrected formulation.

Abstract

We show by explicit computations that there is a superficial inconsistency between the conventional first law of black hole thermodynamics and Bekenstein-Hawking area law for three types of regular black holes. The corrected form of the first law for these regular black holes is given. The derivation relies on the general structure of the energy-momentum tensor of the matter fields. When the black hole mass parameter $M$ is included in the energy-momentum tensor, the conventional form of the first law should be modified with an extra factor. In this case, the black hole mass $M$ can no longer be considered as the internal energy of the regular black holes.