Modified first law of charged dilaton black hole

TL;DR

This paper extends the first law of black hole thermodynamics for charged dilaton black holes by including the dilaton coupling parameter as a thermodynamic variable, providing a geometric interpretation and analyzing its implications.

Contribution

It introduces a new thermodynamic term for the dilaton coupling parameter and offers a geometric interpretation as a conserved charge, enriching extended black hole thermodynamics.

Findings

The first law is extended to include a da term with conjugate Psi^A.

Psi^A is explicitly derived and interpreted as a conserved Noether charge.

The Smarr relation remains unchanged despite the extension.

Abstract

We investigate the thermodynamics of a charged dilaton black hole arising from Einstein-Maxwell-dilaton theory, where the dilaton couples exponentially to the Maxwell field via a dimensionless parameter a. Treating a as a continuous solution parameter, we extend the black hole first law to include a term Psi^A da, where Psi^A is the thermodynamic potential conjugate to a. We derive Psi^A explicitly through a differential analysis of the mass, charge, and entropy, and confirm its form via an independent Hamiltonian calculation. Additionally, by promoting a to a spacetime-dependent scalar and introducing auxiliary gauge fields, we provide a geometric interpretation of Psi^A as a conserved Noether charge. We further analyze the implications of treating a as a thermodynamic variable within the extended phase space. Despite the modification to the first law, we demonstrate that the Smarr relation remains unaffected due to the dimensionless nature of a, highlighting the distinction between variational and scaling symmetries. Our analysis supports the thermodynamic relevance of coupling constants and enriches the framework of extended black hole thermodynamics.