On Thermodynamics of AdS Black Holes with Scalar Hair

TL;DR

This paper investigates the thermodynamics of AdS black holes with scalar hair, revealing a modified first law that includes a new conjugate pair related to the scalar field and its dual operator.

Contribution

It introduces a holographic renormalization approach to derive the modified first law for hairy AdS black holes, identifying a new thermodynamic pair involving scalar source and response.

Findings

The first law of black hole thermodynamics is modified by an additional term involving scalar field variables.

Explicit hairy black hole solutions confirm the validity of the modified first law.

The new conjugate variables are linked to the asymptotic behavior of the scalar field and its dual operator.

Abstract

It has been known that in the presence of a scalar hair there would be a distinct additional contribution to the first law of black hole thermodynamics. While it has been checked in many examples, a deeper understanding of this issue is necessary. The thermodynamics of AdS black holes in Einstein-scalar gravity is studied by using the standard holographic renormalization procedure and the variation of the Hamiltonian via the Wald formalism. It is found that the first law requires a modification by including an additional term that has a particular form $\sim\left<O\right>\delta \phi_s$, with $\phi_s$ and $\left<O\right>$ a new pair of thermodynamic conjugate variables. $\phi_s$ is the leading source term of the asymptotic fall-off of the scalar field near the AdS boundary, and $\left<O\right>$ is precisely the response of the dual scalar operator from the holographic point of view. Some hairy black holes are constructed explicitly to check the first law of thermodynamics as well as the thermodynamic relations.