Smarr formula for black holes with primary and secondary scalar hair

TL;DR

This paper investigates the thermodynamics of black holes with scalar hair in beyond Horndeski gravity, deriving consistent Smarr formulas and revealing a link between primary and secondary scalar hair solutions.

Contribution

It introduces a new thermodynamic approach to derive the Smarr formula for scalar-hairy black holes, resolving inconsistencies in traditional methods.

Findings

Derived consistent Smarr formulas for scalar-hairy black holes.

Established a connection between primary and secondary scalar hair solutions.

Demonstrated the failure of traditional thermodynamics and provided a new resolution.

Abstract

In this work, we revisit the thermodynamics of black holes endowed with primary and secondary scalar hair in the shift and ${\rm Z}_2$ symmetric subclass of beyond Horndeski gravity. Under a specific fine-tuning of the scalar parameter $q$ in terms of the black hole mass, the singular black-hole solution with primary scalar hair reduces to the regular Bardeen solution featuring secondary scalar hair. We first demonstrate that the traditional thermodynamic approach fails to yield a consistent Smarr formula for both solutions under consideration. To address this issue, we adopt the approach introduced in [Phys$.$Rev$.$Lett. 132 (2024) 19, 191401], and we derive both the first law of black hole thermodynamics and the Smarr formula, offering a consistent thermodynamic description for scalar-hairy black holes. As an additional outcome, our analysis reveals a connection between the solutions with primary and secondary scalar hair.