Smarr formula for black holes with primary and secondary scalar hair

Abstract

In this work, we revisit the thermodynamics of black holes endowed with primary and secondary scalar hair in the shift and ${\mathrm{Z}}_2$ 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.

6 pages

References in corpus (6)

• Healthy theories beyond Horndeski
• Corrected form of the first law of thermodynamics for regular black holes
• Black holes with primary scalar hair
• Coupling Constants as Conserved Charges in Black Hole Thermodynamics
• Compact objects with primary hair in shift and parity symmetric beyond Horndeski gravities
• Thermodynamics of black holes featuring primary scalar hair

Cited by in corpus (1)

• Theoretical filters for shift-symmetric Horndeski gravities