Holography of AdS Hairy Black Holes and Cardy-Verlinde Formula

TL;DR

This paper explores the holographic properties of AdS dyonic hairy black holes, examining how boundary conformal symmetry affects the applicability of the Cardy-Verlinde formula and providing explicit calculations for specific cases.

Contribution

It demonstrates that the Cardy-Verlinde formula does not hold for hairy black holes with broken conformal symmetry, but is restored when symmetry is preserved, using counterterms and superpotential methods.

Findings

Cardy-Verlinde formula fails for hairy black holes with broken conformal symmetry.

Formula is valid when magnetic or electric charge vanishes, restoring symmetry.

Results align with superpotential counterterm calculations in symmetric cases.

Abstract

We discuss some aspects related to holography of Anti-de Sitter (AdS) dyonic hairy black holes, which break the conformal symetry of the boundary. We use counterterms for the scalar field that satisfies mixed boundary conditions to compute the Euclidean action and dual stress tensor. We apply these results to show that the Cardy-Verlinde formula is not satisfied. However, when the magnetic (or electric) charge vanishes, the conformal symmetry is preserved and the entropy of the black hole can be put in the Cardy-Verlinde form. In our analysis, there is no need of adding extra finite counterterms and, in this particular case, we explicitly show that our results match the ones obtained from using the superpotential as a counterterm.