Thermodynamic instability of charged dilaton black holes in AdS spaces

TL;DR

This work analyzes the thermodynamic stability of $n+1$-dimensional charged dilaton black holes in AdS by deriving quasilocal mass via Brown–York subtraction with counterterms, and obtaining a Smarr-type relation $M(S,Q)$. It demonstrates that the dilaton coupling $α$ crucially shapes stability: there exists a maximum $α_{\max}$ such that $(\partial^{2}M/\partial S^{2})_{Q}>0$ for $α<α_{\max}$, while larger $α$ yields thermodynamic instability; for small $α$ there is no Hawking–Page transition. The analysis, valid for general $n$, shows stability depends on horizon parameters and dimensionality, with exact AdS behavior for $n\ge5$ and approximate AdS for $n=3,4$. The findings highlight the dilaton field as a key driver of phase structure in AdS black holes and motivate exploration of dynamical stability alongside thermodynamics.

Abstract

We study thermodynamic instability of a class of $(n+1)$-dimensional charged dilaton black holes in the background of anti-de Sitter universe. We calculate the quasilocal mass of the AdS dilaton black hole through the use of the subtraction method of Brown and York. We find a Smarr-type formula and perform a stability analysis in the canonical ensemble and disclose the effect of the dilaton field on the thermal stability of the solutions. Our study shows that the solutions are thermally stable for small $α$, while for large $α$ the system has an unstable phase, where $α$ is a coupling constant between the dilaton and matter field.

Figures (6)

• Figure 1: T versus $\rho_{+}$ for $b=0.2$, $l=1$ and $\alpha=0.1,$$n=4$ (solid line), $n=5$ (bold line), and $n=6$ (dashed line).
• Figure 2: T versus $\rho_{+}$ for $b=0.2$, $l=1$ and $n=5$. $\alpha=0.1$ (solid line), $\alpha=1$ (bold line), and $\alpha=2$ (dashed line).
• Figure 3: $(\partial ^{2}M/\partial S^{2})_{Q}$ versus $\alpha$ for $b=0.2$, $l=1$ and $\rho_{+}=0.4$., $n=5$ (solid line), $n=6$ (bold line), and $n=7$ (dashed line).
• Figure 4: $(\partial ^{2}M/\partial S^{2})_{Q}$ versus $\alpha$ for $b=0.2$, $l=1$ and $n=5$. $\rho_{+}=0.5$ (solid line), $\rho_{+}=0.55$ (bold line), and $\rho_{+}=0.6$ (dashed line).
• Figure 5: $(\partial ^{2}M/\partial S^{2})_{Q}$ versus $\rho_{+}$ for $b=0.2$, $l=1$ and $n=5$. $\alpha=0.5$ (solid line), $\alpha=2$ (bold line), and $\alpha=5$ (dashed line).