Anti-de Sitter Black Holes, Thermal Phase Transition and Holography in Higher Curvature Gravity

TL;DR

This work analyzes anti-de Sitter black holes in Einstein–Gauss–Bonnet and generic R^2 gravity, focusing on thermodynamics, phase transitions, and holographic connections. It derives explicit expressions for mass, temperature, entropy, and free energy, showing that Gauss-Bonnet terms can enable Hawking-Page transitions for hyperbolic horizons (k = −1) and that R^2 corrections yield finite-coupling effects with consistent Wald entropy in appropriate limits. The study also establishes bulk–boundary relations through brane cosmology, derives FRW-type equations, and discusses entropy bounds in holography, highlighting how higher-curvature corrections alter the naive AdS/CFT expectations. Overall, the results illuminate how higher-curvature terms influence black-hole thermodynamics and holographic interpretations beyond the leading large-N, strong-coupling regime.

Abstract

We study anti-de Sitter black holes in the Einstein-Gauss-Bonnet and the generic R^2 gravity theories, evaluate different thermodynamic quantities, and also examine the possibilities of Hawking-Page type thermal phase transitions in these theories. In the Einstein theory, with a possible cosmological term, one observes a Hawking-Page phase transition only if the event horizon is a hypersurface of positive constant curvature (k=1). But, with the Gauss-Bonnet or/and the (Riemann)^2 interaction terms, there may occur a similar phase transition for a horizon of negative constant curvature (k=-1). We examine the finite coupling effects, and find that N>5 could trigger a Hawking-Page phase transition in the latter theory. For the Gauss-Bonnet black holes, one relates the entropy of the black hole to a variation of the geometric property of the horizon based on first law and Noether charge. With (Riemann)^2 term, however, we can do this only approximately, and the two results agree when, r_H>>L, the size of the horizon is much bigger than the AdS curvature scale. We establish some relations between bulk data associated with the AdS black hole and boundary data defined on the horizon of the AdS geometry. Following a heuristic approach, we estimate the difference between Hubble entropy {\cal S}_H and Bekenstein-Hawking entropy {\cal S}_{BH} with (Riemann)^2 term, which, for k=0 and k=-1, would imply {\cal S}_{BH}\leq {\cal S}_H.

Figures (13)

• Figure 1: Einstein gravity $(\hat{\alpha}=0$): The inverse temperature ($\beta_0$) versus the horizon radius ($r_H$). The three curves above from up to down correspond respectively to the cases $k=-1$, $k=0$ and $k=+1$. The vertical line passing through $x\approx 0.18$ asymptotes to the $k=-1$ curve.
• Figure 2: Gauss-Bonnet black hole: inverse temperature ($\beta_0$) versus horizon radius ($r_H$) for $k=1$ (from up to down: $d+1=4,~5,~6$, and $10$). We have fixed $\hat{\alpha}/\ell^2=(d-2)(d-3)\alpha\kappa_{d+1}/\ell^2$ at $\alpha\kappa_{d+1}/\ell^2=(0.2)/81$. For $d+1=5$, a new phase of locally stable small black hole is seen, while for $d\neq 4$ the thermodynamic behavior is qualitatively similar to that of $\hat{\alpha}=0$ case.
• Figure 3: Gauss-Bonnet black hole: inverse temperature vs horizon radius for $k=0$, $d=4$, and $\hat{\alpha}/\ell^2=(0.7)/(0.9)^2$. Only for $\hat{\alpha}=\ell^2$, $T_H=T_c$, and hence $F=0$, otherwise free energy is always negative, since $\hat{\alpha}<\ell^2$ should hold for $\Lambda<0$.
• Figure 4: Gauss-Bonnet black hole: free energy vs horizon radius for $k=0$ and $d=4$. The curve with $F>0$ corresponds to $\hat{\alpha}/\ell^2=(0.7)/(0.836)^2$, which gives a dS solution, since $\ell^2>\hat{\alpha}$ and hence $\Lambda>0$. The other three curves with $F<0$ from up to down correspond respectively to $\hat{\alpha}/\ell^2=(0.7)/(0.84)^2,\,(0.7)/(0.85)^2$ and $0.7/(0.9)^2$.
• Figure 5: Gauss-Bonnet black hole: inverse temperature vs horizon radius for the case $k=1$ in $d=4$ and $\hat{\alpha}/\ell^2=(0.3)/64$. The upper curve corresponds to $T_c^{-1}$ and the lower one to $T_H^{-1}$. The region where $T_H$ exceeds $T_c$ is shown.