AdS and Lifshitz Black Holes in Conformal and Einstein-Weyl Gravities

TL;DR

This work analyzes four-dimensional gravity theories augmented by higher-curvature terms to uncover AdS and Lifshitz vacua and their black-hole realizations. In conformal gravity, AdS black holes acquire an additional hair parameter that mandates an extended first law with new conjugate pairs, while Lifshitz black holes (notably at $z=4$ and $z=0$) are linked to parameter boundaries of AdS solutions; a conformal boundary term further affects thermodynamics. For Einstein-Weyl gravity, exact solutions are not generally available, but a detailed horizon/asymptotic analysis together with numerical integration reveals AdS and Lifshitz black holes beyond Schwarzschild–AdS, including extremal configurations with AdS$_2$ near-horizon regions. Overall, the work highlights a rich landscape of AdS/Lifshitz black holes in extended gravity and their potential holographic implications, including relativistic-to-Lifshitz phase transitions in dual field theories.

Abstract

We study black hole solutions in extended gravities with higher-order curvature terms, including conformal and Einstein-Weyl gravities. In addition to the usual AdS vacuum, the theories admit Lifshitz and Schrödinger vacua. The AdS black hole in conformal gravity contains an additional parameter over and above the mass, which may be interpreted as a massive spin-2 hair. By considering the first law of thermodynamics, we find that it is necessary to introduce an associated additional intensive/extensive pair of thermodynamic quantities. We also obtain new Liftshitz black holes in conformal gravity and study their thermodynamics. We use a numerical approach to demonstrate that AdS black holes beyond the Schwarzschild-AdS solution exist in Einstein-Weyl gravity. We also demonstrate the existence of asymptotically Lifshitz black holes in Einstein-Weyl gravity. The Lifshitz black holes arise at the boundary of the parameter ranges for the AdS black holes. Outside the range, the solutions develop naked singularities. The asymptotically AdS and Lifshitz black holes provide an interesting phase transition, in the corresponding boundary field theory, from a relativistic Lorentzian system to a non-relativistic Lifshitz system.

Figures (4)

• Figure 1: The metric functions $a(r)$ and $f(r)$ for the asymptotically Lifshitz black hole.
• Figure 2: The asymptotic forms for $a(r)/r^4$ and $f(r)/r^2$, illustrating the $z=2$ Lifshitz behaviour.
• Figure 3: The metric functions $a(r)$ and $f(r)$ for the asymptotically AdS black hole.
• Figure 4: The asymptotic forms for $a(r)/r^2$ and $f(r)/r^2$, illustrating the AdS behaviour.