A Lifshitz Black Hole in Four Dimensional R^2 Gravity

TL;DR

The paper investigates a four-dimensional higher-derivative gravity theory with a negative cosmological constant and shows that both Lifshitz and Schrödinger vacua exist for arbitrary dynamical exponent z. It constructs an analytic Lifshitz black hole at z = 3/2 by tuning couplings in the action and analyzes its thermodynamics, finding a nonzero temperature but zero entropy, mirroring a BTZ-like behavior in 3D massive gravity at a special coupling. The results contribute to non-relativistic holography by providing a concrete higher-dimensional example and highlight subtle boundary-condition effects that may reveal a nontrivial dual theory. The work suggests further exploration of additional Lifshitz solutions and boundary conditions to understand the dual field theory in the quantum critical regime.

Abstract

We consider a higher derivative gravity theory in four dimensions with a negative cosmological constant and show that vacuum solutions of both Lifshitz type and Schrödinger type with arbitrary dynamical exponent z exist in this system. Then we find an analytic black hole solution which asymptotes to the vacuum Lifshitz solution with z=3/2 at a specific value of the coupling constant. We analyze the thermodynamic behavior of this black hole and find that the black hole has zero entropy while non-zero temperature, which is very similar to the case of BTZ black holes in new massive gravity at a specific coupling. In addition, we find that the three dimensional Lifshitz black hole recently found by E. Ayon-Beato et al. has a negative entropy and mass when the Newton constant is taken to be positive.