A Lifshitz Black Hole in Four Dimensional R^2 Gravity

TL;DR

This paper constructs and analyzes a four-dimensional higher derivative gravity model with Lifshitz and Schr"odinger solutions, discovering an analytic Lifshitz black hole with unique thermodynamic properties, including zero entropy at finite temperature.

Contribution

It presents an analytic Lifshitz black hole solution in four-dimensional R^2 gravity with detailed thermodynamic analysis, expanding understanding of higher derivative gravity solutions.

Findings

Black hole with Lifshitz asymptotics and zero entropy

Existence of vacuum Lifshitz and Schr"odinger solutions with arbitrary z

Lifshitz black hole has non-zero temperature but zero entropy

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 Schr\"{o}dinger 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.