Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent

TL;DR

This paper numerically investigates black hole solutions in Lifshitz spacetimes with various critical exponents, revealing different behaviors for z>2 and z<2, and analyzing their thermodynamic stability.

Contribution

It provides the first detailed numerical analysis of black holes in Lifshitz backgrounds across a range of critical exponents, highlighting stability and solution structure differences.

Findings

For z>2, Lifshitz fixed point is repulsive; for z<2, it is attractive.

A continuous family of black holes exists for z>2, satisfying finite energy conditions.

Thermodynamic instability may occur for z approximately less than 1.761.

Abstract

Recently, a class of gravitational backgrounds in 3+1 dimensions have been proposed as holographic duals to a Lifshitz theory describing critical phenomena in 2+1 dimensions with critical exponent $z\geq 1$. We numerically explore black holes in these backgrounds for a range of values of $z$. We find drastically different behavior for $z>2$ and $z<2$. We find that for $z>2$ ($z<2$) the Lifshitz fixed point is repulsive (attractive) when going to larger radial parameter $r$. For the repulsive $z>2$ backgrounds, we find a continuous family of black holes satisfying a finite energy condition. However, for $z<2$ we find that the finite energy condition is more restrictive, and we expect only a discrete set of black hole solutions, unless some unexpected cancellations occur. For all black holes, we plot temperature $T$ as a function of horizon radius $r_0$. For $z\lessapprox 1.761$ we find that this curve develops a negative slope for certain values of $r_0$ possibly indicating a thermodynamic instability.