Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent

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≥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 r₀. For z⪅1.761 we find that this curve develops a negative slope for certain values of r₀ possibly indicating a thermodynamic instability.