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 $z\geq 1$. 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.