Caged Black Holes: Black Holes in Compactified Spacetimes II - 5d Numerical Implementation

TL;DR

This paper introduces a novel numerical method to find static black hole solutions in five-dimensional compactified spacetimes, confirming their existence and exploring their thermodynamics and stability near critical mass.

Contribution

It presents the first convergent numerical approach for 5d black holes in compactified spaces, validating theoretical predictions and analyzing stability near the Gregory-Laflamme threshold.

Findings

Solutions approach 5d Schwarzschild for small black holes

Existence of black holes in compactified spacetime confirmed

Indications of a phase transition near critical mass

Abstract

We describe the first convergent numerical method to determine static black hole solutions (with S^3 horizon) in 5d compactified spacetime. We obtain a family of solutions parametrized by the ratio of the black hole size and the size of the compact extra dimension. The solutions satisfy the demanding integrated first law. For small black holes our solutions approach the 5d Schwarzschild solution and agree very well with new theoretical predictions for the small corrections to thermodynamics and geometry. The existence of such black holes is thus established. We report on thermodynamical (temperature, entropy, mass and tension along the compact dimension) and geometrical measurements. Most interestingly, for large masses (close to the Gregory-Laflamme critical mass) the scheme destabilizes. We interpret this as evidence for an approach to a physical tachyonic instability. Using extrapolation we speculate that the system undergoes a first order phase transition.