Analytic approximation to 5 dimensional Black Holes with one compact dimension

TL;DR

This paper develops an analytic approximation for five-dimensional black holes with one compact dimension, analyzing their geometric and thermodynamic properties and comparing them to black strings.

Contribution

It introduces a fourth-order expansion method to approximate black hole solutions in a compactified space, providing insights into their properties and phase transitions.

Findings

Black holes fill the compact dimension at a specific size ratio

Entropy of black holes and black strings are approximately equal at a critical point

Analytic approximation matches numerical results within a certain parameter range

Abstract

We study black hole solutions in $R^4\times S^1$ space, using an expansion to fourth order in the ratio of the radius of the horizon, $\mu$, and the circumference of the compact dimension, $L$. A study of geometric and thermodynamic properties indicates that the black hole fills the space in the compact dimension at $\epsilon(\mu/L)^2\simeq0.1$. At the same value of $\epsilon$ the entropies of the uniform black string and of the black hole are approximately equal.