A critical dimension in the black-string phase transition

TL;DR

This paper investigates the phase transition between uniform and non-uniform black strings in higher-dimensional spacetimes, revealing a critical dimension where the transition order changes, with implications for topology and stability.

Contribution

It constructs static perturbative solutions around the instability point and identifies a critical dimension where the phase transition order shifts.

Findings

Instability mass follows an exponential law with dimension.

Critical dimension for transition order is found to be 13.

Transition is first order for dimensions ≤13, higher order for >13.

Abstract

In spacetimes with compact dimensions there exist several black object solutions including the black-hole and the black-string. These solutions may become unstable depending on their relative size and the relevant length scale set by the compact dimensions. The transition between these solutions raises puzzles and addresses fundamental questions such as topology change, uniquenesses and cosmic censorship. Here, we consider black strings wrapped over the compact circle of a $d$-dimensional cylindrical spacetime. We construct static perturbative non-uniform string solutions around the instability point of a uniform string. First we compute the instability mass for a large range of dimensions, $d$, and find that it follows essentially an exponential law $\gamma^d$, where $\gamma$ is a constant. Then we determine that there is a critical dimension, $d_*=13$, such that for $d\leq d_*$ the phase transition between the uniform and the non-uniform strings is of first order, while for $d>d_*$, it is, surprisingly, of higher order.