Critical Dimension in the Black-String Phase Transition

In spacetimes with compact dimensions, there exist several black object solutions including the black hole and the black string. They may become unstable depending on their relative size and the length scales in 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 nonuniform strings around the marginally stable uniform string. First, we compute the instability mass for a large range of dimensions and find that it follows an exponential law ${\gamma }^d$, where $\gamma <1$ is a constant. Then we determine that there is a critical dimension, $d_{*}=13$, such that for $d\le d_{*}$ the phase transition is of first order, while for $d>d_{*}$ it is, surprisingly, of higher order.