Critical Dimension in the Black-String Phase Transition

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 γ d , where γ is a constant. Then we determine that there is a critical dimension, d * = 13, such that for d ≤ 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.

Thermodynamics of the near-extremal NS5-brane

“…Note that the non-uniform phase is mapped from the neutral non-uniform black string phase in six dimensions. The solutions for the neutral non-uniform black string were obtained numerically in [33] and the behavior near the Gregory-Laflamme mass was found in [34].…”

“…(4.6) This was computed using results of [34]. The entropy function for the entire nonuniform phase, as displayed in Fig.…”

Black Holes in Higher-Dimensional Gravity

“…Since in many cases the Einstein equations become too complicated to be amenable to analytical methods, even after using symmetries and ansätze, the only way to proceed in the non-linear regime is to try to solve them numerically. Especially for KK black holes these techniques have been successfully applied for non-uniform black strings [37,38,39,40,41,42] and localized black holes [46,47,48] (see Sec. 3).…”

“…The non-uniform black string phase emerges from the uniform black string phase at the Gregory-Laflamme point, which is determined by the (time-independent) threshold mode where the instability sets in. An interesting property that has been found in this context is the existence of a critical dimension [39] where the transition of the uniform black string into the non-uniform black string changes from first order into second order. Moreover, it has been shown [55,56,57,41] that the localized black hole phase meets the non-uniform black string phase in a horizon-topology changing merger point.…”

“…But there are many more phases of KK black holes, which in recent years have been uncovered by a combination of perturbative techniques (matched asymptotic expansion), numerical methods and exact solutions. These phases include non-uniform black strings (see [37,38,39,40,41,42] for numerical results), localized black holes (see [43,31,32,34,33,44,35,45] for analytical results and [46,47,48] for numerical solutions) and bubble-black hole sequences [49]. Here recent progress [35] includes the construction of small mass multi-black hole configurations localized on the circle which in some sense parallel the multi-black hole configurations obtained for rotating black holes.…”