Speculative generalization of black hole uniqueness to higher dimensions

TL;DR

This work analyzes why the standard black hole uniqueness theorem does not straightforwardly carry over to higher dimensions, illustrated by the 5d black-string and black-ring cases. It advocates a speculative generalization that requires specifying horizon topology in addition to global charges and, possibly, applies only to stable solutions. The paper discusses the variety of possible horizon topologies in higher dimensions and argues that cobordism and boundary-data considerations prevent a simple no-hair statement. It further argues that a stability requirement could reconcile multiple coexisting solutions, linking to phase structure and positivity arguments. Overall, it frames a plausible, testable direction for extending black-hole uniqueness to higher dimensions.

Abstract

A straightforward generalization of the celebrated uniqueness theorem to dimensions greater than four was recently found to fail in two pure gravity cases - the 5d rotating black ring and the black string on R^{3,1} * S^1. Two amendments are suggested here (without proof) in order to rectify the situation. The first is that in addition to specifying the mass and angular momentum (and gauge charges) one needs to specify the horizon topology as well. Secondly, the theorem may survive if applied exclusively to stable solutions. Note that the latter is at odds with the proposed stable but non-uniform string.

Figures (2)

• Figure 1: A suggested phase diagram for the black-hole black-string transition [12]. $\mu$ is the dimensionless parameter $G_5\, M/L^2$, and $\lambda$ is an order parameter (measure of non-uniformity). In the range $\mu_{GL} \le \mu \le \mu_{\hbox{merger}}$ (at least) three solutions coexist. Solid lines are stable phases while dashed ones are unstable. The solutions to the left of the merger point are black strings while those to the right are black holes. Dotted lines denote transitions - a first order transition at $\mu_1$ and tachyonic decay from the two other points. The full phase diagram should include further refinements in the area of the merger transition.
• Figure 2: Rotating solutions in 5d. $(27\, \pi/32\, G)\, J^2\, /M^3$ as a function of some parameter $\nu$. The dashed line is the rotating black hole of [14] while the solid line is the newly discovered black ring [15]. In the range $J_1 \le J \le J_2$ marked by the dotted lines three solutions coexist, denoted I, II and III from right to left. Reproduced with permission from [15].