The Phase Transition between Caged Black Holes and Black Strings - A Review

Abstract

Black hole uniqueness is known to fail in higher dimensions, and the multiplicity of black hole phases leads to phase transitions physics in General Relativity. The black-hole black-string transition is a prime realization of such a system and its phase diagram has been the subject of considerable study in the last few years. The most surprising results seem to be the appearance of critical dimensions where the qualitative behavior of the system changes, and a novel kind of topology change. Recently, a full phase diagram was determined numerically, confirming earlier predictions for a merger of the black-hole and black string phases and giving very strong evidence that the end-state of the Gregory-Laflamme instability is a black hole (in the dimension range 4<D<14). Here this progress is reviewed, illustrated with figures, put into a wider context, and the still open questions are listed.

Figures (29)

• Figure 1: In asymptotically flat 5d spacetime uniqueness is violated by the co-existence of the rotating black hole and the rotating black ring. The figure shows the range of existence of each phase on the dimensionless angular momentum axis. Note that for $1 \le \pi\, J^2/M^3 \le 32/27$ three phases co-exist.
• Figure 2: Definition of coordinates. For backgrounds with a single compact dimension the essential geometry is 2d after suppressing time and angular coordinates in the extended dimensions. The cylindrical coordinates $(r,z)$ are defined such that $z \sim z+L$ is the coordinate along the compact dimension and $r$ is the radial coordinate in the extended spatial directions. For black holes we define another set of local coordinates $(\rho,\chi)$, defined only for $\rho \le L/2$, which are radial coordinates in the 2d plane with origin at the center of the BH.
• Figure 3: The uniform black string. $r_0$ is its Schwarzschild radius.
• Figure 4: A caged black hole (BH). Newtonian equipotential lines are shown.
• Figure 5: (a) The Gregory-Laflamme instability. (b) A non-uniform black string.