Initial Data for Black Holes and Black Strings in 5d

TL;DR

A phase transition is found within the family of hypersurfaces containing apparent horizons of black objects in a 5D spacetime with one coordinate compactified on a circle: the horizon has different topology for different parameters.

Abstract

We explore time-symmetric hypersurfaces containing apparent horizons of black objects in a 5d spacetime with one coordinate compactified on a circle. We find a phase transition within the family of such hypersurfaces: the horizon has different topology for different parameters. The topology varies from $S^3$ to $S^2 \times S^1$. This phase transition is discontinuous -- the topology of the horizon changes abruptly. We explore the behavior around the critical point and present a possible phase diagram.

Figures (3)

• Figure 1: Contours of $\psi$. Upper panel: $\zeta \simeq 1.85$ and there is a non-uniform BS. Bottom panel: $\zeta \simeq 1.7$ and there is a BH. In both plots one observes how the deformation of the contour lines fades asymptotically. We plot also the apparent horizons in both cases. These are designated by thick curves. In the upper plot that corresponds to the BS phase there are two horizons. The inner spherical apparent horizon, designated by the dotted thick curve, and the outer cylindrical horizon, designated by the solid curve.
• Figure 2: The path in the configuration space of the numerical solutions. The BH (BS) phase is displayed in the bottom (top) panel. In both panels the horizontal dashed line that is located at $\zeta_c\simeq 1.78$ designates the $BS\rightarrow BH$ transition point. Above this line there is BS phase, while below the line there is the BH phase. The BS becomes uniform at $\zeta \simeq 3.0$ and the BH becomes spherical at $\zeta \simeq 0.15.$
• Figure 3: The proper distance from the BH to the reflection plane $z=L$ does not decrease to zero but tends to a finite value as we approach the transition point.