Formation of Topological Black holes from Gravitational Collapse

TL;DR

The paper demonstrates that gravitational collapse of pressureless dust in an asymptotically anti-de Sitter spacetime with negative spatial curvature can produce black holes whose event horizons have arbitrary topology, including higher-genus surfaces. By solving the interior dust dynamics and matching to a topological AdS exterior, it shows the quasilocal mass is set by the initial density and can be made arbitrarily small, with collapse completing in finite proper time. A special case of a massless pseudosphere is discussed, revealing a coordinate-equivalent AdS spacetime and a non-singular (Misner-type) center. These results extend topological BTZ-like constructions to 3+1 dimensions and highlight the role of topology and curvature in gravitational collapse.

Abstract

We consider the gravitational collapse of a dust cloud in an asymptotically anti de Sitter spacetime in which points connected by a discrete subgroup of an isometry subgroup of anti de Sitter spacetime are identified. We find that black holes with event horizons of any topology can form from the collapse of such a cloud. The quasilocal mass parameter of such black holes is proportional to the initial density, which can be arbitrarily small.

Figures (4)

• Figure 1: The pseudosphere is one half of the hyperboloid. The axes do not represent any coordinates in particular. Beneath the pseudosphere is the Poincaré disk, the center of which is the origin. [Balazs and Voros, p121]
• Figure 2: The identification of the octagon is shown. Opposite sides of the octagon in 3A will be identified. The sides are drawn straight for clarity. Dashed lines indcate where sides have been sewn together. First, sides $1$ and $1^\prime$ are identified, folding the top and bottom of the octagon away from view (3B). Sides $2$ and $2^\prime$ are brought together to form a torus with a diamond shaped hole, as in 3C. Next, sides $3$ and $3^\prime$ are stretched out and joined for 3D. The loop is lengthened along the direction of identification 3 and bent until $4$ and $4^\prime$ meet, forming a second torus. Finally the topology is deformed to the preferred shape, seen in 3F. Identification of a polygon of genus $g$ will clearly result in $g$ attached tori or, equivalently, a $g$-holed pacifier.
• Figure 3: The collapse time times ${\sqrt{\Lambda \over 3}}$ is shown as a function of $B$ and $v_0$. Increasing either $B$ or $v_0$ speeds collapse.
• Figure 4: A sample collapse, showing $x_h$ shrinking as time increases. In this case, $v_0=1$ and $B=.5$. This smooth collapse, with no bouncing, guarentees that $\tilde{t}_h < \tilde{t}_c$.