Black Holes and Wormholes in 2+1 Dimensions

TL;DR

The paper investigates black holes and wormholes in 2+1 dimensional gravity with negative cosmological constant by modeling spacetimes as quotients of anti-de Sitter space. It develops a geometric, combinatorial approach using identifications on the AdS boundary, represented in the Poincaré disk, to construct BTZ black holes, multi-black-hole configurations, and various topologies, including surfaces with multiple asymptotic regions. It also extends the discussion to angular momentum, demonstrating rotating BTZ spacetimes and the possibility of rotating multi-hole configurations. The work highlights the horizon structure and the role of conformal infinity as a tool for understanding causal structure in locally AdS spacetimes, with implications for quantum gravity models.

Abstract

Vacuum Einstein theory in three spacetime dimensions is locally trivial, but admits many solutions that are globally different, particularly if there is a negative cosmological constant. The classical theory of such locally "anti-de Sitter" spaces is treated in an elementary way, using visualizable models. Among the objects discussed are black holes, spaces with multiple black holes, their horizon structure, closed universes, and the topologies that are possible.

Figures (17)

• Figure 1: Conformal diagrams of the static (or sausage) coordinates of Eq (\ref{['statmet']}) in sections of AdS space. ( a) The $\chi,\,t$ section, both sides of the origin. The right half is, for example, $\theta = 0$, and the left half, $\theta=\pi$. ( b) The section $t=$ const is the 2D space of constant negative curvature, conformally represented as a Poincaré disk (see section 2.2). The conformal factors are different in the two sections, so they do not represent sections of one three-dimensional conformal diagram. (For the latter see Fig. 4b)
• Figure 2: Conformal diagrams of the "Schwarzschild" coordinates of Eq (\ref{['schwc']}) in sections of AdS space. ( a) An $r,\,t$ section, continued across the $r=\ell$ coordinate singularity. The outer vertical lines correspond to $r=\infty$. The dotted curves show a few of the surfaces $\tau=$ const for the coordinates of Eq (\ref{['cmc']}), with limits at $\tau = \pm\pi\ell/2$. ( b) An $r,\,\phi$ section ($r>\ell$) is a two dimensional space of constant negative curvature, conformally represented as a Poincaré disk (see below). The approximately vertical curves are lines of constant $r$; they are equidistant in the hyperbolic metric. The approximately horizontal curves are lines of constant $\phi$; they are geodesics in the hyperbolic metric. The outer circle corresponds to $r=\infty$
• Figure 3: Conformal diagram of the "extremal" Schwarzschild coordinates of Eq (\ref{['horo']}) in sections of AdS space. ( a) An $r,\,t$ section. ( b) An $r,\,\phi$ section. The lines $r=$ const are horocycles of the Poincaré disk.
• Figure 4: AdS space in stereographic projection. ( a) The hyperboloid is 2-dimensional AdS space embedded in 3-dimensional flat space as in Eq (\ref{['ads']}), restricted to $Y=0$. It is projected from point P onto the plane 1 ($U=\ell$). The image of point A in the hyperboloid is point B in the plane. The part of the hyperboloid that lies below plane 2 is not covered by the stereographic coordinates. ( b) When plotted in the stereographic coordinates (\ref{['stc']}), AdS space is the interior of a hyperboloid. The boundary of the hyperboloid is (part of) the conformal boundary of AdS space.
• Figure 5: The two-dimensional space $H^2$ of constant curvature $1/\ell^2$ is embedded in flat Minkowski space as one sheet of the hyperboloid of Eq (\ref{['emb']}). Under a stereographic projection from point P to the plane, point A on the hyperboloid is mapped to point B in the plane. Thus the hyperboloid ($H^2$) is mapped onto the Poincaré disk, the interior of the curve marked "limit circle"