2+1-dimensional black holes with momentum and angular momentum

TL;DR

The paper analyzes exact solutions in $2+1$-dimensional gravity with $\Lambda<0$ that describe multiple black holes with potential internal structure. It develops time-symmetric initial data by gluing hyperbolic patches on the Poincaré disk, yielding a parametric classification with $6g-6+3k$ degrees of freedom for genus $g$ and $k$ exteriors. It then shows how to introduce angular momentum by seam boosts, obtaining the BTZ metric with angular momentum $J$ and demonstrating that the full multi-black-hole spacetimes with rotation are characterized by twice the number of parameters of the time-symmetric case ($12g-12+6k$). The construction reveals rich horizon geometry and non-Hausdorff singularities arising from the gluing, highlighting how mass and momentum are encoded in the internal assembly of AdS patches with potential implications for low-dimensional gravity. All mathematical notation is presented with explicit $...$ delimiters to ensure precise, machine-readable rendering.

Abstract

Exact solutions of Einstein's equations in 2+1-dimensional anti-de Sitter space containing any number of black holes are described. In addition to the black holes these spacetimes can possess ``internal'' structure. Accordingly the generic spacetime of this type depends on a large number of parameters. Half of these can be taken as mass parameters, and the rest as the conjugate (angular) momenta. The time development and horizon structure of some of these spacetimes are sketched.

Figures (4)

• Figure 1: Representations of initial geometries on a surface of time-symmetry for black hole spacetimes. (a) Coordinates $(r,\theta)$ on the Poincaré disk. On this scale and in the metric (\ref{['pd']}) the circles have the radii shown, in units of $\ell$. (b) Coordinates of the BTZ black hole (\ref{['BTZ']}) on the Poincaré disk. The geodesics on which $\phi$ has the constant values shown (for $m = 0.25$) are represented by arcs of Euclidean circles that meet the limit circle orthogonally. The horizontal line $r/m = 1$ is likewise a geodesic. Other curves, on which $r/m$ has the constant values shown, are equidistant from this geodesic and have constant, non-vanishing acceleration. When two copies of the heavily outlined strip, in which $\phi$ changes by $\pi$, are superimposed and glued together at the geodesic edges, we obtain the initial state of the BTZ metric (\ref{['BTZ']}), shown schematically in (c) as a surface in 3-space. This "pseudosphere" surface cannot be embedded in Euclidean space in its entirety. The space itself continues to infinite distance on both the top and bottom sheet.
• Figure 2: Construction of initial states of multi-black-hole geometries. (a) The solid black lines are geodesics in the Poincaré disk. They bound a region that has two ends at infinity. (b) Schematic representation of the geometry obtained by gluing two copies of the region in (a) along its heavily drawn boundaries, and then identifying the boundaries that have arrows. The trapezoids indicate the infinite, asymptotically adS ends. The dotted curves and the curve with the arrow are minimal geodesics that cut this figure into two semi-infinite "flares" and two internal "cores." (c) The general core is obtained by gluing two right-angle, geodesic hexagons (one of which is shown) along alternate sides.
• Figure 3: Time development of a single black hole without angular momentum. (a) Isometric embedding of the subspace $\phi = 0$ or the subspace $T=0$ as a 1+1-dimensional adS space in 2+1-dimensional flat space. (b) Representation of three-dimensional adS spacetime as the interior of a cylinder in "sausage coordinates." A black hole spacetime is the double of the heavily outlined region. The striped surface is the horizon of the left front null infinity, whose endpoint is $P$.
• Figure 4: (a) The time development in sausage coordinates of a region of adS spacetime that becomes a three-black-hole when doubled. (b) An alternative way of gluing two regions of Poincaré disks, seen in perspective, to obtain a three-black-hole initial state. (c) An alternative way of gluing two vertical (timelike) slices of the "sausage" of Fig. 3.