GEOMETRY OF THE 2+1 BLACK-HOLE

The geometry of the spinning black holes of standard Einstein theory i n 2 + 1 dimensions, with a negative cosmological constant, and without couplings to matter, is analyzed in detail. It is shown that the blac k hole arises from identifications of points of anti-de Sitter space b y a discrete subgroup of SO(2,2). The generic black hole is a smooth m anifold in the metric sense. The surface r = 0 is not a curvature sing ularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple exampl es show the regularity of the metric at r = 0 to be unstable: coupling s to matter bring in a curvature singularity there. Kruskal coordinate s and Penrose diagrams are exhibited. Special attention is given to th e limiting cases of (i) the spinless hole of zero mass, which differs from anti-de Sitter space and plays the role of the vacuum, and (ii) t he spinning hole of maximal angular momentum. A thorough classificatio n of the elements of the Lie algebra of SO(2,2) is given in an appendi x.