标题： 2+1黑洞的几何学 显示英文标题

标题： Geometry of the 2+1 Black Hole

摘要： 标准的二维加一维爱因斯坦理论中的旋转黑洞几何，具有负宇宙常数且无物质耦合，在详细分析了这一几何结构。 证明了黑洞来源于由$SO(2,2)$的一个离散子群对反德西特空间点的识别。 一般的黑洞在度规意义上是一个光滑流形。 表面$r=0$不是一个曲率奇点，而是一个因果结构的奇点。 继续越过它会导致闭合类时线的出现。 然而，简单的例子表明度规在$r=0$处的正则性是不稳定的：物质耦合会在此处引入曲率奇点。 展示了克鲁斯卡尔坐标和彭罗斯图。 特别关注了以下极限情况：(i) 零质量无自旋黑洞，它不同于反德西特空间并扮演真空的角色；(ii) 自旋最大角动量的旋转黑洞。 附录中给出了$SO(2,2)$李代数元素的彻底分类。 显示英文摘要

摘要： The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti-de Sitter space by a discrete subgroup of $SO(2,2)$. The generic black hole is a smooth manifold in the metric sense. The surface $r=0$ is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at $r=0$ to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the 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) the spinning hole of maximal angular momentum . A thorough classification of the elements of the Lie algebra of $SO(2,2)$ is given in an Appendix.