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The black hole considered here is a solution of the standard Einstein equations with a negative cosmological constant and without matter. Its properties are in very close correspondence with its counterpart in four spacetime dimensions. Models for black holes in two spacetime dimensions can also be constructed [for an early attempt, see, ] and have been the object of considerable interest recently in connection with string theory see, 44, 314, E. Witten, ibid
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(1986)
Phys. Rev. D
, vol.33
, pp. 319
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Brown, J.D.1
Henneaux, M.2
Teitelboim, C.3
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33751180214
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ibid, The action is then not the one of Einstein's theory and includes, in the string case, a dilaton field as an essential ingredient. A related string-inspired model in three spacetime dimensions (the ``black string'') has also been considered , B368, 444, J. H. Horne, G. T. Horowitz
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(1992)
Nucl. Phys.
, vol.45
, pp. 1005
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Callan, C.-G.1
Giddings, S.B.2
Harvey, J.A.3
Strominger, A.4
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For signature ( + + + ) there is a theorem stating that any geodesically complete space of constant negative curvature is a quotient of ``Euclidean anti–de Sitter'' (i.e., the three-dimensional Lobachevsky plane) by a discrete subgroup of O(3,1). [W. P. Thurston, ``Geometry and Topology on Three-Manifolds'' (unpublished)]. S. Carlip informs us that an equally clear-cut theorem does not seem to be available for signature ( - + + ) [see, in this context, G. Mess, ``Lorentz Spacetimes of Constant Curvature,'' IHES/M/90/28 report (unpublished)]. However, as persuasively argued by E. Witten (private communication), was natural to expect the black-hole geometry to also be a quotient and this, indeed, turned out to be the case.
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The condition for the quotient space to be smooth is that the action of the isometry group generated by xi be ``properly discontinuous,'' see S. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, Cambridge, England, 1973). As discussed in Appendix B, in the black-hole case this condition is met when the angular momentum is different from zero. However, when J = 0, the group action is not properly discontinuous and the singularity is of the type present in Taub-NUT space.
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One may regard 2+1 gravity as a Chern-Simms theory for SO(2,2) [, ]. The holonomies of that approach are then exp( 2 π n ξ ) with the Killing vector xi used in the identification regarded as an element of the Lie algebra of SO(2,2). This correspondence appears to hold under rather general conditions on the manifold [see, e.g., the work by Mess 4] and also S. Carlip, Class. Quantum Grav., 8, 5, For the black-hole geometry, the Chern-Simons holonomies have been evaluated directly by D. Cangemi, M. Leblanc, and R. B. Mann, ``Gauge Formulation of the Spinning Black Hole in (2+1)-Dimensional Anti–de Sitter Space,'' MIT Report CTP#2162, (1992) (unpublished). Their results agree with our expressions (3.19) (applicable when | J | < M l ). However, since ξ cdot ξ =0 is a smooth surface (recall Sec. IIIB 2) there is no source there for the curvature tensor (or anywhere else, as a matter of fact). One gets a nontrivial holonomy—which can be thought of as a [Truncated]
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(1988)
Nucl. Phys.
, vol.311 B
, pp. 46
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Witten, E.1
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20
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84927276363
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C. W. Misner, in Relativity Theory and Astrophysics I: Relativity and Cosmology, edited by J. Ehlers, Lectures in Applied Mathematics, Vol. 8 (American Mathematical Society, Providence, RI, 1979), p. 160.
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