Gauge formulation of the spinning black hole in (2+1)-dimensional anti-de Sitter space.

We compute the group element of SO(2,2) associated with the spinning black hole found by Bañados, Teitelboim and Zanelli in (2+1)-dimensional anti-de Sitter space-time. We show that their metric is built with SO(2,2) gauge invariant quantities and satisfies Einstein’s equations with negative cosmological constant everywhere except at r = 0. Moreover, although the metric is singular on the horizons, the group element is continuous and possesses a kink there. Submitted to: Physical Review Letters WATPHYS-TH92/09 CTP#2162 November 1992 * This work is supported in part by funds provided by the U. S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069, by the Natural Sciences and Engineering Research Council of Canada (M.L. and R.B.M.) and by the Swiss National Science Foundation (D.C.). 0 The desire to have an interesting and mathematically tractable setting for studying quantum gravity can in part be fulfilled by investigating the properties of lower dimensional black holes [1]. Recently interest in this subject has increased because of the discovery by Bañados, Teitelboim and Zanelli [2] (BTZ) of a spinning black hole solution in (2+1) dimensions. The aim of this note is to associate an element of the anti-de Sitter group with this solution and to discuss its physical characteristics. We use a Chern-Simons formulation of gravity [3] with the gauge field A = ePa + ωJa (1) decomposed in a basis of the Lie algebra so(2,2) [Ja, Jb] = ǫab Jc , [Ja, Pb] = ǫab Pc , [Pa, Pb] = 1 l2 ǫab Jc (2) [a, b, c = 0, 1, 2; ǫ = 1 and indices are lowered with the metric ηab = diag(−1, 1, 1)]. In the absence of matter, the equations of motion F = dA+A∧A = 0 (3) imply the Einstein’s equations with negative cosmological constant Λ = −l−2 provided we identify eμ with the Dreibein (gμν = e a μe b νηab) and ω a μ with the spin-connection [de a + ǫbc ω b∧ e = 0 is obtained from Eq. (3)]. When Eq.(3) holds, the gauge field is pure gauge A = U−1dU (4) given by an element U of the SO(2,2) group. Here we shall obtain the group element U associated with the spinning black hole metric of Ref. [2] dsBTZ = −Ndt +N−2dr2 + r(Ndt+ dφ) (5) 1 −∞ < t < ∞ , 0 < r < ∞ , 0 ≤ φ ≤ 2π with lapse and angular shift functions N(r) = −M + r 2 l2 + J 4r2 = 1 l2 1 r2 ( r − r + ) ( r − (J r+) ) N(r) = − J 2r2 = −J l (r+ l 2 1 r2 . (6) where J = Jl 2r + and r + = Ml 2 {