Quantum stress tensor in the three-dimensional black hole.

The quantum stress tensor < Tμν > is calculated in the 2+1 dimensional black hole found by Banados, Teitelboim, and Zanelli. The Greens function, from which < Tμν > is derived, is obtained by the method of images. For the non-rotating black hole, it is shown that < Tμν > is finite on the event horizon, but diverges at the singularity. For the rotating solution, the stress tensor is finite at the outer horizon, but diverges near the inner horizon. This suggests that the inner horizon is quantum mechanically unstable against the formation of a singularity. Recently, Banados, Teitelboim, and Zanelli [1] found a black hole solution in 2 + 1 dimensions which shares many of the features of its 3 + 1 dimensional counterpart [2]. In particular, the static solution has a singularity and event horizon, while the rotating black hole like Kerr possesses outer and inner horizons and an ergosphere. Asymptotically, however, the 2 + 1 solution is not flat, but approaches anti-deSitter space [3]. 2 + 1 dimensions provides a simpler setting than 3+1 and possibly a more realistic one than 1+1 [4] in which to study the quantum properties of black holes, and specifically, the endpoint of black hole evaporation. Such an investigation should begin with the quantum stress tensor < Tμν > which describes the quantum effects of the black hole on a propagating field in a way that allows one to analyze the back reaction. Provided it can be properly renormalized, < Tμν > is a well defined local quantity in contrast to particle number which is not, in general, a meaningful concept in curved spacetime. Another motivation for studying < Tμν > in the rotating black hole is to investigate the quantum stability of the inner horizon. The maximally extended Reissner-Nordstrom and Kerr solutions include an infinite number of asymptotic regions which in principle could be accessed. However, it has been shown that since the inner horizon is an infinite blueshift surface, classical perturbations will diverge there [5], and the associated back reaction will produce a singularity [6]. Quantum effects for the 1+1 dimensional analog of the Reissner-Nordstrom solution were investigated in [7] where it was shown that < Tμν > diverges near the inner horizon. Attempts to include quantum corrections in 3 + 1 dimensions [8] are somewhat inconclusive suggesting that the classical instability either is enhanced or is dampened resulting in a regular spacetime. In this paper, the exact expression for the quantum stress tensor is found for the rotating 2 + 1 dimensional black hole and is shown to diverge near the inner horizon. An estimation of the back reaction suggests that the inner horizon will be replaced by a curvature singularity. We use units in which h̄ = c = G = 1. The 2+1 dimensional black hole solution found by Banados, Teitelboim, and Zanelli [1] is most easily described as three dimensional anti-deSitter space (ADS3) identified under a discrete subgroup of its isometry group. Recall that ADS3 is the three dimensional hypersurface −T 2 1 +X 1 − T 2 2 +X 2 = −l (1) imbedded in four dimensional flat space with metric ηab ds = −dT 2 1 + dX 1 − dT 2 2 + dX 2 (2)