Vacuum polarization on topological black holes

Abstract

We investigate quantum effects on topological black hole space-times within the framework of quantum field theory on curved space-times. Considering a quantum scalar field, we extend a recent mode-sum regularization prescription for the computation of the renormalized vacuum polarization to asymptotically anti-de Sitter black holes with nonspherical event horizon topology. In particular, we calculate the vacuum polarization for a massless, conformally-coupled scalar field on a fourdimensional topological Schwarzschild-anti-de Sitter black hole background, comparing our results with those for a spherically-symmetric black hole.(where G µν is the Einstein tensor, Λ the cosmological constant and g µν the metric tensor) the RSET governs the back-reaction of the quantum field on the space-time Vacuum polarization on topological black holes 2 geometry. The stress-energy tensor operatorT µν involves products of field operators evaluated at the same space-time point and is therefore divergent. In the point-splitting approach pioneered by DeWitt and Christensen [1][2][3], this divergence is regularized by considering the operators acting on two closely-separated space-time points. The RSET is computed by subtracting off the divergences, which are purely geometrical in nature and independent of the quantum state under consideration. The parametrix encoding these geometric divergent terms was originally constructed using a DeWitt-Schwinger expansion [1][2][3]. This prescription was made more precise by Wald [4,5] who gave a set of axioms that (almost) uniquely determine the RSET. Wald further showed that encoding the divergences using Hadamard elementary solutions, of which the DeWitt-Schwinger expansion can be thought of as a special case, produced an RSET satisfying these axioms. The Hadamard prescription provides a more elegant and general prescription than the DeWitt-Schwinger approach in that it can be applied for arbitrary field mass and arbitrary dimensions (see, for example, [6]).In practice, the computation of the RSET on space-times other than those with maximal symmetry (see, for example, [7,8]) is a challenging task. Of particular interest are black hole space-times, and there is a long history of RSET computations on asymptotically flat Schwarzschild black hole backgrounds, for both quantum scalar fields [9-13] and fields of higher spin [14][15][16][17]. In four space-time dimensions, Anderson, Hiscock and Samuel (AHS) [18,19] have developed a general methodology for finding the RSET on a static, spherically-symmetric black hole. Their method makes heavy use of WKB approximations and the RSET is given as a sum of two parts, the first of which is analytic and the second of which requires numerical computation. More recently, Levi, Ori and collaborators [20,21] have developed a new method (dubbed "pragmatic mode-sum regularization" [22]) for finding the RSET which does not rely on WKB approximations and has the advantage that it can be applied to stationary as well as static black holes [21].Given the challenges ...

Quantum field theory on global anti-de Sitter space-time with Robin boundary conditions

“…Vacuum polarization on spherically symmetric, asymptotically-AdS black holes has been computed for both massless, conformally coupled scalar fields [67] and more general scalar fields [62] satisfying Dirichlet boundary conditions. Static, vacuum, asymptotically-AdS black holes are not necessarily spherically symmetric, and the vacuum polarization has also been studied for asymptotically Lifshitz black holes [68] and topological black holes [63], again for Dirichlet boundary conditions. The effect of boundary conditions on the renormalized vacuum polarization on asymptotically-AdS black holes would make for interesting future work.…”

Vacuum polarization on topological black holes with Robin boundary conditions

“…We now turn to the VP for a massless, conformally coupled quantum scalar field Φ on topological black hole spacetimes. We follow the methodology of [68], which is reviewed briefly here. Further details can be found in [68].…”

“…In [68], we studied the VP for a massless, conformallycoupled scalar field on topological black hole backgrounds. Employing a Euclidean approach, we generalized the "extended coordinates" method of [31,33,34] to black holes with flat or hyperbolic horizons and hence considered a field in the Hartle-Hawking state [10].…”

“…For this reason, to have a well-defined quantum field theory, it is necessary to apply boundary conditions to the field [69][70][71][72][73][74][75][76]. In our previous work [68], we considered the simplest boundary conditions for a quantum scalar field, namely Dirichlet boundary conditions, for which the scalar field vanishes on the boundary. However, Dirichlet boundary conditions are not the only possibility.…”

“…The methodology for computing the renormalized VP is outlined in Sec. IV, following [68]. Our numerical results for the VP are presented in Sec.…”