Stress-energy tensors for vector fields outside a static black hole.

Abstract

We obtain new, approximate stress-energy tensors to describe gauge fields in the neighborhood of a Schwarzschild black hole. We assume that the coefficient of ${\ensuremath{\nabla}}^{2}R$ in the trace anomaly is correctly given by $\ensuremath{\zeta}$-function regularization. Our approximation differs from that of Page and of Brown and Ottewill and relies upon a new, improved ansatz for the form of the stress-energy tensor in the ultrastatic optical metric of the black hole. The Israel-Hartle-Hawking thermal tensor is constructed to be regular on the horizon and possess the correct asymptotic behavior. Our approximation of Unruh's tensor is likewise constructed to be regular on the future horizon and exhibit a luminosity which agrees with Page's numerically obtained value. Geometric expressions for the approximate tensors are given, and the approximate energy density of the thermal tensor on the horizon is compared with recent numerical estimates.