A method for computing the stress-energy tensor of quantized scalar fields in static spherically symmetric spacetimes is described. The fields can be massless or massive with an arbitrary coupling < to the scalar curvature. They can be either in a zero temperature vacuum state or a nonzero temperature thermal state. Analytical approximations which apply to all of these cases are obtained. The method is used to numerically compute the components of the stress-energy tensor of massive and massless scalar fields in Schwarzschild and Reissner-Nordstrom spacetimes. The results are compared to the analytical approximations and the accuracy of the analytical approximations is discussed.PACS number(s): 04.62.+v, 04.70.D~

We present a method for the numerical computation of the stress-energy tensor of a quantized scalar field in a general static spherically symmetric spacetime, with or without horizon. The scalar field may have arbitrary curvature coupling and mass. Our method leads in a natural way to a new analytic approximation to the stress-energy tensor for massless scalar fields in these spacetimes. We use the results to compute stress-energy tensors of quantized scalar fields in Schwarzschild and Reissner-Nordstr^^m black hole spacetimes.PACS numbers: 97.60. Lf, 04.20.Fy The discovery by Hawking [1] that black holes emit radiation in a thermal state showed that quantum fields have a profound effect on black hole spacetimes. This discovery placed black hole thermodynamics on a firm foundation; it also led to the conclusion that black holes in isolation (not surrounded by a heat bath) will evolve by evaporation, dwindling until they become objects of such size and mass that quantum gravity is required to adequately describe them.Early calculations of black hole radiance proceeded by studying the scattering of waves (the modes of the quantized field) in the fixed background spacetime. This approach allows one to obtain information about particle production. However, it does not give information about vacuum polarization effects nor does it tell one how the spacetime geometry near a black hole is changed by the stress energy of the quantum fields. The latter is especially important because the thermodynamic properties and evolutionary history of a black hole are determined by its spacetime geometry.One way to obtain more information about quantum effects is to compute the stress-energy tensor of the quantized field, (T^y). Knowledge of iT^y) gives information on vacuum polarization and particle production in a manner independent of any particular choice of mode decomposition, unlike, e.g., the particle number operator. Further, if iT^y) can be computed for a general class of spacetimes then the semiclassical backreaction equations G^y -STriT^y)( 1) can be solved for that class of spacetimes. In this Letter we present a numerical method which allows for the computation of (T^y) for a quantized scalar field with arbitrary mass and curvature coupling in a general static spherically symmetric spacetime. Included in this class of spacetimes are the Schwarzschild and Reissner-Nordstr^m spacetimes which describe static uncharged and charged black holes, respectively. A similar method of calculating the vacuum polarization, (0^), in a general static spherical spacetimes, was presented in Ref.[2]. The fact that (T^y) can now be computed in these spacetimes will make it possible to obtain static spherically symmetric solutions to the semiclassical backreaction equations. Such solutions will give substantial insight into the question of how quantum effects distort the spacetime geometry near a static black hole which is in thermal equilibrium with its surroundings. They will also provide self-consistent equilibria for the study of blac...

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