Stress-energy tensor of quantized scalar fields in static black hole spacetimes

Abstract

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...

Stress tensor from the trace anomaly in Reissner-Nordström spacetimes

“…Several approaches to approximating T a b have been discussed previously in the literature, developed with important special cases in mind, such as static geometries with a timelike Killing field [16,19,20,21,22,23,24,25]. These approximations are quite successful for regular states in the Schwarzschild geometry, but fail when compared to the numerical results for T a b in the charged RN spacetimes.…”

“…[2] for the quantum stress tensor based on the trace anomaly to static, spherically symmetric spacetimes, focusing specifically on electrically charged Reissner-Nordström (RN) black hole spacetimes, and states with regular stress tensors on the RN horizon. Since direct computations of the renormalized stress tensor expectation value T b a have been carried out in both Schwarzschild and ReissnerNordström (RN) spacetimes for free fields of various spin [8,9,10,11,12,13,14,15,16,17, 18], we will be able to compare the results of the new approximation scheme to these direct computations of T a b . Since the exact effective action of quantum matter generally contains terms which are not determined by the anomaly, using it to compute the stress tensor expectation value is certainly an approximation, which will differ from the direct evaluation of T a b in terms of mode sums in general.…”

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“…Here q denotes expectation value taken with respect to some quantum state with symmetry commensurate with that of the background spacetime. Studies of semiclassical Einstein equation have been carried out in the last two decades by many authors for cosmological [4] and black hole spacetimes [5,6,7,8,9,10,11,12,13,14]. In the analogy with the open system dynamics described in Section 2.1: Eq.…”

“…Jensen and Ottewill [18] have computed the vacuum stress-energy of a massless vector field in Schwarzschild. Approximation methods have been developed by Page, Brown, and Ottewill [19,20,21] for conformally invariant fields in Schwarzschild spacetime, Frolov and Zel'nikov [22] for conformally invariant fields in a general static spacetime, Anderson, Hiscock and Samuel [13] for massless arbitrarily coupled scalar fields in a general static spherically symmetric spacetime. Furthermore the DeWitt-Schwinger approximation has been derived by Frolov and Zel'nikov [23,24] for massive fields in Kerr spacetime, Anderson Hiscock and Samuel [13] for a general (arbitrary curvature coupling and mass) scalar field in a general static spherically symmetric spacetime and have applied their method to the Reissner-Nordström geometry [14].…”

“…The effect of particle creation on the background spacetime -known as the backreaction problem -becomes important when the energy reaches the Planck scale, as in the early universe [4] and at the final stages of black hole evolution [5,6,7,8,9,10,11,12,13,14,15]. Investigation of backreaction problems started with the study of the renormalization or regularization of the energy momentum tensor in curved spacetimes in the late 70's.…”