A mode-sum prescription for the renormalized stress energy tensor on black hole spacetimes

Abstract

In this paper, we describe an efficient method for computing the regularized stress-energy tensor of a quantum scalar field in spherically-symmetric black hole spacetimes. The method relies on two key ingredients, (i) a judicious set of expansion parameters for the singular field of the stressenergy tensor and (ii) analytic representations of the singular field as a mode-sum which can be subtracted from the stress-energy tensor mode-by-mode. We demonstrate the utility of the method by computing the regularized stress-energy tensor for a scalar field with arbitrary mass and coupling in the Schwarzschild black hole spacetime. We further apply our results to discuss the null energy condition and to solve for the semi-classical backreaction.

Breaking Buchdahl: Ultracompact stars in semiclassical gravity

“…Had we inserted Eq. (11) in the angular components of the AHS-RSET as well, we would have obtained a quantity that does not satisfy (12). This implies that one can decide to reduce the order of the ⟨ T t t ⟩ and ⟨ T θ θ ⟩ components, or of the ⟨ T r r ⟩ and ⟨ T θ θ ⟩ ones, instead of the ones we have selected above (⟨ T t t ⟩ and ⟨ T r r ⟩).…”

“…Even in spacetimes with spherical symmetry, obtaining the RSET is a computationally expensive task that requires computing a vast number of field modes with high accuracy [10,11]. Finding self-consistent solutions to Eq.…”

Hadamard states on spherically symmetric characteristic surfaces, the semi-classical Einstein equations and the Hawking effect

“…Analytic calculations are available in two-dimensional toy models (see [41]), but there is no guarantee that these results carry over to more realistic (four dimensional) models. We should also take note of numerical approaches to calculating the expectation values of locally covariant non-linear observables for quantum fields on black hole space-times, such as the methods described in [42,43], as well as a lattice approach to semi-classical gravity recently proposed in [44]. However, in the former two approaches a full analysis of the stress-tensor of a quantum field on a dynamical black hole has, to the knowledge of the authors, not yet been carried out to satisfactory degree, while in the latter approach we feel some additional work is required to ensure that the stress tensor calculated using this method is (approximately) locally covariant in the relevant regimes.…”