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arXiv:2607.07583

Vacuum polarization and renormalized stress-energy tensor of spherical thin shells

Julio Arrechea, Cormac Breen, Adrian Ottewill, Lorenzo Pisani, Peter Taylor

gr-qchep-thquant-phSubmitted Jul 8, 2026• Announced Jul 9, 2026
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Abstract

We provide a thorough study of the properties of the Boulware vacuum in the spacetime of a spherical, static thin shell with a Minkowski interior. To this end, we calculate the renormalized vacuum polarization and stress-energy tensor of massless scalar fields via the extended-coordinate prescription, paying particular attention to their scaling as the shell approaches the black hole limit. Near the surface of the thin shell, we obtain the expected leading-order singular behavior of both quantities via two independent methods: a high-frequency approximation for the modes, and a weak-field approximation. At the center of the shell we find non-local, Casimir-like contributions that remain finite in the black hole limit, and whose backreaction effects we compute via the semiclassical Einstein equations. Away from these regions amenable to analytic treatment, we obtain numerical results for a wide range of shell compactnesses and field couplings. In the black hole limit, we show that the vacuum polarization and renormalized stress-energy tensor outside the shell quickly approach the ones generated by a Schwarzschild black hole, suggesting a possible universality in the vacuum outside highly compact horizonless objects. This work addresses the conceptual and technical aspects necessary for computing renormalized expectation values in matter configurations, laying the foundations for future explorations on the subject.

Comments
32 pages, 10 figures

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We provide a thorough study of the properties of the Boulware vacuum in the spacetime of a spherical, static thin shell with a Minkowski interior. To this end, we calculate the renormalized vacuum polarization and stress-energy tensor of massless scalar fields via the extended-coordinate prescription, paying particular attention to their scaling as the shell approaches the black hole limit. Near the surface of the thin shell, we obtain the expected leading-order singular behavior of both quantities via two independent methods: a high-frequency approximation for the modes, and a weak-field approximation. At the center of the shell we find non-local, Casimir-like contributions that remain finite in the black hole limit, and whose backreaction effects we compute via the semiclassical Einstein equations. Away from these regions amenable to analytic treatment, we obtain numerical results for a wide range of shell compactnesses and field couplings. In the black hole limit, we show that the vacuum polarization and renormalized stress-energy tensor outside the shell quickly approach the ones generated by a Schwarzschild black hole, suggesting a possible universality in the vacuum outside highly compact horizonless objects. This work addresses the conceptual and technical aspects necessary for computing renormalized expectation values in matter configurations, laying the foundations for future explorations on the subject.
We investigate a quantum massive scalar field in the interior of a charged and spherically-symmetric (Reissner-Nordström) black hole. We examine the behaviour for varying values of the black hole charge, field mass and coupling constant when the field is in two quantum states: Hartle-Hawking (representing a black hole in thermal equilibrium) and Unruh (representing a black hole evaporating via the emission of Hawking radiation). We show that the vacuum polarization as well as the angular components of the quantum stress-energy tensor diverge on the Cauchy horizon, in stark contrast to what happens for massless fields. We also calculate the energy fluxes in Eddington-Finkelstein coordinates {u,v}\{u,v\}{u,v}. We show that these fluxes on the Cauchy horizon do not generically vanish. This implies, in particular, that in regular, Kruskal coordinates {U,V}\{U,V\}{U,V} the ingoing flux diverges like V−2V^{-2}V−2 on the Cauchy horizon (where V=0V=0V=0). This divergence suggests that its backreaction via the semiclassical Einstein equations would yield a strong singularity, as opposed to its weaker, classical counterpart. Interestingly, there are exceptions, in which the energy fluxes in Eddington-Finkelstein coordinates vanish: (i) in the extremal limit (where the black hole is maximally charged); (ii) certain fine-tuned regions of parameter space, where the fluxes change sign.
We study the finite-temperature Casimir effect for a massless scalar field confined between two parallel plates in a Schwarzschild-like wormhole spacetime. Imposing Dirichlet boundary conditions, we compute the renormalized Casimir free energy in the comoving frame. We find that the thermal correction to the renormalized Casimir free energy decreases gradually with the temperature and becomes independent of the background geometry in this frame. Thermodynamic quantities derived from the Casimir free energy, namely, the renormalized Casimir entropy, internal energy, and heat capacity at constant volume, exhibit distinct temperature dependence. At low temperatures, all thermodynamic quantities recover the expected behavior, consistent with the fundamental laws of thermodynamics. These results provide a compact framework for analyzing quantum vacuum forces in gravitational backgrounds.
We investigate how the compactification of some spatial dimensions in Rindler spacetime affects the vacuum expectation values (VEVs) of the field squared and the energy-momentum tensor for a charged scalar field prepared in the Fulling-Rindler vacuum state. A toral compactification is considered with quasi-periodicity conditions on the field operator along the compact dimensions. The phases of these conditions are interpreted in terms of the magnetic flux enclosed by the compact dimensions. For a general number of spatial dimensions, the components of the VEVs are explicitly separated, corresponding to the expectation values in the Minkowski vacuum. As a limiting case, the VEVs are retrieved in Rindler spacetime with trivial topology. We demonstrate that, for non-zero phases in the periodicity conditions, the vacuum energy-momentum tensor exhibits off-diagonal components with indices in the compact subspace. Near the Rindler horizon, the leading terms in the asymptotic expansions of the field squared and the diagonal components of the energy-momentum tensor coincide with the corresponding VEVs in Rindler spacetime with trivial topology. The off-diagonal components of the energy-momentum tensor vanish on the Rindler horizon. For small accelerations, the difference between the VEVs in the Fulling-Rindler and Minkowski vacua is exponentially small. The exception to this is a massless field with zero phases in the periodicity conditions. In this special case, the difference decays according to a power law as a function of acceleration. As an application, we consider the VEVs near the horizon of cylindrical black holes and topological black holes with toroidal horizons.
We investigate the combined effects of spatial curvature and topology on the properties of the vacuum state for a charged scalar field localized on the (2+1)-dimensional Beltrami pseudosphere, assuming that the field obeys quasiperiodicity condition with constant phase. As important local characteristics of the vacuum state the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor are evaluated. The contributions in the VEVs coming from geometry with an uncompactified azimuthal coordinate are divergent, whereas the compact counterparts are finite and are analysed both numerically and asymptotically. For small values of proper radius of the compactified dimension, the leading terms of topological contributions are independent of the field mass and curvature coupling parameter, increasing by a power-law. In the opposite limit, the VEVs decay following a power-law in the general case. In the special case of a conformally coupled massless field the behavior is different. Unlike the VEV of field squared and vacuum energy density, the radial and azimuthal stresses are increasing by absolute value. As a consequence, the effects of nontrivial topology are strong for the stresses in this case at small values of radial coordinate.
We apply the complex scaling method to black-hole perturbations in four-dimensional Schwarzschild--de~Sitter (dS) spacetimes. The method converts the outgoing-wave boundary-value problem into a non-Hermitian spectral problem and enables quasinormal-mode poles and the rotated continuum to be treated in a common framework. We focus in particular on the continuum level density, which characterizes the continuum response beyond isolated quasinormal-mode frequencies. Using Regge--Wheeler-type perturbation equations for scalar, electromagnetic, and gravitational fields, we investigate how a nonzero cosmological constant modifies the pole and continuum sectors. We also discuss a possible extension to string-inspired coupled-channel systems, and illustrate that higher-dimensional dS black holes can be treated within the same framework, at least in tensor- and vector-type sectors. Our results indicate that complex scaling offers a useful spectral framework for analyzing both quasinormal modes and continuum response in black-hole physics.
Black holes are usually described either as classical geometries on which quantum fields propagate or as objects whose complete quantum description lies beyond semiclassical gravity. Hawking radiation already shows, however, that a black hole is not an exactly isolated system: the horizon carries a thermal scale and, after coarse graining, may be associated with fluctuations, dissipation and environmental correlations. Whether this horizon-scale openness can be encoded directly in the gravitational background, before any stochastic treatment of the metric, remains an open question. Here we answer it by introducing, for the first time, a conserved Hawking-scale anisotropic source representing the coarse-grained near-horizon atmosphere and solving the Einstein equations for a static black hole at fixed ADM mass. The dressed geometry has a renormalized mass function, a self-consistently shifted horizon and a modified surface gravity. Because the same lapse governs the horizon and timelike geodesics, the thermal response is accompanied by a correlated orbital restructuring: the effective potential is renormalized, the ISCO moves inward and the near-horizon residence time increases. The construction adds no Langevin force to a fixed Schwarzschild background; rather, it defines a deterministic dressed mean geometry, providing a controlled background on which stochastic metric fluctuations may subsequently be treated in the Einstein-Langevin framework. This establishes the deterministic mean-field sector as the necessary first step for future stochastic gravity programs addressing horizon-scale openness.