Pointwise Energy Conditions Research Articles

A semiclassical singularity theorem

Quantum fields do not satisfy the pointwise energy conditions that are assumed in the original singularity theorems of Penrose and Hawking. Accordingly, semiclassical quantum gravity lies outside their scope. Although a number of singularity theorems have been derived under weakened energy conditions, none is directly derived from quantum field theory. Here, we employ a quantum energy inequality satisfied by the quantized minimally coupled linear scalar field to derive a singularity theorem valid in semiclassical gravity. By considering a toy cosmological model, we show that our result predicts timelike geodesic incompleteness on plausible timescales with reasonable conditions at a spacelike Cauchy surface.

Quantum stress tensor of massive vector field in the spacetime of a pointlike global monopole

The components of the renormalized quantum stress tensor for a massive vector field in the spacetime of a pointlike global monopole are determined analytically in the Schwinger–DeWitt approximation. The general results are employed to investigate the pointwise energy conditions for the quantized matter field and it is shown that they are all violated by the massive vector field.

Renormalized quantum stress–energy tensor for massive spinor fields around global monopoles

The renormalized quantum stress–energy tensor [Formula: see text] for a massive spinor field around global monopoles is constructed within the framework of Schwinger–DeWitt approximation, valid whenever the Compton length of the quantum field is much less than the characteristic radius of the curvature of the background geometry. The results obtained show that the quantum massive spinor field in the global monopole spacetime violates all the pointwise energy conditions.

Vacuum polarization of the quantized massive scalar field in the global monopole spacetime: The renormalized quantum stress energy tensor

This paper is devoted to the construction of the renormalized quantum stress energy tensor $\left<T_{\mu}^{\nu}\right>_{ren}$ for a massive scalar field with arbitrary coupling to the gravitational field of a pointlike global monopole, using the Schwinger-DeWitt approximation, up to second order in the inverse mass $\mu$ of the field. The given stress energy tensor is constructed by functional differentiation with respect to the metric tensor of the one-loop effective action of sufficiently massive scalar field, such that the Compton length of the quantum field is much less than the characteristic radius of the curvature of the background geometry. The results are obtained for a general curvature coupling parameter $\xi$, and specified to the more physical cases of minimal and conformal coupling, showing that in this specific cases, the quantum massive scalar field in the global monopole spacetime violates all the pointwise energy conditions.

Palatini wormholes and energy conditions from the prism of general relativity

Wormholes are hypothetical shortcuts in spacetime that in general relativity unavoidably violate all of the pointwise energy conditions. In this paper, we consider several wormhole spacetimes that, as opposed to the standard designer procedure frequently employed in the literature, arise directly from gravitational actions including additional terms resulting from contractions of the Ricci tensor with the metric, and which are formulated assuming independence between metric and connection (Palatini approach). We reinterpret such wormhole solutions under the prism of General Relativity and study the matter sources that thread them. We discuss the size of violation of the energy conditions in different cases and how this is related to the same spacetimes when viewed from the modified gravity side.

Charged analogue of well behaved neutral spheres: an algorithmic approach

We establish a systematic algorithmic approach that generates new classes of solutions to the Einstein-Maxwell system in static spherically symmetric spacetime from well known uncharged solutions. A particular case is shown to satisfy all major physical features of a realistic charged star including the standard point-wise energy conditions of normal matter. The solution matches smoothly with the exterior Reissner-Nordström metric at the pressure free interface. The study, which is reported for a particular choice of gravitational potential, encourages similar approaches to study electrification of well known physically realistic uncharged models.

Quantum stress tensor for a massive vector field in the space-time of a cylindrical black hole

The components of the renormalized quantum Energy-Momentum tensor for a massive vector field coupled to the gravitational field configuration of a static Black-String are analytically evaluated using the Schwinger-DeWitt approximation. The general results are employed to investigate the pointwise energy conditions for the quantized matter field, and it is shown that they are violated at some regions of the spacetime, in particular the horizon of the black hole.

Averaged energy inequalities for the nonminimally coupled classical scalar field

The stress-energy tensor for the classical nonminimally coupled scalar field is known not to satisfy the pointwise energy conditions of general relativity. In this paper we show, however, that local averages of the classical stress-energy tensor satisfy certain inequalities. We give bounds for averages along causal geodesics and show, e.g., that in Ricci-flat background spacetimes, ANEC and AWEC are satisfied. Furthermore we use our result to show that in the classical situation we have an analogue to the phenomenon of quantum interest. These results lay the foundations for analogous energy inequalities for the quantized nonminimally coupled fields, which will be discussed elsewhere.

Gravitational vacuum polarization. IV. Energy conditions in the Unruh vacuum

Building on a series of earlier papers [gr-qc/9604007, gr-qc/9604008, gr-qc/9604009], I investigate the various point-wise and averaged energy conditions in the Unruh vacuum. I consider the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field, work in the test-field limit, restrict attention to the Schwarzschild geometry, and invoke a mixture of analytical and numerical techniques. I construct a semi-analytic model for the stress-energy tensor that globally reproduces all known numerical results to within 0.8%, and satisfies all known analytic features of the stress-energy tensor. I show that in the Unruh vacuum (1) all standard point-wise energy conditions are violated throughout the exterior region--all the way from spatial infinity down to the event horizon, and (2) the averaged null energy condition is violated on all outgoing radial null geodesics. In a pair of appendices I indicate general strategy for constructing semi-analytic models for the stress-energy tensor in the Hartle-Hawking and Boulware states, and show that the Page approximation is in a certain sense the minimal ansatz compatible with general properties of the stress-energy in the Hartle-Hawking state.

Gravitational vacuum polarization. I. Energy conditions in the Hartle-Hawking vacuum.

It is well-known that gravitationally induced vacuum polarization often violates the point-wise energy conditions and sometimes violates the averaged energy conditions. In this paper I begin a systematic attack on the question of where and by how much the various energy conditions are violated. I work in the test-field limit, and focus on conformally coupled massless scalar fields in Schwarzschild spacetime, using the Hartle--Hawking vacuum. I invoke a mixture of analytical and numerical techniques, and critically compare the qualitative behaviour to be expected from the Page approximation with that adduced from the numerical calculations of Anderson, Hiscock, and Samuel. I show that the various point-wise energy conditions are violated in a series of onion-like layers located between the unstable photon orbit and the event horizon, the sequence of violations being DEC, WEC, and (NEC+SEC). Furthermore the ANEC is violated for *some* of the null geodesics trapped in this region. Having established the basic machinery in this paper, the Boulware vacuum will be treated in a companion paper, while other exensions should be straightforward.

Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum.

Building on techniques developed in the preceding paper, I investigate the various pointwise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field in the Boulware vacuum. I work in the test-field limit, restrict attention to the Schwarzschild geometry, and invoke a mixture of analytical and numerical techniques. In contradistinction to the case of the Hartle-Hawking vacuum, wherein violations of the energy conditions were confined to the region between the event horizon and the unstable photon orbit, I show that in the Boulware vacuum (1) all standard (pointwise and averaged) energy conditions are violated throughout the exterior region, all the way from spatial infinity down to the event horizon, and (2) outside the event horizon the standard pointwise energy conditions are violated in a maximal manner: They are violated at all points and for all null or timelike vectors. (The region inside the event horizon is considerably messier and of dubious physical relevance. Nevertheless, the standard pointwise energy conditions seem to be violated even inside the event horizon.) I argue that this is highly suggestive evidence, pointing to the fact that general self-consistent solutions of semiclassical quantum gravity might not satisfy the energy conditions and may in fact for certain quantum fields and certain quantum states violate all the energy conditions.