Averaged Energy Condition Research Articles

Efficient Computation of Null Affine Parameters

Finding affine parameters for null geodesics is often of considerable physical importance, especially when studying null geodesics or dealing with conservation laws and/or averaged energy conditions. But explicitly finding null affine parameters is also often quite tedious and can sometimes even be somewhat tricky. Herein we shall demonstrate that the existence of a conformally related spacetime containing a conformal Killing vector, timelike in the domain of outer communication, is quite sufficient to define a preferred set of spatial three-slices—on which a well-defined “affine” three-metric can be introduced to capture the notion of affine null parameter—before explicitly finding the null geodesics. The construction depends on the properties of conformal transformations and on the conserved quantity associated with the conformal Killing vector. Having the affine null parameter in hand before attempting to find the actual null geodesics often quite radically simplifies other parts of the analysis. We emphasize that the successful identification of affine null parameters is a general-purpose tool of wide applicability in both general relativistic and astrophysical settings.

Averaged energy conditions for vector fields

In this work we shall obtain sufficient conditions for the appearance of singularities in gravitational theories which propagate an extra vector degree of freedom, based on the known relaxations of the singularity theorems. We study the cases of a general Proca field and a vector theory with stable self-derivative interactions. In this study we show several cases of singularities that usually would be considered as potentially singularity-free, since they violate the usual point-like energy conditions.

Evaporation of four-dimensional dynamical black holes sourced by the quantum trace anomaly

We study the evaporation of a four-dimensional spherically symmetric black hole formed in a gravitational collapse. We analyze the back-reaction of a massless quantum scalar field conformally coupled to the scalar curvature by means of the semiclassical Einstein equations. We show that the evaporation is linked to an ingoing negative energy flux at the dynamical horizon and that this flux is induced by the quantum matter trace anomaly outside the black hole horizon whenever a suitable averaged energy condition is satisfied. For illustrative purposes, we evaluate the negative ingoing flux and the corresponding rate of evaporation in the case of a null radiating star described by the Vaidya spacetime.

Averaged energy conditions and bouncing universes

The dynamics of bouncing universes is characterized by violating certain coordinate invariant restrictions on the total energy-momentum tensor, customarily referred to as energy conditions. Although there could be epochs where the null energy condition is locally violated, it may perhaps be enforced in an averaged sense. Explicit examples of this possibility are investigated in different frameworks.

Singularity theorems from weakened energy conditions

We establish analogues of the Hawking and Penrose singularity theorems based on (a) averaged energy conditions with exponential damping; (b) conditions on local stress–energy averages inspired by the quantum energy inequalities satisfied by a number of quantum field theories. As particular applications, we establish singularity theorems for the Einstein equations coupled to a classical scalar field, which violates the strong energy condition, and the nonminimally coupled scalar field, which also violates the null energy condition.

Energy conditions, traversable wormholes and dust shells

Firstly, we review the pointwise and averaged energy conditions, the quantum inequality and the notion of the “volume integral quantifier,” which provides a measure of the “total amount” of energy condition violating matter. Secondly, we present a specific metric of a spherically symmetric traversable wormhole in the presence of a generic cosmological constant, verifying that the null and the averaged null energy conditions are violated, as was to be expected. Thirdly, a pressureless dust shell is constructed around the interior wormhole spacetime by matching the latter geometry to a unique vacuum exterior solution. In order to further minimize the usage of exotic matter, we then find regions where the surface energy density is positive, thereby satisfying all of the energy conditions at the junction surface. An equation governing the behavior of the radial pressure across the junction surface is also deduced. Lastly, taking advantage of the construction, specific dimensions of the wormhole, namely, the throat radius and the junction interface radius, and estimates of the total traversal time and maximum velocity of an observer journeying through the wormhole, are also found by imposing the traversability conditions.

Averaged energy conditions in 4D evaporating black hole backgrounds

Using Visser's semi-analytical model for the stress-energy tensor corresponding to the conformally coupled massless scalar field in the Unruh vacuum, we examine, by explicitly evaluating the relevant integrals over half-complete geodesics, the averaged weak (AWEC) and averaged null (ANEC) energy conditions along with Ford-Roman quantum inequality-type restrictions on negative energy in the context of four dimensional evaporating black hole backgrounds. We find that in all cases where the averaged energy conditions fail, there exist quantum inequality bounds on the magnitude and duration of negative energy densities.

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. 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.

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. III. Energy conditions in the (1+1)-dimensional Schwarzschild spacetime.

Building on a pair of earlier papers, I investigate the various point-wise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally-coupled massless scalar field in the in the (1+1)-dimensional Schwarzschild spacetime. Because the stress-energy tensors are analytically known, I can get exact results for the Hartle--Hawking, Boulware, and Unruh vacua. This exactly solvable model serves as a useful sanity check on my (3+1)-dimensional investigations wherein I had to resort to a mixture of analytic approximations and numerical techniques. Key results in (1+1) dimensions are: (1) NEC is satisfied outside the event horizon for the Hartle--Hawking vacuum, and violated for the Boulware and Unruh vacua. (2) DEC is violated everywhere in the spacetime (for any quantum state, not just the standard vacuum states).

Averaged energy conditions and evaporating black holes.

In this paper the averaged weak and averaged null energy conditions, together with uncertainty-principle-type restrictions on negative energy (``quantum inequalities''), are examined in the context of evaporating black hole backgrounds in both two and four dimensions. In particular, integrals over only half-geodesics are studied. We determine the regions of the spacetime in which the averaged energy conditions are violated. In all cases where these conditions fail, there appear to be quantum inequalities which bound the magnitude and extent of the negative energy, and hence the degree of the violation. The possible relevance of these results for the validity of singularity theorems in evaporating black hole spacetimes is discussed. \textcopyright{} 1996 The American Physical Society.

Remarks on the averaged null energy condition in quantum field theory.

The averaged null energy condition has been recently shown to hold for linear quantum fields in a large class of spacetimes. Nevertheless, it is easy to show by using a simple scaling argument that ANEC as stated cannot hold generically in curved four-dimensional spacetime, and this scaling argument has been widely interpreted as a death-blow for averaged energy conditions in quantum field theory. In this note I propose a simple generalization of ANEC, in which the right-hand-side of the ANEC inequality is replaced by a finite (but in general negative) state-independent lower bound. As long as attention is focused on asymptotically well-behaved spacetimes, this generalized version of ANEC is safe from the threat of the scaling argument, and thus stands a chance of being generally valid in four-dimensional curved spacetime. I argue that when generalized ANEC holds, it has implications for the non-negativity of total energy and for singularity theorems similar to the implications of ANEC. In particular, I show that if generalized ANEC is satisfied in static traversable wormhole spacetimes (which is likely but remains to be shown), then macroscopic wormholes (but not necessarily microscopic, Planck-size wormholes) are ruled out by quantum field theory.

Averaged null energy condition and difference inequalities in quantum field theory.

Recently, Larry Ford and Tom Roman have discovered that in a flat cylindrical space, although the stress-energy tensor itself fails to satisfy the averaged null energy condition (ANEC) along the (non-achronal) null geodesics, when the ``Casimir-vacuum" contribution is subtracted from the stress-energy the resulting tensor does satisfy the ANEC inequality. Ford and Roman name this class of constraints on the quantum stress-energy tensor ``difference inequalities." Here I give a proof of the difference inequality for a minimally coupled massless scalar field in an arbitrary two-dimensional spacetime, using the same techniques as those we relied on to prove ANEC in an earlier paper with Robert Wald. I begin with an overview of averaged energy conditions in quantum field theory.

Averaged energy conditions and quantum inequalities.

In this paper, connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy condtions, and quantum inequality restrictions (uncertainty-principle-type inequalities) on negative energy. In two- and four-dimensional Minkowski spacetime, we examine quantized, free massless, minimally coupled scalar fields. In a two-dimensional spatially compactified Minkowski universe, we derive a covariant quantum inequality-type bound on the difference of the expectation values of ${\mathit{T}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$${\mathit{u}}^{\mathrm{\ensuremath{\mu}}}$${\mathit{u}}^{\ensuremath{\nu}}$ in an arbitrary quantum state and in the Casimir vacuum state. From this bound, it is shown that the difference of expectation values also obeys AWEC- and ANEC-type integral conditions. This is surprising, since it is well known that the expectation value of ${\mathit{T}}_{\mathrm{\ensuremath{\mu}}\ensuremath{\nu}}$${\mathit{u}}^{\mathrm{\ensuremath{\mu}}}$${\mathit{u}}^{\ensuremath{\nu}}$ in the renormalized Casimir vacuum state alone satisfies neither quantum inequalities nor averaged energy conditions. Such ``difference inequalities,'' if suggestive of the general case, might represent limits on the degree of energy condition violation that is allowed over and above any violation due to negative energy densities in a background vacuum state. In our simple two-dimensional model, they provide physically interesting examples of new constraints on negative energy which hold even when the usual AWEC, ANEC, and quantum inequality restrictions fail. In the limit when the size of the space is allowed to go to infinity, we derive quantum inequalities for timelike and null geodesics which, in appropriate limits, reduce to AWEC and ANEC in ordinary two-dimensional Minskowski spacetime. Lastly, we also derive a covariant quantum inequality bound on the energy density seen by an arbitrary inertial observer in four-dimensional Minskowski spacetime. The bound implies that any inertial observer in flat spacetime cannot see an arbitrarily large negative energy density which lasts for an arbitrarily long period of time. From this latter bound, we derive AWEC and ANEC.

Averaged energy conditions for free scalar fields in flat spacetime.

This paper initiates a research program to determine whether, and in what situations, quantum field theory enforces averaged energy conditions on the renormalized stress-energy tensors of quantum fields. This program is motivated by the important roles of averaged energy conditions in general-relativistic singularity theorems, and in preventing the existence of classical, traversable wormholes and wormhole-induced closed timelike curves. As a first step in this research program, this paper shows that a quantized, free scalar field in Minkowski space-time has the following properties: The weak-energy condition is satisfied for a wide class of states when averaged along a complete null geodesic, but it can be violated when averaged along a nongeodesic curve. If the curvature coupling constant in the scalar wave equation is restricted to a certain range, which includes conformal coupling, then the strong-energy condition is satisfied for the same wide class of states when averaged along a complete timelike geodesic. It is shown, further, that this enforcement of energy conditions is not universally true in all spacetimes: by closing up Minkowski spacetime in a spatial direction (e.g., by identifying x=0 with x=L), one can produce quantum states of a free scalar field that violate the averaged weak-energy condition.

Energy conditions and spacetime singularities

In this paper, a number of theorems are proven which collectively show that singularities will occur in spacetime under weaker energy conditions than the strong energy condition. In particular, the Penrose theorem, which uses only the weak energy condition but which applies only to open universes, is extended to all closed universes which have a Cauchy surface whose universal covering manifold is not a three-sphere. Furthermore, it is shown that the strong energy condition in the Hawking-Penrose theorem can be replaced by the weak energy condition and the assumption that the strong energy condition holds only on the average. In addition, it is demonstrated that if the Universe is closed, then the existence of singularities follows from the averaged strong energy condition alone. It is argued that any globally hyperbolic spacetime which satisfies the weak energy condition and which contains a black hole must be null geodesically incomplete.