Standard Energy Conditions Research Articles

Gravitational-wave dynamics and black-hole dynamics: second quasi-spherical approximation

Gravitational radiation with roughly spherical wavefronts, produced by roughly spherical black holes or other astrophysical objects, is described by an approximation scheme. The first quasi-spherical approximation, describing radiation propagation on a background, is generalized to include additional non-linear effects, due to the radiation itself. The gravitational radiation is locally defined and admits an energy tensor, satisfying all standard local energy conditions and entering the truncated Einstein equations as an effective energy tensor. This second quasi-spherical approximation thereby includes gravitational radiation reaction, such as the back-reaction on the black hole. With respect to a canonical flow of time, the combined energy–momentum of the matter and gravitational radiation is covariantly conserved. The corresponding Noether charge is a local gravitational mass-energy. Energy conservation is formulated as a local first law relating the gradient of the gravitational mass to work and energy-supply terms, including the energy flux of the gravitational radiation. Zeroth, first and second laws of black-hole dynamics are given, involving a dynamic surface gravity. Local gravitational-wave dynamics is described by a non-linear wave equation. In terms of a complex gravitational-radiation potential, the energy tensor has a scalar-field form and the wave equation is an Ernst equation, holding independently at each spherical angle. The strain to be measured by a distant detector is simply defined.

Classical scalar fields and the generalized second law

It has been shown that classical non-minimally coupled scalar fields can violate all of the standard energy conditions in general relativity. Violations of the null and averaged null energy conditions obtainable with such fields have been suggested as possible exotic matter candidates required for the maintenance of traversable wormholes. In this paper, we explore the possibility that if such fields exist, they might be used to produce large negative energy fluxes and macroscopic violations of the generalized second law (GSL) of thermodynamics. We find that it appears to be very easy to produce large magnitude negative energy fluxes in flat spacetime. However we also find, somewhat surprisingly, that these same types of fluxes injected into a black hole do {\it not} produce violations of the GSL. This is true even in cases where the flux results in a decrease in the area of the horizon. We demonstrate that two effects are responsible for the rescue of the GSL: the acausal behavior of the horizon and the modification of the usual black hole entropy formula by an additional term which depends on the scalar field.

Singularities and Scalar Fields: Matter Theory and General Relativity

Philosophers of physics should be more attentive to the role energy conditions play in General Relativity. I review the changing status of energy conditions for quantum fields—presently there are no singularity theorems for semiclassical General Relativity. So we must reevaluate how we understand the relationship between General Relativity, Quantum Field Theory, and singularities. Moreover, on our present understanding of what it is to be a physically reasonable field, the standard energy conditions are violated classically. Thus the singularity theorems are unavailable for classical General Relativity. Our understanding of singularities in General Relativity turns on delicate issues of what it is to be a matter field—issues distinct from the content of the theory.

Scalarfields, energy conditions and traversable wormholes

We describe the different possibilities that a simple and apparentlyquite harmless classical scalar field theory provides to violate theenergy conditions. We demonstrate that a non-minimally coupled scalarfield with a positive curvature coupling ξ>0 can easily violateall the standard energy conditions, up to and including the averagednull energy condition (ANEC). Indeed, this violation of the ANECsuggests the possible existence of traversable wormholes supported bynon-minimally coupled scalars. To investigate this possibility wederive the classical solutions for gravity plus a general (arbitraryξ) massless non-minimally coupled scalar field, restrictingattention to the static and spherically symmetric configurations.Among these classical solutions we find an entire branch oftraversable wormholes for every ξ>0. (This includes andgeneralizes the case of conformal coupling ξ = (1/6) weconsidered in1999 Phys. Lett. B 466 127-34.) For these traversablewormholes to exist we demonstrate that the scalar field must reachtrans-Planckian values somewhere in the geometry. We discuss how thiscan be accommodated within the current state of the art regardingscalar fields in modern theoretical physics. We emphasize that thesescalar field theories, and the traversable wormhole solutions wederive, are compatible with all known experimental constraints fromboth particle physics and gravity.

Brane surgery: energy conditions, traversable wormholes, and voids

Branes are ubiquitous elements of any low-energy limit of string theory. We point out that negative tension branes violate all the standard energy conditions of the higher-dimensional spacetime they are embedded in; this opens the door to very peculiar solutions of the higher-dimensional Einstein equations. Building upon the (3+1)-dimensional implementation of fundamental string theory, we illustrate the possibilities by considering a toy model consisting of a (2+1)-dimensional brane propagating through our observable (3+1)-dimensional universe. Developing a notion of "brane surgery", based on the Israel--Lanczos--Sen "thin shell" formalism of general relativity, we analyze the dynamics and find traversable wormholes, closed baby universes, voids (holes in the spacetime manifold), and an evasion (not a violation) of both the singularity theorems and the positive mass theorem. These features appear generic to any D-brane model that permits negative tension branes: This includes the Randall--Sundrum models and their variants.

Tolman wormholes violate the strong energy condition

For an arbitrary Tolman wormhole, unconstrained by symmetry, we shall define the bounce in terms of a three-dimensional edgeless achronal spacelike hypersurface of minimal volume. (Zero trace for the extrinsic curvature plus a "flare-out" condition.) This enables us to severely constrain the geometry of spacetime at and near the bounce and to derive general theorems regarding violations of the energy conditions--theorems that do not involve geodesic averaging but nevertheless apply to situations much more general than the highly symmetric FRW-based subclass of Tolman wormholes. [For example: even under the mildest of hypotheses, the strong energy condition (SEC) must be violated.] Alternatively, one can dispense with the minimal volume condition and define a generic bounce entirely in terms of the motion of test particles (future-pointing timelike geodesics), by looking at the expansion of their timelike geodesic congruences. One re-confirms that the SEC must be violated at or near the bounce. In contrast, it is easy to arrange for all the other standard energy conditions to be satisfied.

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.

Is a horizon-free cosmology possible?

In a space of FRLW models, filled with a single fluid such that its energy-momentum tensor satisfies standard energy conditions and the dynamics of the universe is described by a second order differential equation,the existence of particle horizons is a generic property. To remove particle horizons one must invest nearly as much as to remove them together with the initial singularity.

The closed-universe recollapse conjecture

It is widely believed that all expanding S3 closed universes that satisfy the standard energy conditions recollapse to a second singularity. We show that this is false even for Friedmann universes: they construct an ever-expanding S3 Friedmann universe in which the matter tensor satisfies the strong, weak and dominant energy conditions and the generic condition. We prove a general recollapse theorem for Friedmann universes: if the positive pressure criterion, dominant energy condition and matter regularity condition hold, then an S3 Friedmann universe must recollapse. We show that all known vacuum solutions with Cauchy surface topology S3 or S2×S1 recollapse, and we conjecture that this is a property of all vacuum solutions of Einstein's equations with such Cauchy surfaces. The authors consider a number of Kantowski–Sachs and Bianchi IX universes with various matter tensors, and formulate a new recollapse conjecture for matter-filled universes.

Space-times with intrinsic symmetries on the three-spaces t=constant

We consider metrics which possess a priori certain ‘‘intrinsic symmetries’’ on the three-spaces t=const. In the vacuum case, we obtain a generalization of Birkhoff’s theorem and a set of solutions with a translational isometry operating on the whole space-time. In the nonvacuum case, we assume a perfect fluid matter content and a fluid flow orthogonal to the three-spaces t=const, and obtain several exact analytical solutions, some of them satisfying the standard energy conditions. In particular, a stiff equation of state is obtained in some cases, and we also have found particular solutions where a plane, spherical, or hyperbolic intrinsic symmetry is manifest.

Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes

The present work extends our short communication L. Andersson, M. Mars and W. Simon, Local existence of dynamical and trapping horizons, Phys. Rev. Lett. 95 (2005), 111102. For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime, we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear nonself-adjoint elliptic operators, we introduce the stability operator Lv and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of Lv. The main result shows that given a strictly stable MOTS S0 ⊂ Σ0 in a spacetime with a reference foliation Σt, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S0. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.