Averaged Energy Conditions and Quantum Inequalities

Connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy conditions, and quantum inequality restrictions on negative energy for free massless scalar fields. In a two-dimensional compactified Minkowski universe, we derive a covariant quantum inequality-type bound on the difference of the expectation values of the energy density 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. In contrast, it is well-known that the stress tensor in the Casimir vacuum state alone satisfies neither quantum inequalities nor averaged energy conditions. Such difference inequalities 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 Minkowski spacetime. We also derive a quantum inequality bound on the energy density seen by an inertial observer in four-dimensional Minkowski 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.Comment: 20pp, plain LATEX, TUTP-94-1

Nonorientable spacetime tunneling

Misner space is generalized to have the nonorientable topology of a Klein bottle, and it is shown that in a classical spacetime with multiply connected space slices having such a topology, closed timelike curves are formed. Different regions on the Klein bottle surface can be distinguished which are separated by apparent horizons fixed at particular values of the two angular variables that eneter the metric. Around the throat of this tunnel (which we denote a Klein bottlehole), the position of these horizons dictates an ordinary and exotic matter distribution such that, in addition to the known diverging lensing action of wormholes, a converging lensing action is also present at the mouths. Associated with this matter distribution, the accelerating version of this Klein bottlehole shows four distinct chronology horizons, each with its own nonchronal region. A calculation of the quantum vacuum fluctuations performed by using the regularized two-point Hadamard function shows that each chronology horizon nests a set of polarized hypersurfaces where the renormalized momentum-energy tensor diverges. This quantum instability can be prevented if we take the accelerating Klein bottlehole to be a generalization of a modified Misner space in which the period of the closed spatial direction is time-dependent. In this case, the nonchronal regions and closed timelike curves cannot exceed a minimum size of the order the Planck scale.Comment: 11 pages, RevTex, Accepted in Phys. Rev.

String Supported Wormhole Spacetimes and Causality Violations

We construct a static axisymmetric wormhole from the gravitational field of two Schwarzschild particles which are kept in equilibrium by strings (ropes) extending to infinity. The wormhole is obtained by matching two three-dimensional timelike surfaces surrounding each of the particles and thus spacetime becomes non-simply connected. Although the matching will not be exact in general it is possible to make the error arbitrarily small by assuming that the distance between the particles is much larger than the radius of the wormhole mouths. Whenever the masses of the two wormhole mouths are different, causality violating effects will occur.Comment: 12 pages, LaTeX, 1 figur

Evolving Lorentzian Wormholes

Evolving Lorentzian wormholes with the required matter satisfying the Energy conditions are discussed. Several different scale factors are used and the corresponding consequences derived. The effect of extra, decaying (in time) compact dimensions present in the wormhole metric is also explored and certain interesting conclusions are derived for the cases of exponential and Kaluza--Klein inflation.Comment: 10 pages( RevTex, Twocolumn format), Two figures available on request from the first author. transmission errors corrected

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.Comment: 11 pages, 3 figures, Revtex

A Superluminal Subway: The Krasnikov Tube

The ``warp drive'' metric recently presented by Alcubierre has the problem that an observer at the center of the warp bubble is causally separated from the outer edge of the bubble wall. Hence such an observer can neither create a warp bubble on demand nor control one once it has been created. In addition, such a bubble requires negative energy densities. One might hope that elimination of the first problem might ameliorate the second as well. We analyze and generalize a metric, originally proposed by Krasnikov for two spacetime dimensions, which does not suffer from the first difficulty. As a consequence, the Krasnikov metric has the interesting property that although the time for a one-way trip to a distant star cannot be shortened, the time for a round trip, as measured by clocks on Earth, can be made arbitrarily short. In our four dimensional extension of this metric, a ``tube'' is constructed along the path of an outbound spaceship, which connects the Earth and the star. Inside the tube spacetime is flat, but the light cones are opened out so as to allow superluminal travel in one direction. We show that, although a single Krasnikov tube does not involve closed timelike curves, a time machine can be constructed with a system of two non-overlapping tubes. Furthermore, it is demonstrated that Krasnikov tubes, like warp bubbles and traversable wormholes, also involve unphysically thin layers of negative energy density, as well as large total negative energies, and therefore probably cannot be realized in practice.Comment: 20 pages, LATEX, 5 eps figures, uses \eps

The Quantum Interest Conjecture

Although quantum field theory allows local negative energy densities and fluxes, it also places severe restrictions upon the magnitude and extent of the negative energy. The restrictions take the form of quantum inequalities. These inequalities imply that a pulse of negative energy must not only be followed by a compensating pulse of positive energy, but that the temporal separation between the pulses is inversely proportional to their amplitude. In an earlier paper we conjectured that there is a further constraint upon a negative and positive energy delta-function pulse pair. This conjecture (the quantum interest conjecture) states that a positive energy pulse must overcompensate the negative energy pulse by an amount which is a monotonically increasing function of the pulse separation. In the present paper we prove the conjecture for massless quantized scalar fields in two and four-dimensional flat spacetime, and show that it is implied by the quantum inequalities.Comment: 17 pages, Latex, 3 figures, uses eps

Spacetime Information

In usual quantum theory, the information available about a quantum system is defined in terms of the density matrix describing it on a spacelike surface. This definition must be generalized for extensions of quantum theory which do not have a notion of state on a spacelike surface. It must be generalized for the generalized quantum theories appropriate when spacetime geometry fluctuates quantum mechanically or when geometry is fixed but not foliable by spacelike surfaces. This paper introduces a four-dimensional notion of the information available about a quantum system's boundary conditions in the various sets of decohering histories it may display. The idea of spacetime information is applied in several contexts: When spacetime geometry is fixed the information available through alternatives restricted to a spacetime region is defined. The information available through histories of alternatives of general operators is compared to that obtained from the more limited coarse- grainings of sum-over-histories quantum mechanics. The definition of information is considered in generalized quantum theories. We consider as specific examples time-neutral quantum mechanics with initial and final conditions, quantum theories with non-unitary evolution, and the generalized quantum frameworks appropriate for quantum spacetime. In such theories complete information about a quantum system is not necessarily available on any spacelike surface but must be searched for throughout spacetime. The information loss commonly associated with the ``evolution of pure states into mixed states'' in black hole evaporation is thus not in conflict with the principles of generalized quantum mechanics.Comment: 47pages, 2 figures, UCSBTH 94-0

On the warp drive space-time

In this paper the problem of the quantum stability of the two-dimensional warp drive spacetime moving with an apparent faster than light velocity is considered. We regard as a maximum extension beyond the event horizon of that spacetime its embedding in a three-dimensional Minkowskian space with the topology of the corresponding Misner space. It is obtained that the interior of the spaceship bubble becomes then a multiply connected nonchronal region with closed timelike curves and that the most natural vacuum allows quantum fluctuations which do not induce any divergent behaviour of the re-normalized stress-energy tensor, even on the event (Cauchy) chronology horizon. In such a case, the horizon encloses closed timelike curves only at scales close to the Planck length, so that the warp drive satisfies the Ford's negative energy-time inequality. Also found is a connection between the superluminal two-dimensional warp drive space and two-dimensional gravitational kinks. This connection allows us to generalize the considered Alcubierre metric to a standard, nonstatic metric which is only describable on two different coordinate patchesComment: 7 pages, minor comment on chronology protection added, RevTex, to appear in Phys. Rev.

Averaged Energy Conditions and Evaporating Black Holes

In this paper the averaged weak (AWEC) and averaged null (ANEC) 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.Comment: Sections 2.1 and 2.2 have been revised and some erroneous statements corrected. The main conclusions and the figures are unchanged. 27 pp, plain Latex, 3 figures available upon reques