Bounds on negative energy densities in flat spacetime

We generalise results of Ford and Roman which place lower bounds -- known as quantum inequalities -- on the renormalised energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in $d$-dimensional Minkowski space ($d\ge 2$) for the free real scalar field of mass $m\ge 0$. We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in 2-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan's optimal bound by a factor of 3/2.Comment: REVTeX, 13 pages and 2 figures. Minor typos corrected, one reference adde

Semiclassical Gravity Theory and Quantum Fluctuations

We discuss the limits of validity of the semiclassical theory of gravity in which a classical metric is coupled to the expectation value of the stress tensor. It is argued that this theory is a good approximation only when the fluctuations in the stress tensor are small. We calculate a dimensionless measure of these fluctuations for a scalar field on a flat background in particular cases, including squeezed states and the Casimir vacuum state. It is found that the fluctuations are small for states which are close to a coherent state, which describes classical behavior, but tend to be large otherwise. We find in all cases studied that the energy density fluctuations are large whenever the local energy density is negative. This is taken to mean that the gravitational field of a system with negative energy density, such as the Casimir vacuum, is not described by a fixed classical metric but is undergoing large metric fluctuations. We propose an operational scheme by which one can describe a fluctuating gravitational field in terms of the statistical behavior of test particles. For this purpose we obtain an equation of the form of the Langevin equation used to describe Brownian motion.Comment: In REVTEX. 20pp + 4 figures(not included, available upon request) TUTP-93-

The Effects of Stress Tensor Fluctuations upon Focusing

We treat the gravitational effects of quantum stress tensor fluctuations. An operational approach is adopted in which these fluctuations produce fluctuations in the focusing of a bundle of geodesics. This can be calculated explicitly using the Raychaudhuri equation as a Langevin equation. The physical manifestation of these fluctuations are angular blurring and luminosity fluctuations of the images of distant sources. We give explicit results for the case of a scalar field on a flat background in a thermal state.Comment: 26 pages, 1 figure, new material added in Sect. III and in Appendices B and

Quantum Inequalities on the Energy Density in Static Robertson-Walker Spacetimes

Quantum inequality restrictions on the stress-energy tensor for negative energy are developed for three and four-dimensional static spacetimes. We derive a general inequality in terms of a sum of mode functions which constrains the magnitude and duration of negative energy seen by an observer at rest in a static spacetime. This inequality is evaluated explicitly for a minimally coupled scalar field in three and four-dimensional static Robertson-Walker universes. In the limit of vanishing curvature, the flat spacetime inequalities are recovered. More generally, these inequalities contain the effects of spacetime curvature. In the limit of short sampling times, they take the flat space form plus subdominant curvature-dependent corrections.Comment: 18 pages, plain LATEX, with 3 figures, uses eps

Restrictions on Negative Energy Density in Flat Spacetime

In a previous paper, a bound on the negative energy density seen by an arbitrary inertial observer was derived for the free massless, quantized scalar field in four-dimensional Minkowski spacetime. This constraint has the form of an uncertainty principle-type limitation on the magnitude and duration of the negative energy density. That result was obtained after a somewhat complicated analysis. The goal of the current paper is to present a much simpler method for obtaining such constraints. Similar ``quantum inequality'' bounds on negative energy density are derived for the electromagnetic field, and for the massive scalar field in both two and four-dimensional Minkowski spacetime.Comment: 17 pages, including two figures, uses epsf, minor revisions in the Introduction, conclusions unchange

Gravitons and Lightcone Fluctuations II: Correlation Functions

A model of a fluctuating lightcone due to a bath of gravitons is further investigated. The flight times of photons between a source and a detector may be either longer or shorter than the light propagation time in the background classical spacetime, and will form a Gaussian distribution centered around the classical flight time. However, a pair of photons emitted in rapid succession will tend to have correlated flight times. We derive and discuss a correlation function which describes this effect. This enables us to understand more fully the operational significance of a fluctuating lightcone. Our results may be combined with observational data on pulsar timing to place some constraints on the quantum state of cosmological gravitons.Comment: 16 pages and two figures, uses eps

Quantum Field Theory Constrains Traversable Wormhole Geometries

Recently a bound on negative energy densities in four-dimensional Minkowski spacetime was derived for a minimally coupled, quantized, massless, scalar field in an arbitrary quantum state. The bound has the form of an uncertainty principle-type constraint on the magnitude and duration of the negative energy density seen by a timelike geodesic observer. When spacetime is curved and/or has boundaries, we argue that the bound should hold in regions small compared to the minimum local characteristic radius of curvature or the distance to any boundaries, since spacetime can be considered approximately Minkowski on these scales. We apply the bound to the stress-energy of static traversable wormhole spacetimes. Our analysis implies that either the wormhole must be only a little larger than Planck size or that there is a large discrepancy in the length scales which characterize the wormhole. In the latter case, the negative energy must typically be concentrated in a thin band many orders of magnitude smaller than the throat size. These results would seem to make the existence of macroscopic traversable wormholes very improbable.Comment: 26 pages, plain LaTe

Gravitational vacuum polarization III: Energy conditions in the (1+1) 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).Comment: 7 pages, ReV_Te

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

Dynamic wormholes

A new framework is proposed for general dynamic wormholes, unifying them with black holes. Both are generically defined locally by outer trapping horizons, temporal for wormholes and spatial or null for black and white holes. Thus wormhole horizons are two-way traversible, while black-hole and white-hole horizons are only one-way traversible. It follows from the Einstein equation that the null energy condition is violated everywhere on a generic wormhole horizon. It is suggested that quantum inequalities constraining negative energy break down at such horizons. Wormhole dynamics can be developed as for black-hole dynamics, including a reversed second law and a first law involving a definition of wormhole surface gravity. Since the causal nature of a horizon can change, being spatial under positive energy and temporal under sufficient negative energy, black holes and wormholes are interconvertible. In particular, if a wormhole's negative-energy source fails, it may collapse into a black hole. Conversely, irradiating a black-hole horizon with negative energy could convert it into a wormhole horizon. This also suggests a possible final state of black-hole evaporation: a stationary wormhole. The new framework allows a fully dynamical description of the operation of a wormhole for practical transport, including the back-reaction of the transported matter on the wormhole. As an example of a matter model, a Klein-Gordon field with negative gravitational coupling is a source for a static wormhole of Morris & Thorne.Comment: 5 revtex pages, 4 eps figures. Minor change which did not reach publisher