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Title: Averaged null energy condition in spacetimes with boundaries

Abstract

The averaged null energy condition (ANEC) requires that the average along a complete null geodesic of the projection of the stress-energy tensor onto the geodesic tangent vector can never be negative. It is sufficient to rule out many exotic phenomena in general relativity. Subject to certain conditions, we show that the ANEC can never be violated by a quantized minimally coupled free scalar field along a complete null geodesic surrounded by a tubular neighborhood in which the geometry is flat and whose intrinsic causal structure coincides with that induced from the full spacetime. In particular, the ANEC holds in flat space with boundaries, as in the Casimir effect, for geodesics which stay a finite distance away from the boundary.

Authors:
; ;  [1];  [2];  [2]
  1. Department of Mathematics, University of York, Heslington, York, YO10 5DD (United Kingdom)
  2. (United States)
Publication Date:
OSTI Identifier:
21010920
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.75.025007; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; CASIMIR EFFECT; COSMOLOGY; DISTANCE; GENERAL RELATIVITY THEORY; GEODESICS; GEOMETRY; SCALAR FIELDS; SPACE-TIME; VECTORS

Citation Formats

Fewster, Christopher J., Olum, Ken D., Pfenning, Michael J., Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, and Department of Physics, United States Military Academy, West Point, New York 10996. Averaged null energy condition in spacetimes with boundaries. United States: N. p., 2007. Web. doi:10.1103/PHYSREVD.75.025007.
Fewster, Christopher J., Olum, Ken D., Pfenning, Michael J., Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, & Department of Physics, United States Military Academy, West Point, New York 10996. Averaged null energy condition in spacetimes with boundaries. United States. doi:10.1103/PHYSREVD.75.025007.
Fewster, Christopher J., Olum, Ken D., Pfenning, Michael J., Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, and Department of Physics, United States Military Academy, West Point, New York 10996. Mon . "Averaged null energy condition in spacetimes with boundaries". United States. doi:10.1103/PHYSREVD.75.025007.
@article{osti_21010920,
title = {Averaged null energy condition in spacetimes with boundaries},
author = {Fewster, Christopher J. and Olum, Ken D. and Pfenning, Michael J. and Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155 and Department of Physics, United States Military Academy, West Point, New York 10996},
abstractNote = {The averaged null energy condition (ANEC) requires that the average along a complete null geodesic of the projection of the stress-energy tensor onto the geodesic tangent vector can never be negative. It is sufficient to rule out many exotic phenomena in general relativity. Subject to certain conditions, we show that the ANEC can never be violated by a quantized minimally coupled free scalar field along a complete null geodesic surrounded by a tubular neighborhood in which the geometry is flat and whose intrinsic causal structure coincides with that induced from the full spacetime. In particular, the ANEC holds in flat space with boundaries, as in the Casimir effect, for geodesics which stay a finite distance away from the boundary.},
doi = {10.1103/PHYSREVD.75.025007},
journal = {Physical Review. D, Particles Fields},
number = 2,
volume = 75,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}
  • We show, on four-dimensional Minkowski spacetime, that {l angle}{psi}{vert bar}{ital T}{sub {mu}{nu}}{vert bar}{psi}{r angle}, the renormalized expectation value in a general quantum state {vert bar}{psi}{r angle} of the stress-energy tensor for electromagnetism, satisfies the averaged-null-energy condition, i.e., that {integral}{ital d}{lambda}{l angle}{psi}{vert bar}{ital T}{sub {mu}{nu}}{vert bar}{psi}{r angle}{ital t}{sup {mu}}{ital t{nu}}{ge}0 where this integral is along complete null geodesics with an affine parameter {lambda} and tangent vector {ital t}{sup {mu}}.
  • For a large class of quantum states, all local (pointwise) energy conditions widely used in relativity are violated by the renormalized stress-energy tensor of a quantum field. In contrast, certain nonlocal positivity constraints on the quantum stress-energy tensor might hold quite generally, and this possibility has received considerable attention in recent years. In particular, it is now known that the averaged null energy condition, the condition that the null-null component of the stress-energy tensor integrated along a complete null geodesic is non-negative for all states, holds quite generally in a wide class of spacetimes for a minimally coupled scalar field.more » Apart from the specific class of spacetimes considered (mainly two-dimensional spacetimes and four-dimensional Minkowski space), the most significant restriction on this result is that the null geodesic over which the average is taken must be achronal. Recently, Ford and Roman have explored this restriction in two-dimensional flat spacetime, and discovered that in a flat cylindrical space, although the stress energy tensor itself fails to satisfy the averaged null energy condition (ANEC) along the (nonachronal) 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 (globally hyperbolic) two-dimensional spacetime, using the same techniques as those we relied on to prove the ANEC in an earlier paper with Wald. I begin with an overview of averaged energy conditions in quantum field theory.« less
  • It is by now well known that the standard local (i.e., pointwise) energy conditions always can be violated in quantum field theory in curved (and flat) spacetime, even when these energy conditions hold for the corresponding classical field. Nevertheless, some global constraints on the stress-energy tensor may exist. Indeed recent work has shown that the averaged null energy condition (ANEC), which requires the positivity of energy suitably averaged along null geodesics, holds for a wide class of states of a minimally coupled scalar field on Minkowski spacetime, and also (in the massless case) on a wide class of states inmore » curved two-dimensional spacetimes satisfying certain asymptotic regularity properties. In this paper, we strengthen these results by proving that, for the massless scalar field in an arbitrary globally hyperbolic two-dimensional spacetime, the ANEC holds for all Hadamard states along any complete, achronal null geodesic. In our analysis, the general, algebraic notion of state'' is used, so, in particular, it is not even assumed that our state belongs to any Fock representation. Our proof shows that the ANEC is a direct consequence of the general positivity condition which must hold for the two-point function of any state. Our results also can be extended (with a restriction on states) to the massive scalar field in two-dimensional Minkowski spacetime and (with an additional restriction on states) to the (massless or massive) minimally coupled scalar field on four-dimensional Minkowski spacetime. In the case of a (curved) four-dimensional spacetime with a bifurcate Killing horizon, our proof also extends to establish the ANEC for the null geodesic generators of the horizon (provided that there exists a stationary Hadamard state of the field). This latter result implies that the ANEC must hold for the massive Klein-Gordon field in de Sitter spacetime.« less
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  • The averaged null energy condition (ANEC) requires that the integral over a complete null geodesic of the stress-energy tensor projected onto the geodesic tangent vector is never negative. This condition is sufficient to prove many important theorems in general relativity, but it is violated by quantum fields in curved spacetime. However there is a weaker condition, which is free of known violations, requiring only that there is no self-consistent spacetime in semiclassical gravity in which ANEC is violated on a complete, achronal null geodesic. We indicate why such a condition might be expected to hold and show that it ismore » sufficient to rule out closed timelike curves and wormholes connecting different asymptotically flat regions.« less