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Does backreaction enforce the averaged null energy condition in semiclassical gravity?

Eanna Flanagan, Robert Wald

TL;DR

This work investigates whether backreaction in semiclassical gravity enforces the averaged null energy condition (ANEC). By performing a perturbative analysis about Minkowski spacetime for a free massless scalar with arbitrary curvature coupling and applying Simon's reduction of order, the authors show that although the ANEC integral along a geodesic can be negative, a transversely smeared ANEC integral with Planck-scale width is strictly nonnegative for nearly all physically reasonable perturbations. The first-order ANEC vanishes for pure incoming states, necessitating a second-order analysis in which smeared ANEC remains positive in the long-wavelength regime; this supports Ford and Roman's intuition that macroscopic traversable wormholes cannot arise in semiclassical gravity without Planck-scale structure. The results illuminate how nonlocal energy constraints arise in semiclassical gravity and clarify the treatment of higher-derivative pathologies in the semiclassical Einstein equation.

Abstract

The expected stress-energy tensor <T_{ab}> of quantum fields generically violates the local positive energy conditions of general relativity. However, <T_{ab}> may satisfy some nonlocal conditions such as the averaged null energy condition (ANEC), which would rule out traversable wormholes. Although ANEC holds in Minkowski spacetime, it can be violated in curved spacetimes if one is allowed to choose the spacetime and quantum state arbitrarily, without imposition of the semiclassical Einstein equation G_{ab} = 8 π<T_{ab}>. In this paper we investigate whether ANEC holds for solutions to this equation, by studying a free, massless scalar field with arbitrary curvature coupling in perturbation theory to second order about the flat spacetime/vacuum solution. We "reduce the order" of the perturbation equations to eliminate spurious solutions, and consider the limit in which the lengthscales determined by the incoming state are much larger than the Planck length. We also need to assume that incoming classical gravitational radiation does not dominate the first order metric perturbation. We find that although the ANEC integral can be negative, if we average the ANEC integral transverse to the geodesic with a suitable Planck scale smearing function, then a strictly positive result is obtained in all cases except for the flat spacetime/vacuum solution. This result suggests --- in agreement with conclusions drawn by Ford and Roman from entirely independent arguments --- that if traversable wormholes do exist as solutions to the semiclassical equations, they cannot be macroscopic but must be ``Planck scale''. A large portion of our paper is devoted to the analysis of general issues concerning the nature of the semiclassical Einstein equation and of prescriptions for extracting physically relevant solutions.

Does backreaction enforce the averaged null energy condition in semiclassical gravity?

TL;DR

This work investigates whether backreaction in semiclassical gravity enforces the averaged null energy condition (ANEC). By performing a perturbative analysis about Minkowski spacetime for a free massless scalar with arbitrary curvature coupling and applying Simon's reduction of order, the authors show that although the ANEC integral along a geodesic can be negative, a transversely smeared ANEC integral with Planck-scale width is strictly nonnegative for nearly all physically reasonable perturbations. The first-order ANEC vanishes for pure incoming states, necessitating a second-order analysis in which smeared ANEC remains positive in the long-wavelength regime; this supports Ford and Roman's intuition that macroscopic traversable wormholes cannot arise in semiclassical gravity without Planck-scale structure. The results illuminate how nonlocal energy constraints arise in semiclassical gravity and clarify the treatment of higher-derivative pathologies in the semiclassical Einstein equation.

Abstract

The expected stress-energy tensor <T_{ab}> of quantum fields generically violates the local positive energy conditions of general relativity. However, <T_{ab}> may satisfy some nonlocal conditions such as the averaged null energy condition (ANEC), which would rule out traversable wormholes. Although ANEC holds in Minkowski spacetime, it can be violated in curved spacetimes if one is allowed to choose the spacetime and quantum state arbitrarily, without imposition of the semiclassical Einstein equation G_{ab} = 8 π<T_{ab}>. In this paper we investigate whether ANEC holds for solutions to this equation, by studying a free, massless scalar field with arbitrary curvature coupling in perturbation theory to second order about the flat spacetime/vacuum solution. We "reduce the order" of the perturbation equations to eliminate spurious solutions, and consider the limit in which the lengthscales determined by the incoming state are much larger than the Planck length. We also need to assume that incoming classical gravitational radiation does not dominate the first order metric perturbation. We find that although the ANEC integral can be negative, if we average the ANEC integral transverse to the geodesic with a suitable Planck scale smearing function, then a strictly positive result is obtained in all cases except for the flat spacetime/vacuum solution. This result suggests --- in agreement with conclusions drawn by Ford and Roman from entirely independent arguments --- that if traversable wormholes do exist as solutions to the semiclassical equations, they cannot be macroscopic but must be ``Planck scale''. A large portion of our paper is devoted to the analysis of general issues concerning the nature of the semiclassical Einstein equation and of prescriptions for extracting physically relevant solutions.

Paper Structure

This paper contains 37 sections, 290 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction and Summary
  2. Brief overview of the issues addressed here
  3. Nonlocal constraints on the stress-energy tensor
  4. Brief summary of our assumptions and results
  5. Conventions
  6. semiclassical gravity with backreaction
  7. The classical equations
  8. The status of the semiclassical equations and the local curvature ambiguity
  9. The semiclassical equations
  10. Derivation of the equations
  11. Pure states versus mixed states
  12. Gauge freedom and transformations
  13. The linearized stress tensor and the explicit form of the first order perturbation equation
  14. Explicit form of the second order perturbation equation in the "long-wavelength" limit
  15. Dealing with the pathological solutions: Simon's prescription(s).
  16. ...and 22 more sections

Figures (3)

  • Figure : A summary of our results for the ANEC integral in the different cases. "Pure to first order" indicates that the two point function of the scalar field is pure to first order in $\varepsilon$. "Always $>0$" means that the smeared ANEC integral is always strictly positive for all solutions of the equations except for the trivial, flat spacetime/vacuum solution.
  • Figure : A summary of our results for the ANEC integral in the different cases. "Pure to first order" indicates that the two point function of the scalar field is pure to first order in $\varepsilon$. "Always $>0$" means that the smeared ANEC integral is always strictly positive for all solutions of the equations except for the trivial, flat spacetime/vacuum solution.
  • Figure : A summary of our results for the ANEC integral in the different cases. "Pure to first order" indicates that the two point function of the scalar field is pure to first order in $\varepsilon$. "Always $>0$" means that the smeared ANEC integral is always strictly positive for all solutions of the equations except for the trivial, flat spacetime/vacuum solution.