Does back reaction enforce the averaged null energy condition in semiclassical gravity?

Abstract

The expectation value $〈{T}_{\mathrm{ab}}〉$ of the renormalized stress-energy tensor of quantum fields generically violates the classical, local positive energy conditions of general relativity. Nevertheless, it is possible that $〈{T}_{\mathrm{ab}}〉$ may still satisfy some nonlocal positive energy conditions. Most prominent among these nonlocal conditions is the averaged null energy condition (ANEC), which states that $\ensuremath{\int}〈{T}_{\mathrm{ab}}〉{k}^{a}{k}^{b}d\ensuremath{\lambda}>~0$ along any complete null geodesic, where ${k}^{a}$ denotes the geodesic tangent, with affine parameter $\ensuremath{\lambda}$. If the ANEC holds, then traversable wormholes cannot occur. However, although the ANEC holds in Minkowski spacetime, it is known that the ANEC 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, ${G}_{\mathrm{ab}}=8\ensuremath{\pi}〈{T}_{\mathrm{ab}}〉$. In this paper, we investigate whether the ANEC holds for self-consistent solutions of the semiclassical Einstein equation. We study a free, linear, massless scalar field with arbitrary curvature coupling in the context of perturbation theory about the flat spacetime/vacuum solution, and we modify the perturbed semiclassical equations by the "reduction of order" procedure to eliminate spurious solutions. We also restrict attention to the limit in which the length scales determined by the state and metric are much larger than the Planck length. At first order in the metric and state perturbations, and for pure states of the scalar field, we find that the ANEC integral vanishes, as it must for any positivity result to hold. For mixed states, the ANEC integral can be negative. However, we prove that if we average the ANEC integral transverse to the geodesic, using a suitable Planck scale smearing function, a strictly positive result is obtained in all cases except for the trivial flat spacetime/vacuum solution. Similar results hold for pure states at second order in perturbation theory, when we additionally specialize to the situation where incoming classical gravitational radiation does not dominate the first-order metric perturbation. These results suggest---in agreement with conclusions drawn by Ford and Roman from entirely independent arguments---that if traversable wormholes do exist as self-consistent solutions of the semiclassical equations, they cannot be macroscopic but must be "Planck scale." In the course of our analysis, we investigate a number of more general issues relevant to doing perturbative expansions of the semiclassical equations off of flat spacetime, including an analysis of the nature of the semiclassical Einstein equation and of prescriptions for extracting physically relevant solutions. A large portion of our paper is devoted to the treatment of these more general issues.