DOES BACK-REACTION ENFORCE THE AVERAGED NULL ENERGY CONDITION IN SEMICLASSICAL GRAVITY

The expectation value [T-ab] of the renormalized stress-energy tensor of quantum fields generically violates the classical, local positive e nergy conditions of general relativity. Nevertheless, it is possible t hat [T-ab] may still satisfy some nonlocal positive energy conditions. Most prominent among these nonlocal conditions is the averaged null e nergy condition (ANEC), which states that integral[T-ab]k(a)k(b)d lamb da greater than or equal to 0 along any complete null geodesic, where k(a) denotes the geodesic tangent, with affine parameter lambda. If th e ANEC holds, then traversable wormholes cannot occur. However, althou gh the ANEC holds in Minkowski spacetime, it is known that the ANEC ca n be violated in curved spacetimes if one is allowed to choose the spa cetime and quantum state arbitrarily, without imposition of the semicl assical Einstein equation, G(ab)=8 pi[T-ab]. In this paper, we investi gate whether the ANEC holds for self-consistent solutions of the semic lassical Einstein equation. We study a free, linear, massless scalar f ield with arbitrary curvature coupling in the context of perturbation theory about the flat spacetime/vacuum solution, and we modify the per turbed semiclassical equations by the ''reduction of order'' procedure to eliminate spurious solutions. We also restrict attention to the li mit 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 s tate perturbations, and for pure states of the scalar field, we find t hat the ANEC integral vanishes, as it must for any positivity result t o hold. For mixed states, the ANEC integral can be negative. However, we prove that if we average the ANEC integral transverse to the geodes ic, using a suitable Planck scale smearing function, a strictly positi ve result is obtained in all cases except for the trivial hat spacetim e/vacuum solution. Similar results hold for pure states at second orde r in perturbation theory, when we additionally specialize to the situa tion where incoming classical gravitational radiation does not dominat e the first-order metric perturbation. These results suggest-in agreem ent with conclusions drawn by Ford and Roman from entirely independent arguments-that if traversable wormholes do exist as self-consistent s olutions of the semiclassical equations, they cannot be macroscopic bu t must be ''Planck scale.'' In the course of our analysis, we investig ate a number of more general issues relevant to doing perturbative exp ansions of the semiclassical equations off of flat spacetime, includin g an analysis of the nature of the semiclassical Einstein equation and of prescriptions for extracting physically relevant solutions. A larg e portion of our paper is devoted to the treatment of these more gener al issues.