General proof of the averaged null energy condition for a massless scalar field in two-dimensional curved spacetime

Abstract

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 in 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 twodimensional 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.

Local Thermal Equilibrium States and Quantum Energy Inequalities

“…In this case, we call θ the maximal solution of (56) defined by the initial condition, and refer to (a, b) as the maximal domain. The following statement is a variation on a similar result in [42], and it uses a very similar argument, the main difference being that the assumption (58) here is slightly different from that in [42], where the integral is taken over a semi-axis. Note also that our parameter λ corresponds to 1/λ in the notation of [42].…”

“…The validity of (39) is therefore of importance for the properties of the spacetime structure of solutions to the semiclassical Einstein equations. It has been argued in [42] that condition (39) may be replaced by the following condition:…”

“…More precisely, in [42] it has been shown that (42) and (40) imply that the expansion of a congruence of lightlike geodesics around γ becomes singular along γ (in the sense of diverging to −∞ at a finite value of the affine parameter) unless it vanishes identically on γ. (In [42] this argument is given for half-line geodesics, but it carries over to the case at hand as will be shown in our Appendix A. )…”

Averaged null energy condition in spacetimes with boundaries

“…One may also obtain the ANEC as a limiting case of the averaged weak energy condition (AWEC) 2 [which holds in Minkowski space as a consequence of Quantum Energy Inequalities] [11] by considering the null geodesic as a limit of timelike curves; the relevant class of states is not so clearly defined here and quite strong conditions are required at infinity to make this work (see [12] for some comments in this direction). A more general and quite technical argument [13], using techniques of algebraic quantum field theory, establishes the ANEC in four-dimensional Minkowski space for a restricted class of Hadamard states of the minimally coupled scalar field. In particular, this goes beyond the states contained in the usual Fock space.…”

“…In general globally hyperbolic two-dimensional spacetimes, [13] established the ANEC on complete achronal null geodesics for the minimally coupled free scalar field. This holds for arbitrary Hadamard states in the massless case, and a restricted class of Hadamard states 1 The condition used in [1] is in fact the NEC, but one can see that is sufficient to have the ANEC with the average over the path to be traveled.…”