Averaged Energy Conditions and Quantum Inequalities

TL;DR

Ford and Roman establish covariant links between quantum inequalities (QI) restricting negative energy and averaged energy conditions (AWEC/ANEC) for a massless scalar field. By introducing the difference $D\langle T_{\mu\nu}u^{\mu}u^{\nu}\rangle$ relative to the Casimir background in a 2D compactified spacetime, they derive QI bounds and AWEC/ANEC-type integrals that hold even when the background Casimir state violates these conditions, and show that in the $L\to\infty$ limit these reduce to standard 2D AWEC/ANEC and a 4D energy-density bound. They further obtain null-geodesic QIs and ANEC-type results for the difference, indicating robust constraints on negative energy along null paths. A covariant 4D bound on energy density for inertial observers demonstrates that arbitrarily large negative densities sustained for long times are forbidden. Together, these results connect two major approaches to energy-condition violations and suggest fundamental limitations on exotic spacetime constructions within semiclassical gravity.

Abstract

Connections are uncovered between the averaged weak (AWEC) and averaged null (ANEC) energy conditions, and quantum inequality restrictions on negative energy for free massless scalar fields. In a two-dimensional compactified Minkowski universe, we derive a covariant quantum inequality-type bound on the difference of the expectation values of the energy density in an arbitrary quantum state and in the Casimir vacuum state. From this bound, it is shown that the difference of expectation values also obeys AWEC and ANEC-type integral conditions. In contrast, it is well-known that the stress tensor in the Casimir vacuum state alone satisfies neither quantum inequalities nor averaged energy conditions. Such difference inequalities represent limits on the degree of energy condition violation that is allowed over and above any violation due to negative energy densities in a background vacuum state. In our simple two-dimensional model, they provide physically interesting examples of new constraints on negative energy which hold even when the usual AWEC, ANEC, and quantum inequality restrictions fail. In the limit when the size of the space is allowed to go to infinity, we derive quantum inequalities for timelike and null geodesics which, in appropriate limits, reduce to AWEC and ANEC in ordinary two-dimensional Minkowski spacetime. We also derive a quantum inequality bound on the energy density seen by an inertial observer in four-dimensional Minkowski spacetime. The bound implies that any inertial observer in flat spacetime cannot see an arbitrarily large negative energy density which lasts for an arbitrarily long period of time.