Quantum stress-energy tensors and the weak energy condition

Certain processes predicted by quantum field theory, such as the Hawking black-hole evaporation process and radiation by moving mirrors, involve stress-energy tensors which exhibit peculiar properties from the classical point of view. More specifically, these stress-energy tensors do not obey the weak energy condition because they involve negative-energy densities and we show that, as a result, they are nondiagonalizable by a local Lorentz transformation under certain circumstances. In addition, we show that $T_{\mathrm{ab}}$ $U^a$ $U^b$ is not bounded below for all unit timelike vectors $U^a$ and that this is also a property of the stress-energy tensor associated with the Casimir effect. These observations are important in view of the fact that Tipler has shown that if $T_{\mathrm{ab}}$ is diagonalizable (type I) and if $T_{\mathrm{ab}}$ $U^a$ $U^b$ is bounded below, then the weak energy condition is the weakest energy condition that can be defined locally. One might conjecture that the existence of similar (although as yet unknown) quantum processes, in which the weak energy condition is violated locally, could prevent the eventual formation of a singularity in the gravitational collapse of a star. Although we do not present a specific model, it is possible that in such a process the weak energy condition, while violated locally, would still hold on the average. Extending earlier results of Tipler, we show that Penrose’s singularity theorem will still hold if the weak energy condition is replaced by a weaker (nonlocal) energy condition and if the null generic condition holds.