Plate with a hole obeys the averaged null energy condition

The negative energy density of Casimir systems appears to violate general relativity energy conditions. However, one cannot test the averaged null energy condition (ANEC) using standard calculations for perfectly reflecting plates, because the null geodesic would have to pass through the plates, where the calculation breaks down. To avoid this problem, we compute the contribution to ANEC for a geodesic that passes through a hole in a single plate. We consider both Dirichlet and Neumann boundary conditions in two and three space dimensions. We use a Babinet's principle argument to reduce the problem to a complementary finite disk correction to the perfect mirror result, which we then compute using scattering theory in elliptical and spheroidal coordinates. In the Dirichlet case, we find that the positive correction due to the hole overwhelms the negative contribution of the infinite plate. In the Neumann case, where the infinite plate gives a positive contribution, the hole contribution is smaller in magnitude, so again ANEC is obeyed. These results can be extended to the case of two plates in the limits of large and small hole radii. This system thus provides another example of a situation where ANEC turns out to be obeyed when one might expect it to be violated