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.

Figure 1

In free space (left) we have even and odd normal mode wave functions. In the presence of a Dirichlet plate (right), the even functions are replaced by the odd functions with a change of sign crossing the plate.

Figure 2

If there is a hole in the Dirichlet plate (left), the new even functions satisfy Neumann conditions in the hole and Dirichlet conditions elsewhere. The odd functions for a patch with the same shape as the hole (right) are the same except for a change of sign between sides.

Figure 3

Contributions to NEC in two dimensions (left) and three dimensions (right) for a Dirichlet plate with a hole of unit radius, as functions of distance along the axis passing through the center of the hole. Extrapolation is used for points at a distance less than $0.15$ in the left panel and $0.25$ in the right panel. The dotted lines show the perfect mirror result.

Figure 4

Contributions to NEC in two dimensions (left) and three dimensions (right) for a Neumann plate with a hole of unit radius, as functions of distance along the axis passing through the center of the hole. Extrapolation is used for points at a distance less than $0.11$ in the left panel and $0.25$ in the right panel. The dotted lines show the perfect mirror results.