Can 'negative energy' be created by the Casimir Effect?

Question

In several papers about Time machines I have read that the Casimir effect can be used to create negative energy, so the Alcubierre-device or wormholes could be produced.

How can negative energy be created?

Answer 1

The energy density between the Casimir conductors can indeed be positive or negative. The calculation of Casimir energies is often done by noting that the plates impose boundary conditions on the field modes that can exist between them. Therefore in the presence of the plates, a more restricted set of modes is allowed than would be the case if the plates were absent. In the vacuum, each mode $k$ contributes and energy $\frac{1}{2}\omega_k$ to the energy, so plates vs no plates gives different vacuum energy values.

Both cases, plates, and no-plates, of course give infinite answers so some regulation method must be introduced in order to give a finite answer for the difference in energies. The calculation is "reasonably" straightforward for the case of parallel plates, and leads to $$ \mathcal{E} \propto -\frac{1}{L^4} $$ where $L$ is the plate separation. This gives a negative pressure (i.e. attraction) between the plates. The calculation is illustrated in Elizalde & Romeo "Essentials of the Casimir Effect and its Computation" Am J Phys $\bf{59}$ 8 (1991), available here.

If, instead of flat plates, the calculation is done for the two halves of a conducting spherical shell, this time the pressure is positive (repulsive), so the behaviour you get depends strongly on the geometry of your plates.

More interesting from your viewpoint is the interaction between the Casimir energy and the gravitational field. To work this out, you need the energy momentum tensor for the Casimir region. Given this, you can compute (semiclassically) the gravitational effect by taking its expectation value $\langle T_{\mu \nu}\rangle$.