AAWSAP DIRD, Antigravity for Aerospace Applications, March 2010

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are not electromagnetic in nature; that is to say they are non-Maxwellian, and so the
perfect-conductor boundary conditions do not apply to them. It turns out that complete
manifolds exhibit what is called the topological Casimir effect for any non-Maxwellian
fields. In order to define boundary conditions for other fields one replaces the conductor
boundary conditions and Minkowski spacetime by a manifold of the form mx I (that is,
a product space), where mis the real line defining the time dimension for this particular
product space and I is a flat 3-dimensional manifold having any one of the following
topologies: m2 x S1, mx T2, T3, mx K2, and so forth, mbeing the real line that defines
any linear space dimension (for example, m= line, m2 = 2-dimensional plane), Tn being
the n-torus, K2 the 2-dimensional Klein bottle, S1 the circle, and so forth.
The case r = m2 x S1 has the closest resemblance to the electromagnetic Casimir effect,
the difference being that instead of imposing conductor boundary conditions, one
imposes periodic boundary conditions on some of the space coordinates in the 3­
dimensional manifold. When imposing this topological constraint on the field theoretic
calculation of the topological Casimir effect (for linear massless fields), one finds that
the generic expression for the energy density is also PcE = -Ahc I d4 , where
A= ±dr( n 2 I 90), dr is the number of degrees of freedom (for example, helicity states) per
spatial point, the plus sign holds for boson fields (giving a negative energy density) and
the negative sign for fermion fields (giving a positive energy density).
If one were to admit spin structure in the manifolds described above and the field is
spinorial, then there is another important subtlety that must be taken into account
when evaluating TV~; . However, this introduces an additional complexity involving the
relationship between the spin structure and the global structure (that is, the
configuration space or fiber bundle) of the field in question whereby the topology not
only of the base manifold, but of the fiber bundle itself has an effect on r.~; . In addition
to this, there are (compactified) extra-space dimensional quantum field (that is, D­
Brane or "brane world") analogs of the Casimir effect yet to be explored. But a detailed
consideration of these is beyond the scope of this report and will be left for future
investigation.
As a final note, one points out that the methods used to obtain the electromagnetic r;i.;
between parallel plane conductors can also be used when the conductors are not
parallel but are joined together along a line of intersection . If the conductors have
curved surfaces instead, then one obtains results that are similar to the case of
intersecting conductors. These geometries have also been evaluated for the case of
dielectric media. These particular cases will not be considered further since there are
technical subtleties involved that complicate the calculations and application of the
different approaches. This topic will also be left for future investigation.
DYNAMICAL CASIMIR EFFECT: MOVING MIRRORS
Negative energy can be created by a single moving reflecting (conducting) surface
(a.k.a. a moving mirror) . A mirror moving with increasing acceleration generates a flux
of negative energy that emanates from its surface and flows out into the space ahead
of the mirror (Reference 37,90). See Figure 5 (below) for an illustration of this effect.
This is essentially the simple case of an infinite plane conductor undergoing acceleration
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