AAWSAP DIRD, Traversable Wormholes, Stargates, and Negative Energy, April 2010

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UNCLASSIFIED/ fFOA OFFI&il.t..k Y&& 8Ptk\f
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S = - - c- (K' -8'K k)
(2)
J
i
J
k
4rrG
where the second fundamental form K ;i is a matrix that represents the extrinsic
curvature of an (telling how the wormhole throat is curved with respect to the
enveloping four-dimensional spacetime), Sii is the three-dimensional unit matrix, and
K \ is the trace (sum of diagonal matrix elements) of KV K ;i is a diagonal matrix
having the two principal radii of curvature, p1 and p2, of the thin shell as its components
(see Figure 5). S ;i may be interpreted in terms of the thin shell's surface energy density
cr and principal surface tensions, S1 and S2, which are also diagonal matrix components.
thin shell of mass-energy
pl
Figure 5. A Thin Shell of (Localized) Mass-Energy Possessing Two Principal Radii of Curvature, p1 and
p2.
Equation (2) is solved and the components of S ;i are found to be (Reference 8):
(3a)
3 The Latin indices (i, j, k = 0...2) denote three-dimensional hypersurface coordinates, ><° ...x2, such that x1, x2"'
space coordinates and><° = time coordinate.
UNCLASSIFIEQ { {FOR OFFICIAL Y&& 8HLY
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