p2

The resulting spacetime is everywhere Riemann-flat except possibly at the throat. Also, the stressenergy
tensor in this spacetime is concentrated at the throat with a δ-function singularity there. This is a
consequence of the fact that the spacetime metric at the throat is continuous but not differentiable, while
the connection is discontinuous; thus causing the Riemann curvature to possess a δ-function singularity
(causing undesirable gravitational tidal forces) there. The magnitude of this δ-function singularity can be
calculated in terms of the second fundamental form on both sides of the throat, which we presume to be
generated by a localized thin shell of matter-energy. The second fundamental form represents the
extrinsic curvature of the ∂Ω hypersurface (i.e., the wormhole throat), telling how it is curved with respect
to the enveloping four-dimensional spacetime. The form of the geometry is simple, so the second
fundamental form at the throat is calculated to be (McConnell, 1957):
K
i ±
j
⎛κ
0 0 ⎞
0
⎜ ⎟
=± ⎜
0 κ1
0 ⎟
⎜ 0 0 κ ⎟
⎝
2 ⎠
⎛0 0 0
⎜
⎞
⎟
=± ⎜
0 1 ρ1
0 ⎟
⎜0 0 1 ρ ⎟
⎝
2 ⎠
(2.3),
where i,j = 0,1,2 and K i j ± is the second fundamental form. The full 4×4 matrix K α β has been reduced to
3×3 form, as above, for computational convenience because the thin shell (or hypersurface) is essentially
a two-surface embedded in three-space. The overall ± sign in equation (2.3) comes from the fact that a
unit normal points outward from one side of the surface and points inward on the other side. We hereafter
drop the ± sign for the sake of brevity in notation. The quantities κ 0 , κ 1 , and κ 2 measure the extrinsic
curvature of the thin shell of local matter-energy (i.e., the stuff that induces the wormhole throat
geometry). Since the wormhole throat is a space-like hypersurface, we can exclude time-like
hypersurfaces and their components in the calculations. Therefore we set κ 0 = 0 in equation (2.3) because
it is the time-like extrinsic curvature for the time-like hypersurface of the thin shell of matter-energy. As
seen in equation (2.3) κ 1 and κ 2 are simply related to the two principal radii of curvature ρ 1 and ρ 2
(defined to be the eigenvalues of K i j) of the two-dimensional spacelike hypersurface ∂Ω (see Figure 3). It
should be noted that a convex surface has positive radii of curvature, while a concave surface has negative
radii of curvature.
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