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

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There is no singularity of infinitely collapsed matter residing at the wormhole throat.
These requirements then lead us to define a spherically symmetric Lorentzian
spacetime metric, ds2,2 that prescribes the required traversable wormhole geometry
(Reference 1, 3):
ds2 = - e2lf!<rlc 2dt2 +[1-b(r)/rr' dr2 + r2d02
(1)
where standard spherical-polar coordinates are used (r: 2nr = circumference; 0 s 0 s n;
0 s rps 2n), tis time (-oo < t < oo), d<E>2 = dff- + sin2Bdq}, r/1,.r) is the freely specifiable
redshift function that defines the proper time lapse through the wormhole throat, and
b(r) is the freely specifiable shape function that defines the wormhole throat's spatial
(hypersurface) geometry. The throat is spherically shaped. There are a large number of
variations of Equation ( 1), which define traversable wormholes having different
properties. The reader should consult (Reference 3) for further details. By inserting
Equation (1) into the Einstein field equation and cranking through the math, one can
derive the density and flux of energy and momentum (a.k.a. pressure) encoded by T,,v
for the source of matter that is required to produce the traversable wormhole. The
results show that the source of matter must have zero or negative energy density
and/or an outward radial tension (negative pressure) that is larger than the magnitude
of the energy density (Reference 1-3). Travelers moving through the throat at very
high speed will tend to measure a negative energy density. These exotic properties are
required to create and thread open the wormhole, and stabilize it against collapse (see
Section III for more details).
The technical description of a trip through a spherically symmetric traversable
wormhole is simply given by the proper time and/or the proper distance of travel
through its throat as measured by space travelers, while the (radial) travel velocity
through the throat is v = v(r) < c. The proper time of travel as measured by space
travelers going through the wormhole is given by /J., = f(yv)-1dA, where y = [1 ­
(v/c)2J-112 and the integration (over the element of proper distance, d1c) is taken from
the wormhole entrance to its exit. The proper distance of travel as measured by the
space travelers is /J.1c = v!J.,. Remote static observers watching the space travelers go
through the wormhole will measure their travel time to be M = f(ve <P('"l)-1d). and their
travel distance will be t,,.). = vM, where the integration is taken over the same limits as
before.
2 A spacetime metric, ds2, is a Lorentz-invariant distance function between any two points in spacetime that is
defined by ds2 = gµ,dX1'dx '', where 9w is the metric tensor which is a 4x4 matrix that encodes the geometry of
spacetime and dxµ is the infinitesimal coordinate separation between two points.
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