More corrections

On Friday, August 22, 2003, at 03:22 PM, Jack Sarfatti wrote:

Corrections

On Friday, August 22, 2003, at 01:55 PM, Jack Sarfatti wrote:


As a toy model take a generic static spherically symmetric metric
without a black hole horizon since UFOs are obviously not black holes.

The static spherical symmetry assumptions are also no good except as a
very crude first approximation. Alcubierre's free float geodesic warp
drive metric is neither static nor spherically symmetric. The warp drive
is roughly similar to a self-accelerating (from the external observer's
POV) "dipole" with dark energy and dark matter exotic vacuum boundary
layers analogous to positive and negative charge in the electric case.
Of course, unlike exotic vacua dipoles, electric dipoles do not
self-accelerate. The dark energy is what Bondi, Terletski and Forward
called "negative matter" (Journal of Propulsion, Vol 6, No 1 pp 28 - 37
Feb 1990, Robert Forward)

The differential element of curved spacetime is then of the generic form

ds^2 = - e^2phi(r)(cdt)^2 + [1 - b(r)/r]^-1dr^2 + r^2d(spherical polar
angle line element)

phi(r) is the "red shift function", b(r) is the "shape function" of the
UFO, r is the usual Schwarzschild curvature coordinate not the isotropic
radial coordinate.

Suppose we have an exotic vacuum /\zpf boundary layer of thickness d
with zero point energy density field (c^4/8piG)/\zpf with /\zpf a
constant in the thin boundary layer.


I did that too fast. This choice is wrong. The square root is wrong.

Basically c^2/\zpf acts like Grho in Newtonian gravity and in GR.

So we need to solve the problem of a slab of area A thickness d.

The effective G is very strong of course.

Hence phi ~ (/\zpfAd/r)f(d/r, angles)

where d ~ 10^-4 cm is my guess and f is essentially a multipole expansion.

The angular dependence destroys the spherical symmetry of course.

For a signal originating inside the exotic vacuum boundary layer, the
effective r is the spatial coherence length L of the signal.

Assume a flat metric outside the exotic vacuum boundary layer

For r d this happens automatically and smoothly because of the
exponential dependence of the redshift function on d/r

Therefore, the shift in proper time dtau (period of oscillation of the
signal wave) comparing signals from the exotic vacuum region /\zpf =/= 0
to the normal vacuum region /\zpf = 0 where your tape recorder is
located is (within the crude spherical symmetry approximation)

dtau (exotic vacuum)/dtau(normal vacuum) = e^2 (/\zpfAd/L)

where d/r 1 in the normal vacuum where the detector is located to
receive the signal from the exotic vacuum boundary layer at the UFO.

For a geometrodynamic universal red shift of all wave signals of any
physical propagating energy field from metrically engineered space-time
warping

dtau (exotic vacuum)/dtau(normal vacuum) 1 corresponding to the "dark
matter" /\zpf 0 at the bow of the UFO

On the other hand, at the stern of the UFO where we have "dark energy"
/\zpf 0 we get a universal blue shift.

Remember, the dark energy universally repels, the dark matter
universally attracts. Put the two together in the right way and you get
a net unidirectional timelike geodesic motion. Objects in the UFO are in
weightless free float, though to the outside observer the object appears
to accelerate and to traverse distances with effectively superluminal
average speed. The former was shown by Bondi and Terletskii, apparently
independently, some 40 years ago. The latter was shown by Alcubierre
about 10 years ago.

Alcubierre's warp drive metric (Class Quantum Gravity 11 (1994)

is, for vacuum propeller self-motion along the x axis where the UFO
position is xs(t)

ds^2 = - (A^2 - BiB^i)(cdt)^2 + 2Bidx^idt + gijdx^idx^j

i,j = x,y,z

Note the non-static cross term 2Bidx^idt

A = 1 is the free float timelike geodesic condition for the lapse function

Bx = -vs(t)f(rs(t))

By = Bz = 0

vs(t) = dxs/dt

rs(t) = [(x - xs(t))^2 + y^2 + z^2]^1/2

f(rs(t)) = [tanhZ(rs + d) - tanhZ(rs - d)]/2tanh(Zd)

d, Z arbitrary parameters, but d is essentially the thickness of the
exotic vacuum boundary layer

The metric then becomes

ds^2 = - (cdt)^2 + (dx - vxf(rs)dt)^2 + dy^2 + dz^2

That must be made compatible as a solution of the exotic vacuum field
equation

Guv + /\zpfguv = 0

The basic "dipole" distribution of exotic vacuum field /\zpf is shown in
a computer drawing of trace of extrinsic spatial curvature in
Alcubierre's original paper

where essentially /\zpf ~ trace of the extrinsic spatial curvature

to be continued