On Dec 3, 2004, at 8:55 PM, wrote:
"So Jack -- what is the exact GR metric for a *uniformly accelerating
frame of reference*?"
Uniform gravity fields do not exist. They are only useful approximate
fictions!
In the Galilean limit low speed limit it's the one I originally wrote.
ds^2 = -(1 - (gt'/c)^2)(cdt')^2 + 2(gt'/c)dz'(cdt') + dx'^2 + dy'^2 + dz'^2
= -(cdt)^2 + dx^2 + dy^2 + dz^2
where
c^2{LC}^z't't' = g
{LC}^ztt = 0
z' = z - (1/2)gt^2
x = x'
y = y'
t = t'
gt'/c << 1
gz'/c^2 << 1
There is an off-diagonal gravimagnetic term in gu'v' because in this
approximation we have absolute simultaneity. This gravimagnetic term
gets cancelled out as a special relativity effect and in fact it is no
longer possible to have a truly uniform artificial gravity inertial
force in globally flat spacetime. It is also not possible to have it
"sourced" from Tuv(source) =/= 0. Indeed, it is the presence of the
gravimagnetism here that permits the approximation of the uniform
g-field inertial force in the accelerated rocket frame. This comes from
the spacetime asymmetry of the low-speed limit. That asymmetry
disappears as we approach the light cone and the quaint Newtonian notion
of a uniform gravity field disappears with it because of the time
dilation of the speed of light barrier!
In the high speed limit you can use a suitably modified Rindler metric
as Smoot shows for an extended set of non-inertial observers. In both
cases the spacetime is globally flat Minkowski spacetime with NO SOURCES
i.e Tuv = 0, which is the POINT here. You mistakenly think Tuv =/= 0 for
that metric you swiped from Smoot without understanding its physical
meaning.
That is, when gt'/c -> 1
it is NOT possible to have
c^2{LC}^z't't' = g uniformly & constantly over a large spacetime region.
You have horizons and time dilation distortions of the Rindler metric as
George Smoot showed in detail.
The uniform constant gravity field is strictly a Galilean approximation
that cannot be extrapolated into the high speed realm of special relativity!
You have misinterpreted the physical meaning of the Rindler metric as
describing a curved spacetime with Tuv =/= 0 sources giving at least an
approximately uniform gravity field. That is not true.
Also, note you only see gravity fields, i.e. g-fields in non-inertial
off-geodesic LNIFs. One however measures tidal curvature local tensor
fields in LIFs using only geodesic test particles.
George Smoot at UCB in
http://aether.lbl.gov/www/classes/p139/homework/eight.pdf then claims
that different non-inertial point observers at z'(i) i = 1 to N at same
t', each having different local proper accelerations a(i) = c^2/z'(i),
relative to the local inertial (z,t) observers at the same objective
events P, will be at fixed relative distances to each other with
identical good running clocks remaining in synchrony with each
non-inertial observer reporting a weight per unit mass of g.
There are no Tuv sources here. The counter-intuitive artificial gravity
effects here are all inertial forces in globally flat Minkowski created
by the non-gravity forces of ejecting propellant. You could never attain
the Rindler metric if the ships were only using geodesic warp drive.
The OFF-GEODESIC non-inertial frame Rindler coordinates are t' and z',
where t and z are the inertial frame GEODESIC coordinates.
ct' = (c^2/g)tanh^-1(ct/z)
z' = [z^2 - (ct)^2]^1/2
The GLOBALLY FLAT Minkowski 1 + 1 metric
ds^2 = (cdt)^2 - dz^2 = [(gz'/c^2)^2(cdt')^2 - dz'^2
Tuv(source matter) = 0 here everywhere!
The world lines z' = constant for the off-geodesic NON-INERTIAL REST
FRAME are the hyperbolic world lines z = z(t)|z' in the z-t plane with
constant proper acceleration g in the NON-INERTIAL REST FRAME. The
latter only exists by a non-gravity force effect like the expulsion of
propellent in rocket engines along each world line for different fixed
choices of z' = constant in the family of hyperbolas z = z(t)|z'
.
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