## Friday, December 03, 2004

Uniform gravity fields do not exist. They are useful approximate
fictions!

On Dec 3, 2004, at 8:55 PM, iksnileiz@earthlink.net wrote:

"So Jack -- what is the exact GR metric for a *uniformly accelerating
frame of reference*?"

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'