How unlike real elevator (Rindler field) is field from planar mass?

Neil Bates

For years I thought that, except for tidal field and the g = -(c^2)/Z
issue, the essentially "uniform" gravity field near a wide (approx. to
"infinite") planar mass sheet would be just like that of the
accelerating elevator of historic fame in GR. I thought I could use
whatever happened, local neighborhood, in the Rindler field (RF) to
predict what would happen in the field (PF) of the planar mass. Well,
after discussions in a Cosmic Variance thread
(http://cosmicvariance.com/2007/11/13...tum-mechanics/)
with the apparently knowledgeable Greg Egan, that assumption is blown.
In particular, I thought: what could be more assured, than in a uniform
PF, everything would fall at the same rate no matter what its
circumstance. Not just composition etc., but lateral motion as well. A
horizontally traveling bullet hits the ground the same time as a dropped
body, we expect from naive Equivalence Principle. That should apply to
a light beam as well, the double deflection around a radial field being
due to extra "space curvature" effect and not some inherent distinction.

But, Greg says: No, the RF and PF metrics are not the same, and it
matters for *local* experiments. The lab-frame acceleration of a body
in rapid transverse motion in a PF is: g(moving) = g(1 + v^2/c^2). I
thought, WTF?! He says:

Greg Egan: "In the Rindler "elevator", transverse motion is just an
extra degree of freedom that has no effect whatsoever in the Z
direction. In the curved space-time near a planar mass, the geometry is
sliced differently by world lines with different transverse velocities."

That just seems weird to me, but if it is, it is. I still worry about
one thing: energy conservation. He says the change in transverse
velocity compensates so as to get correct impact energy during a fall.
But I had a proposal, which may or may not have the apparent weight
corrected appropriately:

Neil Bates: "... Let's have constrained matter flowing rapidly in a
circular path on the floor. Let's assume, from f = ma in the proper
frame of the mass particles (there should be a consistent relation
between what free acceleration is seen, with force in the constrained
case), that the lab force is now increased to gamma(1+v^2)* m_0*g
instead of the gamma* m_0*g that we expect from increased effective
mass-energy. Now lets lower a shelf holding this ring, extracting work
according to W = - f dot delta h. Then we slow down the ring current,
leaving the same mass energy as before (but now same as rest mass.) Then
we could raise the ring using only gamma* m_0*g delta h, giving us "free
energy" of gamma * v^2 * m_0*g delta h. ..."

Later, I conjectured:

" ... is the correction for free fall equivalent to an effective
"weight" of just gamma*m_0? If not, I don't see how an effective mass of
(1 + v^2) could square with conservation during raising and lowering.
Wow, losing the "elevator" as game for whatever I wanted to think about
is really disappointing (and low to middle-brow science education
doesn't [warn] about this much - I just didn't get into gravity at top
level. ..."

So, what's the scoop on this issue? Why don't we hear more about it (or
am I not looking in all the right places ...)? What else can't I trust
RF to PF equivalence on? Finally, do weak gravitational fields (i.e.,
their effects in every way, including the peculiar velocity-dependent
acceleration) obey superposition, basically? (Like, an effect of
magnitude 3 and one of magnitude 5 might add to 7.99 or 8.02 etc., but
not way off like adding to two or 16 etc.)

tx

Greg's page about gravity etc:

http://gregegan.customer.netspace.ne....html#CONTENTS

Other references to this issue:

http://www.mathpages.com/home/kmath530/kmath530.htm
http://arxiv.org/pdf/gr-qc/0503092
http://arxiv.org/PS_cache/arxiv/pdf/...708.2906v1.pdf