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Re: Equivalence Principle



John Mallinckrodt asked:
... . Can any other impartial observer (Paul?,
David? ...) confirm for me the validity of Desloge's analysis and,
if so, possibly give me a way of understanding, on more conceptual
grounds, how one might understand the distinction between uniform
gravity and uniform acceleration?

Sorry, I don't think I can be of much help here since I have not seen
The paper in question. My guess would be that the argument is as follows:

1. A uniform gravitational field can (using Cartesian-like harmonic
coordinates) be represented by a Levi-Civita connection that is constant
over all of spacetime. For example, a uniform Newtonian g-field has (in
such a coordinate system) connection coefficients that vanish unless the
component's upper (contravariant) index is spatial and its two lower
(covariant) indices both refer to time. Thus of the 4^3 components of the
connection they all vanish except three of them which are just the
(negative) three components of the Newtonian 3-vector g-field. These 3
nonzero components are, by hypothesis, constant.

2. The equation that determines the Riemann curvature tensor R in terms of
the connection involves terms that are linear in the first derivatives of
the connection coefficients *and* terms which are quadratic in the
connection coefficients -- yet without any derivatives taken. Thus if we
have a connection whose coefficients (in a given coordinate system) are
constant then the contribution to the R tensor from the derivative terms
must be zero. This does not necessarily mean, however, that the
contribution from the quadratic terms vanishs since no derivative is taken
for them. Thus it is possible to have a nonzero R tensor (i.e. a true
gravitational field existing) and yet have a connection which is everywhere
constant (i.e. a uniform gravitational field).

A problem I see with this analysis is that if one applies it to the case
of the constant uniform Newtonian 3-vector g-field above and we use the
corresponding connection to evaluate the Riemann R tensor we discover that
all of that tensor's coefficients vanish anyway since the quadratic terms
for R involving products of nonzero connection coefficients are always
matched up in such a way that the nonzero factors are multiplied by other
coefficients that are zero. Thus the uniform Newtonian g-field case
above *does* give a zero R tensor and *is* equivalent to a uniformly
accelerated coordinate system in a flat spacetime.

My guess, therefore, is that Desloge's paper considers other more
complicated cases where the Levi-Civita connection is globally constant
and yet yields a nonzero Riemann tensor. In such a case the corresponding
configuration of the gravitational field, expressed in terms of the
connection, cannot be thought of as a simple Newtonian-like gravitational
g-type 3-vector field and involves contributions to the connection that
have no simple 3-vector analog.

David Bowman
dbowman@gtc.georgetown.ky.us