Evans wrote on:
http://www.atomicprecision.com/blog/2006/09/28/some-comments-on-gauge-invariance-in-ece/
". . .
Cartan geometry is not invariant under change of teread, as asserted
by the so called “reviewer” of the aforegoing message. For example, the
metric is the inner product of two tetrads, and if the tetrad is changed,
the metric is changed. In general relativity, the tetrad is a physically
meaningful field. It will be interesting to see if any kind of sensible
response is obtained from George Ellis. The battle for ECE theory has been
won resoundingly already, here I am seeing just how absurd the physics
establishment can get."
A tetrad at a point P of spacetime is a four-leg
consisting of four vectors ea (a=0,1,2,3) in the tangential space TP of the
manifold at P that are orthonormal wrt the metric (gμν)
at P, i.e. by representing the tetrad vectors
ea = eaμ ∂μ
by the basis vectors ∂μ (μ=0,1,2,3) of the local coordinate basis
we have the orthonormality relations:
gμν eaμebν = ηab
[C1;(3.118)]
where (ηab) = diag(−1, 1, 1, 1) is the Minkowski matrix.
Cf. [C1;(3.118)] and [C2;(J.5)].
Every local Lorentz transform (LLT) Λ=(Λa'a) changes the tetrad vectors but leaving the canonical form
of the metric unaltered:
ea → ea' = Λa'a ea
[C1;(3.125)]
Therefore we have
Λa'a Λb'b ηab =
ηa'b' .
[C1;(3.126)]
to obtain
ηa'b' eμa'eνb' =
Λa'a Λb'b ηab
eμa'eνb'
=
ηab
(Λa'a eμa')
(Λb'b eνb')
=
ηab
eμa eνb = gμν .
[C1] Sean M. Carroll, Lecture Notes on General Relativity, [C2] Sean M. Carroll, Spacetime and Geometry, Comment
Result:
The metric (gμν) (and hence the geometry of the manifold) is not afflicted
by a change of the (local)
tetrad due to an arbitrary LLT.
References
http://arxiv.org/pdf/gr-qc/9712019
Addison & Wesley, ISBN 0-8053-8732-3
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