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 **e**_{a} (a=0,1,2,3) in the tangential space T_{P} of the
manifold at P that are orthonormal wrt the metric (g_{μν})
at P, i.e. by representing the tetrad vectors
**e**_{a} = e_{a}^{μ} **∂**_{μ}
by the basis vectors **∂**_{μ} (μ=0,1,2,3) of the local coordinate basis
we have the orthonormality relations:

g_{μν} e_{a}^{μ}e_{b}^{ν} = η_{ab}
[C1;(3.118)]

where (η_{ab}) = diag(−1, 1, 1, 1) is the Minkowski matrix.
Cf. [C1;(3.118)] and [C2;(J.5)].

Every local Lorentz transform (LLT) Λ=(Λ_{a'}^{a}) changes the tetrad vectors but leaving the canonical form
of the metric unaltered:

**e**_{a} → **e**_{a'} = Λ_{a'}^{a} **e**_{a}
[C1;(3.125)]

Therefore we have

Λ_{a'}^{a} Λ_{b'}^{b} η_{ab} =
η_{a'b'} .
[C1;(3.126)]

to obtain

η_{a'b'} e_{μ}^{a'}e_{ν}^{b'} =
Λ_{a'}^{a} Λ_{b'}^{b} η_{ab}
e_{μ}^{a'}e_{ν}^{b'}
=
η_{ab}
(Λ_{a'}^{a} e_{μ}^{a'})
(Λ_{b'}^{b} e_{ν}^{b'})
=
η_{ab}
e_{μ}^{a} e_{ν}^{b} = g_{μν} .

The metric (g

[C1] Sean M. Carroll, Lecture Notes on General Relativity,

http://arxiv.org/pdf/gr-qc/9712019

[C2] Sean M. Carroll, Spacetime and Geometry,

Addison & Wesley, ISBN 0-8053-8732-3