Comment on Evans' blog publications

Evans wrote on:

". . . 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:

eaea' = 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' = ηaba'a eμa') (Λb'b eνb') = ηab eμa eνb = gμν .

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.


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

[C2] Sean M. Carroll, Spacetime and Geometry,
        Addison & Wesley, ISBN 0-8053-8732-3