Last update: 31.01.2005 19:00
In 1997 S.M. Carroll published a "proof" of the so-called «tetrad postulate» [1; p.91]. He claimed that the covariant derivatives of the tetrad vector components would vanish without any further conditions. The examination of Carroll’s simple "proof" shows that he had proven a well known compatibility condition for the covariant derivative of a vector related to two different frames. With the line "A bit of manipulation allows us to write this relation as the vanishing of the covariant derivatives of the vielbein" he claims that he had therewith proven the universal validity of the «tetrad postulate». However, a reason for the special form of the covariant derivatives of the tetrad vector is missing here and in Carroll’s book  from 2003 also.
M.W. Evans used Carroll’s claim as the basis of his GUFT. But to stop alarming critics of the of the «tetrad postulate» he published a «simple proof of the tetrad postulate» . However, this "proof" is completely identical with S.M. Carroll’s derivation of the compatibility condition [1; (3.132)] and therefore not new at all. And the question why [1; (3.132)] should imply the vanishing of the covariant derivatives [1; (3.133)] remains unanswered as in Carroll’s publications. Evidently Evans feels no need to prove that. In emails he argued that the tetrad vectors are vectors of a special kind; but just in that case a proof of the claimed special form of the tetrad vector derivatives would have been necessary all the more, but is missing.
However, even S.M. Carroll denies the special properties of the tetrad vectors. In [1; p.89]
"The vielbeins eμa thus serve double duty as the components of the coordinate basis vectors in terms of the orthonormal basis vectors, and as components of the orthonormal basis one-forms in terms of the coordinate basis one-forms; while the inverse vielbeins eaμ
And from both marked properties of the vielbein coefficients the received theory [4,5] yields the usual representations of the respective covariant derivatives, which refute the claims of Carroll and Evans of an unrestricted validity of the «tetrad postulate».
S. M. Carroll: Lecture Notes on General Relativity,
S. M. Carroll: Spacetime and Geometry: An Introduction to General Relativity,
M.W. Evans: PROOF OF THE TETRAD POSTULATE
G.W. Bruhn: Covariant Derivatives and the Tetrad Postulate,