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#2 For the moment consider the Levi-Civita metric connection { } rather than the spin connection. It has no torsion and it preserves the lengths of vectors in parallel transport. Suppressing indices with X as the General Coordinate (Gauge) Transformation (GCT) of T4(x):

{ } --> { }' = XXX{ } + Xd'X

d' is not the Cartan d here

The first term on the RHS is the usual third rank GCT tensor transformation.

The second term is the spoiler.

Similarly for the spin-connection S 1-form under Lorentz transformations L of SO1,3

S ---> S' = LLS + LdL

here d is the Cartan exterior derivative & L is a 0-form, S & S' are 1-forms.

S is not an anti-symmetric 2nd rank SO1,3 tensor, though it is a GCT4(x) local scalar gauge invariant.

Note that it's the GLOBAL Lie algebra of T4 that entails the LOCAL matter field source --- Tuv(matter) in Einstein's field equation.

Similarly, adding torsion means that the orbital + spin angular momentum current densities should be a source for a new set of field equations not found in Einstein 1916 GR. This extension (Einstein-Cartan) was done by TWB Kibble in 1961. One can go further and use the entire 15 parameter conformal group and also replace the Poincare sub-group by the de Sitter group with a positive cosmological constant that is the inverse area-entropy of our future cosmological event horizon. Gravity in its fullness is the induced compensating gauge connection of the largest universal spacetime symmetry group - that may be partially spontaneously broken in the vacuum. Conformal dilation invariance is spontaneously broken - massive Higgs amplitude- massless Goldstone phase particles etc.

Returning to

{ } --> { }' = XXX{ } + Xd'X

This is a gauge transformation for the SAME objectively real gravitational field. It represents the two different appearances of that  SAME invariant gravity field seen by TWO LOCALLY COINCIDENT NON-INERTIAL ("fido") FRAME (LNIF) observers measuring the same processes. Each LNIF experiences g-force caused by some non-gravity force. Indeed, there is no such thing as a gravity force according to Einstein's equivalence principle. Therefore, trying to unify gravity as a force with the electro-weak-strong forces should give one pause.

The difference { } - { }' is NEVER a GCT tensor, in fact it is the contingent Xd'X definitely not a GCT tensor.

However, and this is completely independent from the above sentence, the most general connection from locally gauging larger and larger universal spacetime symmetry groups up to the conformal group is of the form

Connection = {LC} + torsion tensor + nonmetricity tensor + ....

i.e. third rank GCT tensors.

Finally the Einstein equivalence principle EEP means that there is a TETRAD transformation T the physically connects COINCIDENT LIF ("frefos") with LNIF ("fidos") such that

{LNIF} ---> TTT{LNIF} + Td'T = 0

From this one concludes that { } does not contain a non-zero GCT third rank tensor piece.

Proof - reducto ad absurdum

Suppose there was such a tensor B =/= 0

{LNIF} = B + {LNIF}*

TTTB + TTT{LNIF}* + Td'T = 0


TTTB = - TTT{LNIF}* + Td'T    (1)

But this is a self-contradiction, because there exists a second T' such that

T'T'T'TTT{LNIF}* + T'T'T'TdT + T'd'T' = 0

on the RHS of (1)

On the other hand the LHS of (1) is a non-zero tensor that must transform as T'T'T'TTTB =/ = 0

Hence the LHS =/= 0 whilst the RHS = 0.

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