The key equation is Rovelli's (2.89) for only the torsion-free curvature-only spin connection in terms of the tetrads. It has quadratic and quartic parts. The quartic part can be put into the desired form, but the quadratic part cannot. Also both parts depend on gradients in the tetrad component fields. It may be that only the torsion part of the spin connection can be put into the Yang-Mills covariant derivative form. I have not yet confirmed that. However, this is really a side issue, as in general we need to treat the 6 spin connection 1-forms S^a^b and the 4 tetrad 1-forms e^a as independent Yang-Mills type compensating local gauge field potentials in which we define the exterior covariant derivative as
D = d + S/\
Suppressing indices for simplicity. This is analogous to a Yang-Mills theory where the curvature two form field is
R = DS
i.e. curvature field 2-form = exterior covariant derivative of the spin connection Yang-Mills potential with itself, i.e. in 1916 GR
R = dS + S/\S
This is completely analogous to the Yang-Mills theory where
F = DA
= dA + A/\A
DF = 0
D*F = J*
DJ* = 0
In 1916 GR
DR = 0
D*R = *J
must translate in ordinary tensor notation to
Guv = kTuv
D*J = 0
Tuv^;v = 0 i.e. local energy-momentum stress current densities conserved - all bets off on global integrals over spacelike surfaces.
All of the above is for zero torsion fields
T = De = 0
This is an auxiliary equation not found in the internal Yang-Mills theories. The theory is more complex of course when T =/= 0 i.e. locally gauging the full 10-parameter Poincare spacetime symmetry group. One must be careful on how to make the analogy of GR with Yang-Mills theories. The analogy is perfect in Utiyama 1956 where there is only S and no e in the sense of the compensating field A where e = I + A because T4 is not locally gauged there. GCTs are put in adhoc - not pretty.
On Sep 28, 2007, at 4:25 PM, Jack Sarfatti wrote:
In trying to make gravity tetrad GR into a formal analog of Yang-Mills I have posited
S^ac = w^acc'e^c'
e^c' are the Einstein tetrad 1-forms
S^ac are the spin-connection 1-forms (involving gradients of the tetrads in 2.88)
Rovelli has (2.88) for example. Now I had thought I had seen the equivalent of S^ac = w^acc'e^c' in Rovelli's book, but now I cannot find it.
Using it, the torsion field 2-form is
T^a = de^a + S^ac/\e^c
= de^a + w^acc'e^c'/\e^c
which is like the Yang-Mills field 2-form
F^a = dA^a + w^acc'A^a/\A^c'
It is not clear that S^ac = w^acc'e^c' is consistent with (2.88)