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YangMills version of General Relativity
Posted:
Sep 29, 2007 8:41 PM


The key equation is Rovelli's (2.89) for only the torsionfree curvatureonly spin connection in terms of the tetrads. It has quadratic and a quartic parts. The quartic part can be put into the 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 YangMills 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 1forms S^a^b and the 4 tetrad 1forms e^a as independent YangMills 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 YangMills theory where the curvature two form field is
R = DS
i.e. curvature field 2form = exterior covariant derivative of the spin connection YangMills potential with itself, i.e. in 1916 GR
R = dS + S/\S
This is completely analogous to the YangMills 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
corresponds to
Tuv^;v = 0 i.e. local energymomentum 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 YangMills theories. The theory is more complex of course when T =/= 0 i.e. locally gauging the full 10parameter Poincare spacetime symmetry group. One must be careful on how to make the analogy of GR with YangMills 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 YangMills I have posited
S^ac = w^acc'e^c'
e^c' are the Einstein tetrad 1forms
S^ac are the spinconnection 1forms (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 2form is
T^a = de^a + S^ac/\e^c
= de^a + w^acc'e^c'/\e^c
which is like the YangMills field 2form
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)



