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Topic: Yang-Mills version of General Relativity
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Jack Sarfatti

Posts: 1,942
Registered: 12/13/04
Yang-Mills version of General Relativity
Posted: Sep 29, 2007 8:41 PM
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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 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 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

corresponds to

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)





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