Re: Yang-Mills version of General Relativity's renormalizable substratum

Hi Happy Jack, been following your stuff.

On Sep 29, 5:42 pm, Jack Sarfatti <sarfa...@xxxxxxxxxxx> wrote:
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.

Not really, IMO, take the integral,

$ (Tuv;v = 0) => constant.

The "global extension" allows the constant
to be either Planck's "h" or more explicit
in angular momentum as "h/2pi".

That's doves well with the quantization of
the Energy Flux in terms of the Action,
rather like Guass's Law quantized, where

energy = D(Action)/D(time)

with "D" being a finite increment, equal
to the said constant.

Suppose for example, you decided to express
your field in terms of invariants, well then
a logic extension of a field would use at
the outset, EMR, but the EMR is a conveyance
of the action invariant, i.e. said constant.

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.

It might be set pretty, have a look at this,

e de = (I+A) d(I+A)

then let X = (I+A) to get,

e de = X dX

and solve e = X + (constant = q).

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)

Dangerous to do tetrad gradients in QT,
you're sliding into continuum thinking.

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)

Well you have an invariant force "F^a" which
contradicts General Relativity, because
you're back to the continuum using "dA^a",
OTOH, staying with Quantized GR is simpler.
Regards
Ken S. Tucker

.

Relevant Pages

  • Yang-Mills version of General Relativity
    ... The key equation is Rovelli's for only the torsion-free curvature-only spin connection in terms of the tetrads. ... 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. ...
    (sci.math)
  • Yang-Mills version of General Relativitys renormalizable substratum
    ... The key equation is Rovelli's for only the torsion-free curvature-only spin connection in terms of the tetrads. ... 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. ...
    (sci.physics.relativity)
  • Einstein-Cartan Tetrads and Spin Connections as Yang-Mills Field Theory
    ... the effect of the equivalence principle makes a difference in comparing gravity fields to Yang-Mills fields. ... S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged. ... That is 1916 GR is really a theory of the spin 1 Yang-Mills curvature tetrad field A with a redundant S as given in Rovelli. ... The key equation is Rovelli's for only the torsion-free curvature-only spin connection in terms of the tetrads. ...
    (sci.math)
  • Einsteins General Relativity as a Spin 1 Yang-Mills Theory
    ... the effect of the equivalence principle makes a difference in comparing gravity fields to Yang-Mills fields. ... S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged. ... That is 1916 GR is really a theory of the spin 1 Yang-Mills curvature tetrad field A with a redundant S as given in Rovelli. ... The key equation is Rovelli's for only the torsion-free curvature-only spin connection in terms of the tetrads. ...
    (sci.physics.relativity)