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Topic: Einstein-Cartan Tetrads and Spin Connections as Yang-Mills Field
Theory

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Jack Sarfatti

Posts: 1,942
Registered: 12/13/04
Einstein-Cartan Tetrads and Spin Connections as Yang-Mills Field
Theory

Posted: Oct 1, 2007 6:24 PM
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v2 (expanded & corrected too many "/\" in R formulae)
On Sep 29, 2007, at 6:22 PM, Jack Sarfatti wrote:

Note also, the effect of the equivalence principle makes a difference in
comparing gravity fields to Yang-Mills fields.

Look at the exterior covariant derivative 1-form

D = (d + S)/\

note that d is usually written without the /\ that is tacitly understood
since

d(p-form) = (p + 1)-form

dual to boundary operator

&(p + 1) dim manifold = p-dim manifold so that Stokes theorem generalizes to

Integral of d(p-form) on p + 1 dim manifold = Integral of (p-form) on
the boundary &(p + 1)manifold

for example when p = 0 in 3D space i.e. fundamental definite integral
formula of calculus

d(0-form) = gradient of a function

p = 1 e.g. loop integral of vector (1-form) field is interior flux
(2-form) integral

d(1-form) - curl of a vector (2-form)

p = 2 Gauss's divergence theorem - end of short digression

d(2-form) = divergence of a 1-form vector field in 3D space.

can generalize to Minkowski spacetime of 1905 SR

Stoke's theorem is key. A star gate traversable wormhole time machine to
past (or equivalently in this regard a weightless zero g-force geodesic
glider warp drive bubble in sense of Alcubierre's toy model metric) held
open by universally antigravitating positive zero point dark energy
density with equal and opposite negative pressure (w = -1 with GR source
factor 1 + 3w) depends on multiply-connected spacetime corresponding to
topological defects in the vacuum ODLRO field whose phase modulation
determines the local curved tetrad fields and the local torsion field
spin connections from localizing the 10-parameter Poincare symmetry
group of special relativistic quantum field theory. The latter is
background-dependent the former is not.

Multiple connectivity means, in our specific problem of metric
engineering practical warp and wormhole, closed 2 surfaces (portals or
wormhole mouths) without boundary that are not boundaries of the 3D
space wormhole tunnel.

?
http://www.kahl.net/astro/graphics/wormhole.gif

the 2D mouth is a circle (1 space dimension removed in picture at a
fixed "time"

the 3D space Dr. Who walks through is the 2D tube in the picture - you
are flatlander bug constrained to surface in the picture.

"dark energy" = "exotic matter"

Since the vacuum ODLRO field is single-valued that means that the
geometrodynamic field area density flux through a nonbounding closed 2D
surface is quantized and this explains the Hawking-Bekenstein formula
corresponding to point gravity monopole defects in the fabric of
dynamical 3D space.

Entropy/kB = Horizon Area/4 Quantum of Area Flux = N

i.e. Newton's Planck area hG/c^3 ~ 10^-66 cm^2

with the world hologram formulae

Size of wavelet Quantum Foam Bubble ~ N^1/6(Quantum of Area)^1/2

i.e. &L ~ N^1/6Lp = (Lp^2L)^1/3

L ~ N^1/2Lp (hologram)

Horizon area ~ (10^28 cm)^2 in our pocket universe on the Cosmic
Landscape with &L ~ 10^-13 cm and observed dark energy density ~
hc/NLp^4 ~ 10^-29 gm/cc


Back to main point

d in a sense is e

i.e. in a flat spacetime in a geodesic GIF coordinate basis

d(0-form) is "4-gradient" d/dx^a on a 0-form function since ea^u =
Kronecker delta

ea = ea^u(d/dx^u)

e^a = e^audx^u

That is we can think of d/\ as e/\

or

D/\ = (e + S)/\

e = I + A


i.e.

D/\ = I/\ + A/\ + S/\

I is when we have globally flat Minkowski spacetime and a Global
Inertial Frame (GIF)

A & S are the compensating geometrodynamic field gauge potentials that
first appear in a Global Non-Inertial Frame

in globally flat Minkowski spacetime where the curvature R^a^b = 0 and
the torsion T^a = 0 but in 1916 GR R^a^b =/= 0 while still T^a = 0.


where

I/\ is in flat Minkowski spacetime of 1905 SR

A comes from using a GNIF and finally from localizing T4 to LIFs & LNIFs.

Note that A induces S that is not independent when the torsion T field
vanishes globally.

In a GNIF

R = D/\S = 0


i.e.

R = (I/\ + A/\ + S/\)S = 0

and also

T = De = 0

i.e.

(I/\ + A/\ + S/\)(I + A) = 0

when rigid T4 is localized to T4(x) then we have possibility that

R = (I/\ + A/\ + S/\)S =/= 0

This notation makes the Yang-Mills field structure more apparent.

Consider

F = (I/\ + A/\ + S/\)(I + A + S) = T + R

OK so now it really looks like Yang-Mills F with A & S appearing
symmetrically!


In 1916 GR S = S(A) is redundant as shown in Rovelli's eq. 2.89

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 (2.89). Hence GR
is renormalizable in t'Hooft's sense including vacuum ODLRO Higgs field
that may give Salam's strong short-range f-gravity in addition to zero
mass geometrodynamic field quanta. Indeed the composite quanta are spin
0, spin 1 and spin 2 from pairs of the fundamental spin 1
geometrodynamic quanta i.e. zero point fluctuations of the uncondensed
part of the post-inflation vacuum ODLRO field. Think of the
geometrodynamic field random "dark energy" quanta like the zero point
motions of helium 4 atoms in the T = 0 ground state that is only 10%
coherent condensate even though the effective superfluid density is 100%.





typo corrected draft 2
On Sep 29, 2007, at 5:37 PM, Jack Sarfatti 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 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

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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