Einstein(1916)-Utiyama(1956)-Kibble(1961)

On Sep 13, 2007, at 5:13 PM, Paul Zielinski wrote:

Jack Sarfatti wrote:
PS Also Paul what you have been looking for is natural in this tetrad
substratum. It is not natural on the metric tensor/Levi-Civita
connection level because of bilinear tetrad cross terms.

Complete Einstein-Cartan tetrad is

e^a = I^a + B^a

Where B^a is the intrinsic curvature tetrad field - without committing
to my specific hologram model where

B^a -- N^-1/3A^a

N = (Closed Surrounding Horizon 2D Surface)/4Lp^2 Bekenstein's formula
Except that as I understand it your I-B tetrad split doesn't correspond
to my decomposition at all, since your I component in isolation
describes a flat intrinsic spacetime geometry,

Yes.

while the B term when added in accounts for the presence of curvature;

Yes

whereas in my proposed L-C decomposition, there is no dependence
whatever on the intrinsic geometry, flat or otherwise, in the
"coordinate" component of the decomposition. So I think we may have our
wires crossed here.

OK - I am just pointing out that the tetrad formalism does have a sharp
clean split between flat and curved/torsioned that is non-perturbative
and background-independent, but that disappears at the bilinear metric
tensor level. Of course inertial g-forces are curvature independent,
they only pertain to the center of mass of an extended test particle,
the curvature is detected only in the relative coordinates of the
extended test particle. In an N-particle system we can separate the CM
coordinate from the relative coordinates. Simplest for idealized 2-body
problem of course.

Granted the precise relationship between the two decompositions should
be carefully investigated, which I plan to do.

OK

Of course you are technically correct about the I-B cross terms that
appear when you recover the LC connection from the tetrads.
No perturbation theory on background dependent Minkowski spacetime is
implied here. That's a Red Herring.
That's "linearized GR", which I suppose corresponds nicely with the
Kraitchnan-Deser-Feynman spin-2 model. I understand that your model does
not rely on linearized GR, and your I + B decomposition has nothing to
do with "perturbations" applied to a flat geometry. Your model is
clearly non-perturbative.

Right.
My model is background-independent in Lee Smolin's sense.
Right -- although your I tetrad component in isolation describes a flat
spacetime geometry, correct?

Yes.

Nothing I say demands B^a I^a as in perturbation theory.
Right.
I^a has all the inertial force effects of non-geodesic frames in
Minkowski spacetime.

But only given a "coordinate" basis, as I understand the tetrad model.

Depends what is meant.

I^a = I^audx^u

I^au is a first-rank linear affine tensor in the u-index

so that I^a is an affine invariant.

In general when we have non-affine transformations, i.e. to accelerated
frames in Minkowski spacetime i.e. no real gravity curvature - forget
torsion for simplicity since it's not in 1916 GR anyway

Then

I^au - I^au' = X^uu'I^au + X^uu'(d/dx^u")X^u"u

for linear affine frame transformations

X^uu'(d/dx^u")X^u"u = 0 by definition

Note that for GR

e^a = I^b + B^b is GCT invariant

i.e. GCT frame transformations include linear affine of 1905 SR +
nonlinear (non-affine = localized T4) - of CMs of spatially extended
detectors.

That is

B^au - B^au' = X^uu'B^au - X^uu'(d/dx^u")X^u"u

so that total Einstein-Cartan e^au is a GCT first rank tensor in u, but
"Yang-Mills" spin 1 Bu is not - like the EM vector potential is not a
tensor under U(1). This is precisely why Kibble's 1961 localized T4 =
Einstein's 1916 GCTs. Very neat. Very pretty - showing natural
Yang-Mills gauge structure is at tetrad level not where Ashtekar & LQG
people put it at Levi-Civita composite level where it has obscure
non-physical excess formal baggage leading to "non-renormalizability"
puzzles. We need tetrads to couple spinor matter fields to gravity, so
why this is not an obvious way to go for the Pundits astounds me.

I^a, B^a & e^a are affine 1st rank tensors under rigid Lorentz group -
zero torsion field case.

I suppose we should play same game there, i.e. e^a still a torsion
localized Lorentz group tensor & I^a & B^a not so, only their sum.

If e^a not a localized Lorentz group tensor then ds^2 = e^aea no longer
invariant in curvature-torsion extension of 1916 GR - very dramatic new
physics if that is the case! This is something to ponder.

Note that

T^a(Minkowski) = dI^a + w^abcI^b/\I^c = 0 zero torsion 2-form in
Minkowski spacetime

and

R^a^b(Minkowski) = d(w^a^bcI^c) + w^ac'cI^c'/\w^b^cc"I^c" = 0 zero
curvature 2-form in Minkowski spacetime

However, cross terms I^a with A^b occur in the general case mixing
inertial with intrinsic effects.

I think this is only when you use a coordinate basis

I don't know what you mean. In 1905 SR - zero curvature, zero torsion

I^au - Kronecker-delta in all global inertial Minkowski geodesic frames
related only by linear affine transformations. It is curvilinear in
accelerated non-geodesic Minkowski frames from the X^uu'(d/dx^u")X^u"u
"inertial g-forces".

-- otherwise your "I" has no discernible relationship to observer
frames, but is simply an arbitrary orthonormal basis in a tangent space
with no intrinsic relationship to spacetime coordinates.

I don't understand your sentence. Tetrad formalism is GCT invariant like
using intrinsic vectors curlA, divA in vector calculus

i.e. e^a = e^audx^u is GCT scalar, i.e. local GCT frame invariant for
all local observers independent of coordinate basis relative to the
4-parameter T4 group. However, it is a first rank tensor for the
6-parameter Lorentz group SO(1,3).

Note Utiyama 1956 localized only SO(1,3) he got torsion induced
curvature and had to stick in Einstein's 1916 GCTs ad-hoc. Kibble 1961
localized complete P10 = T4xSO(1,3) and got GCTs from localized T4
absent in Utiyhama 1956. Kibble's theory larger than Einstein's 1916
curvature only theory. People got confused not understanding that
Utiyama's torsion-induced curvature is not the same as Einstein's
torsion-free curvature. This is trivial using tetrads as Cartan forms.
In terms of crystalography, dislocation defects can induce disclination
defects but not vice versa - or so it appears from

T^a = De^a torsion 2-form

R^a^b = DS^a^b

S^a^b = spin connection 1-form

with Yang-Mills mapping

S^a^b = w^a^bce^c

w^a^bc are the symmetry group G Lie algebra structure constants.

[Q^a,Q^b] = w^a^bcQ^c

for 1916 GR

G = T4

for Utiyama 1956

G = SO(1,3)

for Kibble 1961

G = P10 = T4 x SO(1,3).

Therefore, in general

w^a^bc = w^a^bc(T4) + w^a^bc(SO(1,3))

where

Einstein 1916 GR uses only w^a^bc(T4)

Utiyama 1956 uses only w^a^bc(SO(1,3))

Kibble 1961 uses the full w^a^bc = w^a^bc(T4) + w^a^bc(SO(1,3))

On Sep 13, 2007, at 4:21 PM, Jack Sarfatti wrote:


On Sep 13, 2007, at 12:12 PM, Paul Zielinski wrote:

Jack Sarfatti wrote:
The A^a q-number part is still emergent, it's just that it is the
residual q-number random zero point part.
OK, but then how can you say that this part is equivalent to a quantized
Yang-Mills field of the kind considered by t'Hooft, for purposes of
renormalization?

Because it has a very similar formal structure to the internal symmetry
Yang-Mills quantum field operators.

Positive frequency part creates a q-number A^a quantum out of the
coherent c-number A^a condensate. Negative frequency part puts a quantum
back into the c-number condensate etc. 2 independent polarizations if
massless etc.

Let's just look at the intrinsic q-part, there is a natural "Yang-Mills"
field 2-form

F^a = dA^a + w^ac'cA^c'/\A^c

With Lagrangian density 0-form ~ *[(1/4)*F^a/\Fa]

Note that A^a = A^a(condensate c-number) + A^a(q-number)

so that the bare Hamiltonian from the Lagrangian has quartic terms. Thus
is same formal structure as in Yang-Mills.

Think of sound waves in a crystal. Sound, like gravity and torsion, is
an emergent collective phenomenon out of the individual lattice atoms
right? You can have "classical" "condensate" sound waves (many phonons
in same momentum state - I mean narrow wave packet), but also you can
detect "particle" like phonon quantum effects in the fluctuations - but
the phonon itself is a collective object out of the atomic substratum.
And these phonon quantum effects can be treated as manifestations of an
"emergent" quantized field?
Is that what you mean?

Yes. Sound is an emergent collective phenomenon. At low intensities you
get quantum fluctuations - phonon analog to quantum optics effects
Poisson noise, sub-Poisson et-al. Sound has both classical wavelike
properties and quantized particle phonon properties for different kinds
of experiments. I am saying that both intrinsic tetrad curvature ~ A^a
and intrinsic torsion ~ w^a^bcA^c are both collective emergent both
c-number and q-number like sound is. Sakharov basically had this idea in
1967 though not as detailed.

Note I suppress the possible model-dependent "hologram" N^-1/3 coupling
factors and pure Minkowski I^a terms in the above rough heuristics,