Emergence of Gravity from Vacuum Coherence
There are TWO GCT tensor pieces to AFFINE non-metricity and torsion, both are ZERO in GR, but torsion is not zero in the extended theory e.g. Shipov's.
You get the 4 Einstein-Cartan tetrad 1-forms e^a from locally gauging T4 in all actions. The warped part C^a^a of e^a are the spin 1 compensating gauge potentials where the gauge transformations are simply Einstein's GCTs that define the multi-linear "tensors."
The spin 2 affine connection in general is something like
A^wuv = e^wa(d/dx^u)e^av
Its quadratic in the fundamental spin 1 Yang-Mills fields from locally gauging the 4-parameter translation group T4 in all physical actions of special relativistic fields. That, BTW is a deep formulation of the equivalence principle.
In QM 1 + 1 = 2 + 1 + 0
so, in general, there are spin 2 tensor + spin 1 vector + spin 0 scalar quanta if you quantize this smooth c-number vacuum ODLRO "supersolid" emergent gravity theory.
The torsion connection tensor is
S^wuv = A^w[u,v]
S^wuv =/= 0 when we also locally gauge the 6-parameter Lorentz group SO(1,3) and this corresponds to the 6 1-forms C^[a,b] below in terms of the 8 0-form Goldstone phases of the 9 real component Higgs post-inflation scalar fields.
This is a generalization of the Jahn-Teller effect in a crystal I suppose? My paper with Marshall Stoneham 1967 at Harwell AERE + my 1969 PhD thesis on gauge fields and ODLRO in superfluids + my 1969 paper predicting the supersolid phase of Helium 4 that David Goodstein and Richard Feynman helped me with. I would drive up to Cal Tech from San Diego. Also my 1966 Landau-Ginzburg Mexican Hat potential model of self-trapped laser filaments paper that Ray Chiao said he read when he was doing his famous experiments on them at Berkeley.
The 6 dynamical torsion field spin connection 1-forms (not in 1915 GR) correspond to Gennady Shipov's 10-D oriented point test particle. Einstein's 1915 GR only has a 4D point test particle.
This picture from Shipov shows a primitive 6D Calabi-Yau "fiber" of string theory.
My 9 real Higgs fields fit most naturally into 9D spacelike slices with brane worlds corresponding to stable topological defects from non-trivial homotopy groups with quantized deRham integrals from single-valued Higgs fields as shown in general in the David Thouless book on topological methods in quantum theory.
Therefore, vacuum coherent ODLRO Higgs-Goldstone fields + local gauging of spacetime symmetries (AKA equivalence principle) seems to be the missing organizing idea of string theory. It's been there all the time right under their noses.
On Feb 28, 2007, at 2:16 AM, Paul Zielinski wrote:
"It looks like you are still thinking that the tensor part of LC is supposed to be the "non-metricity tensor of the Levi-Civita connection". Of course, the non-metricity of the LC connection is exactly zero.
The tensor part of LC is not this quantity. It is a tensor X = -Q_A that is equal to the negative of the non-metricity tensor of a different (i.e., non-metric) torsion-free affine connection A =/= LC, where
A = LC + Q_A, and Q_A =/= 0."
There is no physics to this. There is no way to measure Q_A independently. It's not GR. You cannot say you have a new way to reinterpret GR with this formal shell game.
"There is no question that the non-metricity tensor Q_LC of the LC connection is zero. However, it does not follow from this that the LC connection doesn't contain the tensor X = -Q_A."
1. Take the Schwarzschild metric
1. Compute X.
2. Show how to measure X independently.
Meantime all the cool stuff I showed you re: C^a^b went over your head.
Jack Sarfatti wrote:
On Feb 27, 2007, at 10:56 PM, Paul Zielinski wrote:
"OK Jack, fire away. Give me your best shot.
Here is a direct quote from Carroll:
'...any set of connections can be expressed as some fiducial connection plus a tensorial correction.' (Notes on General Relativity, p 59).
Now explain to us why this doesn't apply to the LC connection?"
I told you a jillion times. Read the rest of what he says!
"Clearly, a zero "correction" is no correction at all. For Carroll's statement to be non-trivial, the tensorial correction must be non-vanishing.
Yet you are saying, in effect, that in the case of the LC connection the tensorial correction is always exactly zero. I think you have to prove
this mathematically, in the abstract, without any appeal to any particular physical theory."
"So you are saying that this doesn't apply to the LC connection? Why not?"
Sean Carroll like many others points out that most generally
Affine = LC + Torsion + Nonmetricity
Affine = LC + Torsion + Nonmetricity of Affine
Torsion and non-metricity are GCT 3rd rank tensors.
LC is not a tensor.
GR is precisely that theory in the set of theories with different connections such that
Affine = LC
End of trivial story.
"No. It is a theory such that the metric-compatible affine connection LC, defined on a 4D spacetime manifold, is taken to represent the Einstein field. But this is also true in the alternative model.
None of this has anything to do with the purely mathematical question of whether the LC connection contains a tensor -Q_A for some torsion-free non-metric affine connection A."
Physically LC parallel transports vectors along worldlines. While maintaining the inner products of vectors constant during the parallel transport process. Going around a closed worldline the difference of orientation of the vector with itself is a measure of the average curvature over the area of the loop - shrunk to zero - i.e. crystal disclination defect.
Limit of the ratio of angle deficit to area of shrinking loop is the sectional curvature dimensions 1/Area.
If there is non-metricity the length of the vector will also change in the parallel transport and in a loop you will get a discrepancy in lengths of the vector unless you impose a kind of Bohr-Sommerfeld quantization on the closed loop.
"Exactly. But this has nothing to do with coordinate curvilinearity or with Riemann curvature."
If there is torsion starting with two sides of an infinitesimal parallelogram and parallel transporting one against the other both ways the loop does not close to second order - there is a torsion gap, i.e. crystal dislocation defect.
"Right. If the connection is torsion-free this doesn't happen."
In my theory you have a set of 0-forms the Goldstone phases of the vacuum coherence ODLRO Higgs field.
From the 0-forms comes a set of 1-forms.
There are 10 1-forms.
4 tetrad 1-forms & 6 spin connection 1-forms.
Then you have GR + torsion trivially.
i.e. a theory with
A = LC + Torsion Tensor
beyond 1915 GR of course.
dO-form = 1-form
d^2)-form = 0
However that 1-form need not be exact if the 0-form has a singularity, i.e.
Integral of the 1-form over a non-bounding 1-cycle without boundary = winding number integer
Bohr-Sommerfeld condition from non-trivial 1-homotopy.
Then you can pretend Stoke's theorem works, i.e. flux without flux beyond Wheeler's "charge without charge" - these are 1D string vortex defects.
The integral of the 1-form around the loop is the surface integral of a ghostly 2-form flux through the area of the loop. This is related to the Bohm-Aharonov effect.
Similarly given 2 0-forms A & B, I define the non-closed 1-form as
C = dA/\B - A/\dB 1-form
dC = 2dA/\dB =/= 0 2-form
I can now play the same game one dimension higher! These are 0D point monopole stable defects.
Now we have star gate wormholes!
The mouth of the wormhole is the non-bounding 2-cycle without boundary - the portal like they see on the Skinwalker Ranch!
The deRham integral of 2-form C over the closed 2-surface is now an integer wrapping number ~ Bekenstein BITS for the entropy of black hole event horizons, de Sitter dark energy future observer horizons, Unruh effect, Hawking radiation, you name it. I got it all topologically pre-metrically.
There is again the world hologram ghostly volume 3-form without volume from the actual hologram area 2-form dC.
OK everything I need comes from C as the template.
C^a^b = dA^a/\B^b - A^a/\dB^b
The 4 Einstein curvature tetrad 1-forms are the diagonal matrix elements
The 6 torsion field spin connection 1-forms are the anti-symmetrized off-diagonal matrix elements
C^[a,b] = - C^[b,a]
Einstein's 1915 GR is the limit
C^[a,b] = 0
D = d + W^ac/\
e^a = I^a + C^a^a
The torsion 1-form is
T^a = De^a = 0
This determines the non-dynamical zero torsion spin connections W^ac from
de^a + W^ac/\e^c = 0
The curvature 2-form is
R^a^b = DW^a^b
The Einstein-Hilbert action density is
and the rest is history.
See Rovelli Ch II for the details.
Plus it's obvious how to generalize for torsion.
So I have 8 0-form Goldstone phases from 9 real Higgs fields with vacuum manifold S^8.
This fits most naturally in terms of stable defects into a 9+1 D spacetime. Hey that's interesting. Where have we seen that before?
I want to compactify to 3 large space dimensions so I can use
A^2 = A^aAa
B^2 = B^aBa
take square roots.
And I am back to the Bekenstein world holography bits with point defects in the world crystal's spacelike slices. They are the "lattice points"!
"All very interesting, but I still don't see a mathematical proof of your contention that the LC connection defined on an arbitrary Riemannian manifold does not contain a tensor X_A = -Q_A for each torsion-free non-metric connection A."
Even if it did, who cares? Where's the new physics to that? Where's the beef?
The Question is: What should be The Question?
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein