Sunday, October 30, 2005

Curvature and Torsion as Bohm-Aharonov Effects from Topological Defect Phase Singularities

On Oct 30, 2005, at 12:13 PM, Jack Sarfatti wrote:

On Oct 30, 2005, at 6:25 AM, Gennady Shipov wrote:

Jack Sarfatti

Witten et-al WRONG that GR is not a LOCAL GAUGE THEORY! (when properly
framed of course using P physical coincidence not x = manifold point).
Gennady Shipov
I agree with Witten et-al .
Jack Sarfatti

Is that what you mean? What exactly do you agree with?

Gennady Shipov
Riemann geometry which has been used by Einstein, has no Cartan structural equationsà and cannot be submitted as a group manifold. It is only a shadow of the "true" geometry.

I do not understand what you just wrote. Rovelli has the structural equations in Ch 2 of his online "Quantum Gravity" book. Indeed I have written them down now several times.
has free pdf early version of book.

Rovelli's eqs are essentiall

Given W^IJ = spin-connection 1-form

I,J in tangent space fiber indices raised and lowered with constant flat Minkowski metric nIJ in accord with EEP.

Exterior covariant derivative is

D = d + W/
The curvature 2-form is

R^IJ = dW^IJ + W^IK/\W^KJ

Bianchi identities

DR = 0

Einstein's field eqs.

D*R = *J(Matter)

Local conservation of stress-energy current densities

D*J(Matter) = 0

Einstein-Hilbert Field Lagrangian Density 0-form

L = {IJKL}R^I^J/\e^K/\e^L

{IJKL} is completely antisymmetric tensor

e^I = e^Iudx^u

ds^2 = guvdx^udx^v = eu^InIJev^Jdx^udx^v

e = e^I&I

where &I are basis vector fields in fiber dual to dx^I

So above is standard with zero torsion 2-form assumed, i.e.

T^I = De^I = de^I + W^IJ/\e^J = 0

In my theory, I have additional structure:

e = 1 + B

B = (hG/c^3)^1/2'd'(Goldstone Phase of Charge Neutral Higgs Field Component)

B also comes from local gauging of T4 to Diff(4)

De = dB + W/\(1 + B) = 0

W determined from B completely.

If we also locally gauge O(1,3)

we get in addition to W(Curvature) an S(Torsion)

D' = d + W^IJ/\ + S^IJ/\ = D + S/
We now have to make some choices

R' = D'W = R + S/\W

T' = D'S = dS + 'S/\S' + W/\S

Note T' has 2 indices, T only had 1.

'S/\S' = S^IKS^KJ etc.

Bianchi identities

D'R' = 0

D'T' = 0

Field equations

D'*R' = *J(Translation Sources)

D'*T' = *J(Rotating Sources)

Local conservation of current densities

D*J(Translation) = 0

D*J(Rotation) = 0

OK these are MY NEW CURVATURE-TORSION FIELD EQs (replacing my earlier ones that may not have been quite correct) from LOCAL GAUGING full 10-parameter Poincare space-time symmetry group with spontaneous broken WEAK FORCE internal symmetry supplying the charge neutral Goldstone phase that generates BOTH CURVATURE and TORSION as Bohm-Aharonov "Flux-without-flux" nonlocal SINGULAR GAUGE TRANSFORMATIONS on the charge-neutral Goldstone phase.

B(Curvature) = (hG/c^3)^1/2'd'(Neutral Goldstone Phase of SU(2)weak Higgs Vacuum ODLRO)

'd' means SINGULAR gauge transformation i.e. off the singular topological defect 'd'(Phase) is closed i.e.

d'd'(Phase) = 0 like what looks like an exact 1-form with a 0-form Phase.

Nevertheless when the closed loop 1-cycles surround a topological defect Phase Singularity the NONLOCAL DeRham integral of the 1-form around this singular NON-BOUNDING 1-cycle LOOP not only does not vanish, but it is quantized!

The SINGULAR INTEGRAL of the B-1form about the topological defect is a MEMBER of the HOMOTOPY GROUP ~ Z

i.e. PI1[S1] = Z in this case.

In general PI(surrounding cycle)[Order Parameter Fiber Space] is the physically relevant Homotopy Group.

In simple terms this is Bohr-Sommerfeld quantization from single-valuedness of the "More is different" emergent LOCAL zero thermodynamical entropy vacuum order parameter from the weak force spontaneous broken symmetry giving both gravity and inertia together in ONE UNIFIED MECHANISM consistent with the EQUIVALENCE PRINCIPLE.

"Flux-without-Flux is DEFINING the "circulation integral of the p-form" as equal to the flux integral of its exterior derivative over the surrounded p+1 manifold including its topological defect.

This is essentially the NONLOCAL BOHM-AHARONOV ACTION-AT-A-DISTANCE of the topological defect in which the Goldstone Phase is undefined at the ZEROs of its complementary Higgs Amplitude

i.e. Order Parameter Component = (Higgs Amplitude)e^i(Goldstone Phase) c-number field

However, we still have Number-Phase complementarity for the small vibration quanta either real or virtual.

There are two approaches to the geometry - Riemannian and Cartanian. The Riemannian approach is wrong from the beginning. Only a method of orthogonal reper used by Cartan gives the correct approach, and it means, that torsion of space is different from zero. If torsion is equal to zero we have Minkovski space (Einstein).

Jack Sarfatti

I have proved that GR is a local gauge theory
R = DW

D = d + W/
DR = 0

D*R = *J

D*J = 0

R = Curvature 2-form

above is only for 1915 GR limit of zero torsion 2-form

T = De = D(1 + B) = 0

R is like EM field F in

F = dA

dF = 0

d*F = *j

dj* = 0

Where B(P), W(P), R(P)

P is local coincidence not a bare manifold point x

Technically P = {x} = Coset Orbit of All x in manifold mod Diff(4).

Jack Sarfatti

It's straight-forward to generalize to Shipov's torsion field theory
using the Cartan forms that are local frame invariant automatically.
Gennady Shipov

Using of Cartan forms cannot give generalization of the physical theory.

Formalism of external forms it only the mathematical tool.

Besides my theory is finished with experiment and, even up to technology, but the theory of strings is phenomenological model which after some time willl die.

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