Gennady Shipov, Alex Poltorak, discussion on torsion & non-metricity

On Nov 12, 2005, at 5:24 PM, Alex Poltorak wrote:

Gennady,

Teleparallelism is indeed a metric theory with vanishing covariant derivative of the metric. I never suggested otherwise. I merely said that in my reformulation of GR (which is not Teleparallelism), where gravity is described by nonmetric part of an affine connection induced by the choice of the frame of reference, the Equivalence principle takes form:

[Gravity=nonmetricity] = [Inertia=affine connection]

OK, the way I would say this in my notation is

Alex's theory has Q =/= 0, Gennady's theory has Q = 0.

But what about Gennady's torsion S in Alex's theory?

Also, does Alex have W spin connection = 0?

The most general covariant exterior derivative for the locally-gauged space-time 15 parameter conformal twistor group is the 1-form

D = d + W^ab/\ + S^ab/\ + Q^ab/
Geometrodynamic W^ab ~ {Pa} Lie sub-algebra fragment of 4-parameter T4 is associated with B^a in the "substratum".

Geometrodynamic S^ab ~ {sab} Lie sub-algebra fragment of 6-parameter O(3,1) is associated with T^a in the "substratum".

Geometrodynamic Q^ab ~ {ca, dilation} Lie sub-algebra fragment of 4 + 1-parameter conformal part is associated with C^a, Dilation in the "substratum".

With non-metricity we have a 15-dim parameter configuration space.

What are the possibilities?

Only W =/= 0 i.e. 4-parameter "space-time" of 1915 GR

Only Q =/= 0 i.e. 5-parameter including the dilations.

Only W & S =/= 0 i.e. 10-parameter (Shipov's theory)

Only W & Q =/= 0 i.e. 9-parameter theory

Most general theory has 15-parameters.

The basic physical idea here is to locally gauge ALL continuous space-time symmetry groups, i.e. 15-parameter conformal group.

The above is independent of spontaneous broken vacuum symmetry.

The standard model of leptons and quarks is only formulated for

W = S = Q (2-index 1-forms) = 0 identically globally.

The standard model has U(1)hypercharge SU(2)weakSU(3)strong - but in GUT it has a more general G that breaks to H = U(1)em. There is a general theorem in topological quantum theory that any such G has degenerate macro-quantum coherent vacuum manifold G/H = G/U(1)em ~ S2 with POINT DEFECTS where the Higgs amplitude vanishes and its conjugate Goldstone phase is undetermined. This applies for a 3-vector vacuum order parameter i.e. the 3x3 adjoint irrep of SU(2)weak. The Pioneer anomaly is evidence for the physical reality of this order parameter with a HEDGEHOG point defect of wrapping number -1 at center of Sun. This comes from the fact that the second homotopy group of S2 is Z (the integers). This fact also explains WHY the geometrodynamic flux is quantized in the Hawking-Bekenstein rule Area/Lp^2 ~ Z.

The wrapping number is how many oriented times the S2 in the G/U(1)em vacuum manifold is covered for a single covering of S2 cycle surrounding the point defect in physical space.

One idea here is that the Hawking-Bekenstein geometrodynamical flux quantization has the same kind of explanation as magnetic flux quantization in superconductors, as vorticity quantization in superfluid helium, as resistivity quantization steps in quantum Hall effect in 2D films. Note that fractions would mean that a complete wrapping (or winding depending on dimension of defect) in the physical surrounding circuit only covers a fraction of the G/H ground state manifold.

The point is that Hawking-Bekenstein area quantization means emergent gravity with a local vacuum coherent order parameter that must be the same as in the Higgs mechanism for the origin of the inertia of leptons and quarks in order to obey the equivalence principle.

Therefore, I do not understand the meaning of

[Gravity=nonmetricity] = [Inertia=affine connection]

Because I associate "Inertia" with the "rest mass" m of micro-geon models of leptons & quarks that are only seen in NON-GEODESICS, i.e. in 1915 S = Q = 0 GR

md^2X^u/ds^2 + m(^uvw)(dx^v/ds)(dx^w/ds) = e(dx^v/ds)F^u^v(EM) for example

Eq. for a timelike non-geodesic for a charged test particle of rest mass m.

The rest mass "inertia" m drops out of the geodesic equation. I always mean "W-geodesic" from locally gauging ONLY T4 to Diff(4).


Affine connection in this formulation may have Ricci torsion, but this torsion is not due to the gravity, but due to the rotation of the frame of reference.

The difference between my approach and Teleparallelism is that in my approach gravity is described by nonmetricity while in Teleparallelism it is described by Ricci torsion.

Is "Ricci torsion" equivalent to SPIN-CONNECTION W?

i.e.

W = Wu^a^bdx^u&a&b


The similarity of both approaches lies in the fact that both formulations use a true (1,2) tensor (tensor of nonmetricity in my case and torsion in Teleparallelism) to describe gravity, which avoids the Energy Problem of GR arrizing from using a noncovariant quantity -- Levi Civita connection -- to describes gravitational field. Consequently, in both formulations (nonmetric and Teleparallelism) it is possible to construct a covariant energy-momentum tensor of the gravitaitonal field.

Alex

Alex, how do you get tidal geodesic curvature in your theory?

Presumably you have a curvature 2-form, or, equivalently, a 4th rank curvature tensor somewhere in your theory?

How do you get

Guv(Geometry) + kTuv(Matter) = 0

in your theory?


________________________________

From: Gennady Shipov [mailto:shipov@aha.ru]
Sent: Sat 11/12/2005 7:16 AM
To: Jack Sarfatti; Alex Poltorak

Subject: Re: Physical torsion


Alex

Nonmetricity means, that covariant derivative with respect to the metric tensor is not equal to zero. In a case of Teleparallelism it is equal to zero, therefore Teleparallelism is a metric geometry.

What nonmetricity you are talking about?


Sent: Friday, November 11, 2005 4:44 AM
Subject: Re: Physical torsion


On Nov 10, 2005, at 12:33 PM, Alex Poltorak wrote:


Greetings,

For the most part I agree with Gennady. However, my analysis (published in GR9 and, more recently, GR17) suggests that gravity is not a metric curvature 2-form but rather, it is a tensor of nonmetricity or, in Cartan terminology, the symmetric part of the Ricci 2-form; while the inertia is described as affine connection plus Ricci torsion. the Equivalence Principle, gravity = inertia, is therefore expressed as

[Gravity=nonmetricity] = [Inertia=affine connection+Ricci torsion]

Best regards,

Alex Poltorak


Can you rewrite this in the Cartan notation?

Given the spin connection W

R = DW = (d + W/\)W = Ricci curvature 2-form

i.e.

Rab = DWab = dWab + Wa^c/\Wcb = Ricci 2-form

So what is the formula for the symmetric part of Rab?

What is the "Ricci torsion" formula?

Do you do everything with W from T4, or do you need the S from O(1,3) in the local gauge POV?

How do you define "inertia" physically using the above formal definition?