Einstein's Equivalence Principle is subtle, but not malicious
Following Kiehn's suggestion I am writing all of this up properly for a short book of mathematical lecture notes. If other people want to send in some pedagogical articles I can put them in as well. The book will be registered at Lib of Congress and on Amazon et-al and in all the book catalogs on-line and in book stores.


On Nov 9, 2005, at 12:36 AM, Gennady Shipov wrote:


----- Original Message -----
From: Jack Sarfatti
To: ROBERT BECKER ; James Woodward

Subject: Re: Russian R&D Torsion fields for exotic propulsion concepts

bcc

On Nov 7, 2005, at 10:57 PM, ROBERT BECKER wrote:

Jack,
Much has happened with your theory over the past week since you replied below. I can not keep up with you that fast? but I want to address a couple of points.

Yes indeed! Gennady Shipov made a valuable contribution where I went off-course on pushing the analogy of GR to Yang-Mills too far in too simple a way.

Robert Becker
2. Teleparallelism (TP) -

I can understand why you find the TP assumption not physically well-motivated. However, as I describe in the Mitre Paper in greater detail, there are very strong and profound mathematical reasons for assuming TP. And from this mathematics flows physically significant conclusions.

I see no physical motivation for it as yet. I know Shipov uses it.

Shipov Gennady
"You will find physical motivation here

http://www.shipov.com/science.html

http://www.shipov.com/files/DarkEnergy.pdf "


Robert Becker wrote:
"3. Torsion - In a comment made in one of his letters to you last week, Shipov does seem to sustain my statement (and a pillar of Vargas Theory) that Torsion in the Cartan formalism, has a fundamentally translational character, not associated with rotation or spin."

Yes, I don't understand that. It contradicts Kibble's paper.
Definitely, torsion is translational in the sense of a crystal dislocation.
(Hagen Kleinert?http://www.physik.fu-berlin.de/~kleinert/? )
and curvature is rotational in the sense of disclination i.e. rotation of vector parallel transported round a closed loop. The loop does not close in 2nd order when there is torsion.

Shipov Gennady wrote:
"Sirs, in this question I and Hagen Kleinert do not assert anything new."

OK in that case I am sure what I say is consistent with Kleinert's idea. There seem to be two meanings of "translation". Sure the torsion gap is a dislocation defect that can be thought of as a "translational defect", however it actually comes from locally gauging the 6-parameter Lorentz group O(1,3) group. Those are the 6 parameters in the added dimensions of Gennady's "oriented point manifold" of 6 + 4 = 10 dimensions. This was done explicitly first in the 60's I think by Utiyama who locally gauged O(1,3) but did not locally gauge T4. Utiyamia got torsion without curvature - he put the latter in by hand as I recall. Then Kibble published a paper showing that locally gauging the full 10-parameter Poincare group gives extended GR with both torsion and curvature.
Ref: Lorentz invariance and the gravitational field, TWB Kibble J Math Phys 2, 2, March April 1962 reprinted in "Gauge Theories in the Twentieth Century" ed. J.C. Taylor, Imperial College Press, 2001
"
Cartan writes in the article " On the Generalization of the Notion of the Curvature of Riemann and spaces whit Torsion" (Comptes Rendus Acad. Sci, vol 174, 1922, p.593-595), where he gives physical interpretation to torsion for the first time :

" In the general case, when there is a translation associated with every closed infinitesimally small contour, one can say that the space given differentiates itself from the euclidean space in two manners:

(1) by a curvature in the sense of Riemann, which expresses itself by a rotation;

(2) by a torsion, which expresses itself by a translation."

Yes, but there are two meanings there. The rotation is of the test vector taken around a tiny closed loop. However, the GCT is a 4-parameter group from locally gauging T4, so that when you use the T4 connection around the tiny closed loop you get an effective rotation at the initial = final point P that INDUCES a local Lorentz transformation (space-time rotation) at P. So this is the source of the confusion. T4 & O(1,3) are INTER-LOCKED since the Poincare group is the SEMI-DIRECT PRODUCT of T4 with O(1,3) their Lie algebras do not split off independently of each other into independent sub-algebras, i.e. Poisson brackets of translations with space-time rotations do not vanish. Similarly, the translational torsion gap comes from trying to go around the loop with the additional torsion connection that comes from locally gauging the 6-parameter O(1,3). This dual mixing on the rotations and the translations is because

[T4 translations, O(1,3) rotations] =/= 0

so their effects mix up together like in the uncertainty principle

[P,X] ~ ih

Therefore, this was simply a semantic ambiguity in the use of the "informal language" (Bohm)


Jack Sarfatti

However from the LOCAL GAUGE POV Shipov's extra 6 dimensions are the now variable phases of the 6 generators of the O(1,3) group now locally gauged to give the torsion potential 1-form S where now

D' = D + S/
D = d + W/
R' = D'W = curvature 2-form

T' = D'S = torsion 2-form

D'R' = 0 Bianchi identity

D'T' = 0 Bianchi identity

Now I have to be careful not to make the same mistake(s)


Shipov Gennady (between quotes)

"Jack, be careful .
Everything, that is written above not concern to classical geometry."

The Cartan notation is simply unfamiliar the way the Feynman diagrams were at the beginning. However Rovelli in Ch 2
http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html
makes the connection to classical tensor methods clear.

I am sure my way of doing this is essentially correct, some minor errors along the way perhaps. Once one sees the power of this Cartan notation one is able to analyze novel physical situations almost instantly.
"The definitions accepted in classical geometry the following:

Local translational group T(4)

x(4) -4 translational coordinares (parameters) of local translattional group T(4)

e - tetrad 1-forms - generators of the local group T(4)

De=de - e/\W= 0 (A) "

That's what I already wrote exactly.

" Cartan structural equations of the local group T(4) (Teleparallelism A(4)) , when S - Cartan 2-forms is equal to zero
Here
D = d + W/\"
I wrote that also already.
"W = G + (-W) - connection 1-forms (structural fuctions of the local group T(4))
-W - Ricci torsion 1-forms
De=de - e/\W= S (A')"

I wrote that also, slightly different notation.

So far you have not done anything different from what I already did.

"Cartan structural equations of the local group T(4), when S - Cartan 2-forms is not equal to zero

You can see

Ricci torsion =/= Cartan torsion."

Yes, of course. I have all that trivially.

My VANISHING RIcci torsion 2-form is (my notation not identical to yours)

T = De = D(1 + B) = d(1 + B) + W/\(1 + B) = 0

i.e.

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

that I neatly invert BTW to get

W = -*[dB/\(1 - B)]

* = Hodge dual

This shows WHY the Ricci Spin-Connection has 2 indices, i.e.

W^a^b = -*[dB^a/\(1 - B)^b]

B is from locally gauging T4 to Diff(4)

Next locally gauge 6-parameter O(1,3) to get the compensating torsion 1-form connection with 2-indices S^a^b

D' = d + W/\ + S/
The Cartan torsion 2-form is then

T' = D'e = (d + W/\ + S/\)(1 + B)

= S/\(1 + B) =/= 0

However, I can also make other 2-forms.

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

with curvature-torsion coupling.

I will define this second torsion 2-form as

T" = D'S = (d + W/\ + S/\)S = dS + W/\S + S/\S

I POSIT the following ANSATZ for the dynamical field equations for the extended curvature-torsion theory

D'T' = 0 Bianchi identity I 3-form

D'R' = 0 Bianchi identity II 3-form

D'T" = 0 Bianchi identity III 3-form

G' = D'*W = *J(curvature source) 4-form source equation

O' = D'*S = *J(torsion source) 4-form source equation

Local curvature & torsion sources current density conservation laws are

D*J = 0

i.e. both 5-forms automatically vanish in 3 + 1 space-time

DG' = 0

DO' = 0

Note for comparison internal U(1) EM gauge field theory

A 1-form from local gauging of U(1) on a source field.

F = dA 2-form

dF = 0 Bianchi identity (AKA no magnetic monopoles + Faraday induction motors)

d*F = *J 3-form source eq (AKA Ampere's law + Gauss's law)

dd*F = d*J = 0 4-form

So notice that in the internal symmetry gauge theories the source and current conservation equations are 3-form and 4-form respectively.

THIS IS WHAT THREW ME OFF ORIGINALLY.

The Equivalence Principle CHANGES THINGS in this regard.

For space-time symmetry gauge theories the source equations are 4-form and the local current density conservation equations are 5-forms (automatically zero in 4D space-time).
Shipov wrote: "In the article " On the Generalization of the Notion of the Curvature of Riemann and spaces whit Torsion"
Cartan wrote about Teleparallelism with Ricci torsion: "Mechanically it corresponds to a medium of constant pressure and constant angular momentum." (It is true for homogeneous spaces with Teleparallelism . S.G.)
Using Teleparallelism I managed to connect many abnormal phenomena with Ricci torsion, instead of Cartan torsion which Richard, Hammond at all use.

Local rotational group O(1.3)

e(6) noncholonomic tetrad, which perform 6 - angular coordinares (parameters) of local rotationall group O(1.3)
W - generators of the local group O(1.3).

DW=dW-W/\W=R- dW -W/\W=0 (B)
- Cartan structural equations of the local group O(1.3), when R' - Cartan curvarute 2-forms is equal to zero (Teleparallelism )
Here
R - Riemann curvature plays a role of structural functions of the local group O(1.3)
DW=dW-W/\W=R- dW -W/\W=R' (B')

- Cartan structural equations of the local group O(1.3), when R' - Cartan curvarute 2-forms is not equal to zero."

I will think about what you say here later. I don't know instantly off-hand how to connect these formal points you raise with the physical interpretation. I am not sure if your "W" is same as my "W".

Do you mean in my notation

R' = D'W = DW + S/\W = R + S/\W = 0

I suppose that is what you mean by "TP" (Teleparallism)

In that case

R = - S/\W

That is the geodesic-deviation tidal curvature 2-form of 1915 GR is simply the negative of the exterior product of the torsion and spin connection 1-forms that come from locally gauging the entire 10-parameter Poincare group.

I suppose that is OK as a SPECIAL LIMITING CASE, but WHY DO IT?

I don't see any compelling physical reason to constrict the larger theory down to that sub-theory?

OK if you IMPOSE as an Ansatz

R' = 0

You certainly obey the Bianchi identity

D'R' = 0

Is there any effect of TP as R'= 0 on the source equations?

G' = D'*W = *J(curvature source) 4-form source equation

O' = D'*S = *J(torsion source) 4-form source equation

R' = D'W = 0

Does this tell us anything about

G' = D'*W = *J ?

I don't know instantly off hand.

Jack Sarfatti

The source equations must be

D'*W = *J(curvature)

D'*S = *J(torsion)

since both W & S are connection 1-forms? (they do not map to homogeneous tensors of course)

*W & *S are 3-forms

The source equations are 4-forms.

Robert Becker wrote:
7. Cartan Formalism - I think I follow your derivation below of the emergent gravity theory using this formalism a fair degree. The formalism and derivation do remind me of the Vargas Theory. He starts from the same point as you, with the important difference that he does not add in any fields like the Higgs B term. He also assumes Teleparalellism (TP) rather than vanishing Torsion. Another distinction is that he does not introduce gauge fields because he believes that the physics must unfold on the principal fiber bundle, not on the auxiliary bundles of gauge theory. But you both inevitably end up with Maxwell-like equations from your Theory. The Vargas equations are what I call geometric Einstein-Maxwell equations. From these equations come a geometric interpretation or analogy of the electromagnetic field. So,?Vargas ends up with geometrizing EM, while you produce an emergent gravitation. Now, what would the combination be...

Yes, there are probably interesting connections. I am only using two battle-tested physical ideas here.

1. Locally gauging EVERY almost every continuous symmetry group both internal and space-time.

2. Spontaneous breaking of symmetry. i.e. ground state (or vacuum) instability for critical values of control parameters.

I want to implement the equivalence principle by deriving gravity from the broken symmetry in the weak force in the standard model of leptons and quarks so as to have gravity and inertia two sides of the same coin. I think what Haisch and Puthoff et-al propose is too cheap, too incomplete. They have perhaps a small effect only.


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