Sunday, February 06, 2005

Finsler Geometry? Who ordered that?



Begin forwarded message:

From: Jack Sarfatti
Date: February 6, 2005 12:16:08 PM PST
To: Doc Savage
Subject: Re: Non-holonomy and anisotropy of metrics: NASA Pioneer, preferred frames

bcc: Invisible College

Mathematicians and Theoretical Physicists have opposing tendencies. We theoretical physicists (at least in the tradition of Hans Bethe where I was at Cornell in late 50's and early 60's) aim to interpret the most phenomena with the simplest possible mathematics even if the latter is ugly - Morse and Feshbach not Witten and Greene. :-)
We are not into "Math for math's sake." The beautiful math is slain by the ugly fact - and all that. Physics today has many observational anomalies to be explained. I have drawn the line in the sand. I say only two battle tested ideas are needed to explain everything:

1. Local gauging of symmetries (both internal and spacetime) connecting reference frames introducing compensating gauge connection dynamical fields.

2. Spontaneous breakdown of selected vacuum (virtual quanta) and ground (real quanta) states of selected symmetries yielding preferred frames of reference. (AKA "More is different" in soft-condensed matter physics and "Higgs mechanism" in high-energy physics where the vacuum coherence is Frank Wilczek's "multi-layered multi-colored" cosmic superconducting vacuum field.)

How to apply the above is completely pragmatic, i.e. data-driven.

For example, selection of a preferred direction (angular orientation) in space under the space-space rotation group O(3) in the ferromagnet is the same idea as in the selection of a preferred inertial frame (absolute rest) for the set of Lorentz boosts (space-time rotations rather than space-space rotations i.e. "rapidity" rather than "angle" - hyperbolic sinh(rapidity)/cosh(rapidity) in O(1,1) rather than trigonometric sin(orientation)/cos(orientation) in O(2). The hyperbolic case is what R. Cahill and the Sicilians are claiming empirically. It is the space-time rotation vacuum spontaneous breakdown of Lorentz boost symmetry analog to the space-space rotation ferromagnetic spontaneous breakdown of ground state orientational symmetry. The systems are different, the groups are different, but the idea is the same! Preferred space orientation in the ferromagnetic ground state. Preferred space-time orientation (selected rapidity) in the local space-time region allegedly detected with Michelson interferometers and He-Ne lasers. The physical dynamics is unchanged. There is no dynamical breaking of Lorentz boost symmetry in the Cahill effect anymore than there is dynamical breaking of rotational symmetry in the ferromagnet. Dynamical breaking of symmetry i.e. a new term in the Hamiltonian (or Lagrangian) is NOT what I am talking about with "More is different"/"Higgs Mechanism" spontaneous breakdown of vacuum/ground state symmetry. The distinction is "dynamical" vs "spontaneous". Local gauging is still another form of symmetry breaking. The global symmetry is broken down to a local symmetry, but is restored in a way by the compensating gauge potential connection field relative to a now larger dynamical system. General relativity is an example of local gauging just as the electro-weak/strong force theory is. Only the groups are different. The idea is the same! Spontaneous breakdown of vacuum symmetry is independent of local gauging of global to local symmetries.

In 1916 GR the translation group T4xO(1,3) global symmetry is broken down to local O(1,3) Lorentz group symmetry in the tangent bundle. The compensating gauge-potential connection field is most easily seen in the Cartan tetrad formulation of 1916 GR (e.g. Brazilian paper) that clearly distinguishes appearance (GCT effect) from intrinsic (gauge force without force) effects. The gauge transformations on the non-trivial part of the tetrad correspond to the GCT (Diff(4)) transformation in the geometrodynamic picture. The gauge force without force and geometrodynamic picture/formalisms are physically equivalent - merely a Young Woman/Old Hag Gestalt Shift - a simple two-way mathematical map from one picture to another. That is,

eu^a = (Kronecker Delta)u^a + Bu^a dimensionless Cartan tetrad, Bu^a is the non-trivial intrinsic part

u in base, a in tangent fiber - this is the EEP missing in Hal Puthoff's PV without PV.

Bu = Bu^aPa/h dimension (Length)^-1 = gauge force-without-force compensating potential breaking GLOBAL T4 to LOCAL GCT.

{Pa} = Lie algebra ("mom-energy" of Wheeler) of T4

Bu = (Higgs-Goldstone Macro-Quantum Phase from spontaneous broken U(1) in PV virtual electron-positron pairs),u

,u = ordinary partial derivative

The map from gauge force-without-force picture to Einstein geometrodynamical picture is

guv(LNIF) = [Minkowski(LIF)]uv + Lp^2[Bu,v + Bv,u] (dimensionless)

the above is the elastic Bohm giant pilot-wave hidden variable guidance constraint

expect non-holonomic "Dirac string" phase singularities where

Bu,v - Bv,u =/= 0

(mixed second order ordinary partial derivatives of the Goldstone Vacuum Coherence Phase do not commute)

Note Lp^2 = hG/c^3 quantum of area

Gravity is zero when Lp^2 shrinks to zero for whatever reason i.e. h -> 0, or G -> 0, or c -> infinity.

That is, we cannot then perform the local gauging of T4's global linear transformations down to GCT's local nonlinear transformations in a physically correct way if Lp^2 = 0.

The Ricci rotation coefficients Aa^b^c are global phases in 1916 GR since O(1,3) is not locally gauged.

That is,

Tu = eu^aAa^b^cSbc

Where {Sbc} is the Lie algebra (boosts & rotations) of O(1,3)

is not a dynamically independent torsion field at this stage but is totally dependent in its variation from the T4 derived tetrad eu^a.

Locally gauge O(1,3) to get Shipov's torsion field theory!

There is NO conformal group here yet. However, we can easily extend the Brazilian gauge formalism of GR to include it, i.e. to locally gauge full conformal group and to spontaneously break it in the vacuum only if data so demands!

*However I see no experimental necessity, so far, to go to the conformal group or to Finsler geometries.

i. R. Cahill & the Sicilians report data that requires spontaneous breakdown of O(1,3) vacuum symmetry in space of Earth's neighborhood. No local gauging to get torsion fields is needed.

ii. NASA Pioneer anomaly does not require any new mathematics to explain.

iii. Shipov & Akimov report data that would require local gauging of O(1,3) to get torsion fields with propagation. Richard Hammond also reported data to that effect.

We now have a battle-tested computational paradigm to deal with ALL observational anomalies - is my conjecture. including UFOs as a military threat e.g. ABC's Peter Jennings in
http://abcnews.go.com/Primetime/story?id=468496&page=1

Finsler geometries et-al? Who ordered that? However, in case I am wrong in my sweeping conjecture, it's good to keep one eye on these exotic mathematical flora and fauna. :-)

I have tried to edit the typos here so that I could better understand the message. Please correct any of my edits if they change the meaning. I also make some comments. Thanks.

On Feb 6, 2005, at 7:37 AM, Sergiu Vacaru wrote:

Dear Colleagues,

Thanks for your kind e-mails and information on your duscussions. I hope I could participate in the future.

Here I have some remarks on Finsler geometry:

It is considered to be a generalization of Riemnannian geometry, on tangent bundles, with anisotropies of metric (depending on direction), which is more sophisticated and with less experimental evidence for such extensions.

Anisotropies of metric? Meaning what? Off-diagonal guv? Solutions need not be spherically symmetric nor isotropic.

The most surprising thing is that Finsler-like structures can be modelled on (pseudo) Riemannian spaces (in a more general context of string theory with nontrivial torsion, on Rieman--Cartan manifolds) by generic off--diagonal metrics which can not be diagonalized by coordinate transforms but effectively diagonalized with respect to certain nonholonomic frames (vielbeinds) with associated nonlinear connection structure. The off--diagonal terms define just the coefficients of nonlinear connection (N-connection) and inversely. On a such manifold, a part of the degrees of freedom (coordinates) are holonomic and another ones are nonholonomic (subjected to certain constraints) which is distinguished by the nonlinear connection structure and creates a specific anisotropy of gravitational interactions even in the framework of general relativity. We found a number of exact solutions of the Einstein equations modeling such configurations (deformed black holes, spinor and solitonic configurations, cosmological type solutions). In a particular case, we can take state that the off-diagonal terms are related to the Cartan N-connection from Finsler geometry and the diagonal terms are some Finsler metric coefficients. This allows us to model Finsler structures just on a (pseudo) Riemannian space.

I will need to see the actual math. Sounds interesting.

The conclusion is that we can not avoid Finsler geometry even in the usual (Einstein) gravity if the generic off-diagonal metrics and nonholonomic frames are considered. By the way, Riemann in his 1854 hability thesis considered already Finsler metrics. He had not investigated them considering the constructions to be of higher order sophistications. It was necessary in the elaboration of E. Cartan's moving frame method in order to understand that by nonholonomic frames it is possible to model Finsler structures even by quadratic (metric) forms but with additional nonholonomic structures.

Interesting.

The main problem is that the bulk of monographs on Finsler geometry were written to emphasize more general constructions than the the Riemannian ones, just on tangent bundles. It is realy difficult to see the point on modeling of Finsler structures on the usual Riemannian manifolds. The bulk of discussions and pessimistic conclusions for physics (even of Berstein and C. Will) had been taken with respect to physical models on tangent bundles and without a deep analysis of the N-connection structure. I hope soon to elaborate an e-preprint with details.

My credo is: we can not understand deeply the Einstein's gravity without Finsler geometry.

Finally, I note that Clifford-Finsler spaces result trivially from the Clifford spaces if nonholonomic frames and off-diagonal metrics are considered.

I'm grateful for your time.

SV.

This has piqued my interest in looking at the math. :-)

Carlos Castro escribiĆ³:

Dear Sergiu ( Jack and Tony ) : I included Sergiu Vacaru in this e-mail because he is the expert on non-holonomic and anisotropic deformations of ordinary Riemann-Cartan geometry. He is an expert on Finsler geometries that are far more fundamental than Riemannian geometries. You may understand the recent work by Cahill on anisotropic propagation of light signals within the context of Sergius' more findamental work. This is very common in Finsler geometries. Sergiu gave me the website

http://www.icca7.ups-tlse.fr

where you can register for the Int. Congr. of Clifford Algebras to be hold in Toulouse, France.

In case you wish to attend.
I have to see.
All the best
Carlos

No comments: