Sunday, January 07, 2007

Teleparallelism as Clifford Parallelism

I seem to have rediscovered what Whittaker knew in 1930
I suppose it's obvious but I have not seen this connection made in the little I have read in this subject.

I follow here Roger Penrose in "The Road to Reality" though he does not mention what I am about to show you. In S^2, P^2, S^3 and P^3 two straight lines (geodesics) that start out parallel will converge. This is "geodesic deviation" or "curvature." In L^2 & L^3 the opposite like parallel rays on a converging lens with real focus in geometrical optics - diverge like parallel rays through a convex lens on the emerging side flat on the entering side with a virtual focus. Imagine a straight line l in Euclidean 3D space. Displace line l parallel to itself along direction vector d perpendicular to line l. Rotate line l to line m slightly as the perpendicular displacement d increases. The result is line m skew to original line l. The distance between lines l and m will increase if we move outwards along them in either direction. Now apply this same process to either 3D spherical space S^3 or 3D projective space P^3, but adjust the rate of skewing rotation causing increasing separation such that it exactly compensates the natural convergence in spherical and projective non-Euclidean geometries. This is the Clifford parallelism.

"Teleparallelism" = "Clifford parallelism"

The "skewing" is the Shipov torsion field (in 3+1). Clearly if you try to make a closed loop it will have a torsion gap (dislocation in world crystal lattice).

The 6D "Calabi-Yau" O(1,3) torsion compensates the 4D T4 curvature (note the non-dyamical spin connections for zero torsion make rotational disclination defects round closed loops without torsion gaps to second order Taylor series expansions).

So Shipov's idea is that the torsion field is really the source of the curvature field.

Guv(curvature) = kTuv(torsion)

Therefore, he has to get all of matter from torsion fields in a spinor formalism. Since he has a 6D Calabi-Yau space naturally sitting there perhaps that can be done i.e. superstring theory with Shipov's torsion as the missing organizing idea?

Jack Sarfatti
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein

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