Tuesday, May 03, 2005

Curvature, Torsion & Hyperspace

Also, I do not understand formally his equation

Riemann Curvature = Affine Curvature - (Torsion + Nonmetricity)Curvature

In general

Curvature = 4D Flat Curl of Connection + (Connection)^2

In particular

Riemann Curvature = 4D Flat Curl of (LC) + (LC)^2

EEP says that in LIF

Riemann Curvature = 4D Flat Curl of (LC)

since (LC) = 0 in LIF ALWAYS

Affine Curvature = 4D Flat Curl of Affine Connection + (Affine Connection)^2

= 4D Flat Curl of (LC) Connection + 4D Flat Curl of Torsion Connection +4D Flat Curl of Nonmetricity Connection

+ [(LC) + (Torsion) + (Nonmetricity)]^2

EEP simply tells us that in LIF, (LC) = 0

But in an LNIF clearly there are cross terms and, therefore, the curvature tensor is not a simple sum of its parts. There are interference cross-terms in the LNIF. No way to escape that assuming local gauge principle to get Shipov torsion & Hyperspace non-metricity as DYNAMICAL connection fields. There are also the conformal boosts and dilations waiting to be locally gauged for even MORE dynamical connection fields!

EEP demands (LC) = 0 in an LIF, since in his model

(LC) = (Affine) - (Torsion + Nonmetricity)


That is

(Affine) = (Torsion + Nonmetricity) in the LIF.

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