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localize the full Poincare group.

The spin connection 1-form gets an extra piece K the contortion

A' = A(T4) + K(SO1.3)

Define the curvature 2-form

R' = DA' + A'/\A'

Teleparallelism means

R' = 0

Obviously then the old Einstein curvature F can be expressed in terms of the contortion tensor (relative to GCT of GR) and

T = K/\e =/= 0

because de + A/\e = 0

We now have a true Yang-Mills theory where

the first set of "Maxwell" equations is

D'T = 0 = torsion analogs of Faraday induction and no magnetic monopoles - but for the gravity field

where

D' = d + A + K

the source equations (analog of Ampere's and Gauss's laws) are

D'*T = *J

local conservation of torsion currents

D*J = 0

as Waldyr Rodrigues says this is all equivalent to Einstein's GR

Hagen Kleinert also says you can either use the torsion or the curvature in the teleparallel sense.

Since Woodward needs the EM analogy, this is more rigorous and perhaps can be applied to his experiment.

Note the Yang-Mills Lagrangian torsion density is ~ (1/4)Tr{T/\*T} - it's a non-Abelian gauge theory for gravity.

From:

The local gauge principle with minimal coupling allows one to infer the extended Lagrangian of the original source field + the induced gauge force field. It is basically Einstein's locality principle. Observers that are spacelike separated can orient their detectors independently without changing the physics. Remember this is all classical - no quantum entanglement at this stage.

In general

Local gauging induces a connection 1-form A

This gives a covariant exterior derivative

D = d + A

the "field" F (in GR it's curvature, when A is the spin-connection)

F = DA = dA + A/\A

The torsion is

T = De

where e = the set of 4 tetrads

In analogy with U1 electromagnetism

DF = 0 (a kind of Bianchi identity)

D*F = *J this equation in tetrad notation maps to Einstein's Guv + kTuv = 0

See Rovelli Ch. 2

D*J = 0 is local conservation of stress-energy current densities in the case of T4(x) GR.

Similarly if there is torsion.

To get both torsion and curvature as independent dynamical fields we need to localize the entire Poincare group not just its T4 subgroup.

Category: MyBlog

Written by Jack Sarfatti

Published on Saturday, 06 August 2011 22:05

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