The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Topic: Gravity tetrad as the spin 1 gauge field
Replies: 0  

Advanced Search

Back to Topic List Back to Topic List  
Jack Sarfatti

Posts: 1,942
Registered: 12/13/04
Gravity tetrad as the spin 1 gauge field
Posted: Sep 24, 2007 6:48 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

The principle of local gauge invariance for spin 1 compensating vector
fields that are renormalizable with the Higgs-Goldstone hidden ODLRO
symmetry (t'Hooft, 1973) is battle-tested experimentally (electroweak
force & QCD) and is a direct consequence of Einstein locality for the
Bohmian IT hidden variables when the effect of the BIT quantum (super)
potential is small Q. The effect of the latter is, in essence, a tiny
warp-drive bubble in which the, e.g. electron as a tiny T = 0 Kelvin
non-radiating "black hole", can under the action of Q can literally turn
around in time giving a literal meaning to the Feynman diagrams
(electron positron pair creation & annihilation) as more than simply a
heuristic of perturbation theory.

The Einstein-Cartan "spin 1" tetrad 1-form Lorentz group vector field
decomposes as the local T4(x) minimal coupling to matter fields Psi

e^aPa = I^aPa + A^aPa

i.e. P -> P + A acting on Psi

P = I^aPa

A =A^aPa

{Pa} is the Lie algebra of the universal rigid T4 symmetry group of all
material field actions in 1905 Special Relativity (SR).

This is analogous in QED to the local U1(x) minimal coupling

P -> P + A acting on Psi

Now what is I^a?

I^a are the 4 tetrad 1-form fields I^a = I^audx^u that are AFFINE
INVARIANTS under the restricted flat geodesic Global Inertial Frame
(GIF) transformations in Minkowski spacetime under the 10-parameter
Poincare group. That is, only maps from one GIF to another GIF.

s^2(1905) = I^aIa AFFINE INVARIANT

As soon as one considers a transformation from a GIF to an accelerating
off-(flat) geodesic Global Non-Inertial Frame (GNIF) I^a is no longer
invariant. Note these are constant accelerations the same
everywhere-when (conformal boosts?)

In fact I^a -> I^a + X^a

Therefore, we introduce a compensating gauge potential A^a such that

A^a -> A^a - X^a

to keep the sum e^a = I^a + A^a INVARIANT under this wider NON-AFFINE
group of PHYSICAL FRAME transformations.

s^2("conformal"?) = (I^a + A^a)(I^a + A^a)

This is like in electromagnetism where the gauge transformation is

A -> A - Gradf

This is the first BABY STEP to Einstein's 1916 General Relativity (GR).

There is a SPIN-CONNECTION 1-form

S^a^b = w^a^bce^c = w^a^bc(I^c + A^c)

The curvature field 2-form R^a^b must vanish in the above case

R^a^b = DS^a^b = dS^a^b + S^ac/\S^cb = 0 everywhere-when

The torsion field 2-form T^a must also vanish

T^a = De^a = de^a + S^ac/\e^c

Einstein's 1915 GR locally gauges the rigid 4-parameter T4 translation
subgroup of P10 to T4(x), i.e. locally inhomogeneous displacements where

I^a -> I^a(x)

I^a(x) -> I^a(x) + X^a(x)

A^a -> A^a(x) -> I^a(x) - X^a(x)

Now the HOMOGENEOUS global spacetime interval s connecting pairs of
arbitrarily separated events is localized to the INHOMOGENEOUS local
ds(x) connecting pairs of closely spaced events.


R^a^b(x) =/= 0 is allowed but T^a(x) = 0 is enforced.

i.e. curvature but not torsion in 1916 GR.

Note that all states connected by a redundant gauge transformation (on
same gauge orbit) are the same physical state.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.