Jack SarfattiNOW FOR THE REAL PHYSICAL MEANING OF THE LEVI-CIVITA THEOREM THAT THE DIFFERENCE IN TWO LC CONNECTIONS IS A TENSOR

THAT TENSOR IS THE PROPER ACCELERATION OF THE LOCAL NON-INERTIAL FRAME.

On 1/8/2014 9:55 PM, Jacob Sarfatti wrote:

Its physical significance is pretty simple:

The non-tensor inhomogeneous term in the transformation of the Christoffel symbol connection field is the change in proper acceleration of the lnif.

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More precisely,

What I meant was that the XdX inhomogeneous term in the GCT gauge transformation LNIF -> LNIF' CANCELS OUT in the computation of the proper acceleration of the LNIF detector, which by definition is a property of its REST FRAME, i.e. the reading of the accelerometer clamped to its center of mass. We are here talking about the local rest frame of the detector not that of the test particle the detector is monitoring.

It's this cancellation of the two equal and opposite XdX terms in the respective rest frames of the detectors that maintains the tensor property of the proper accelerations of the two coincident LNIF/LNIF' connected by GCT X.

GCT X is a an element of the local translational gauge group T4(x).

So in EM we have

A -> A' = A + (hc/e)dS

this keeps

P = mV + (e/c)A gauge invariant.

mV -> mV' = mV + hdS

S = quantum phase of wave function of test particle with inertia m and with charge e.

(e/c)A -> (e/c)A' = (e/c)A - hdS

hdS is the momentum carried by a longitudinally polarized virtual photon that is the CONTACT electrical force in quantum field theory.

dP/dt = 0 is ACTION-REACTION principle between electric charge and coincident EM field.

dP/dt = 0 is the real force law

mdV/dt = eE

since E ~ (1/c)dA/dt

Now for gravity, we focus on the proper acceleration of the detector (a rest frame property of the detector) not the test particle.

The GCT X induces an XdX kinetic acceleration term which is canceled by the equal and opposite XdX term in the LC transformation.

This keeps the tensor property intact for

DV^i(LNIF)/ds = {LNIF}^i00

in every rest frame

i.e.

DV^i'(LNIF')/ds = {LNIF}^i'0'0'

The LC connection in flat spacetime is exactly like the LC connection in curved spacetime.

Flat spacetime is simply an unstable solution of Einstein's field equations.

Mathematically the LC connection has zero self curl in flat spacetime. The self curl of the LC connection is the curvature tensor.

The LC connection is not zero in flat spacetime in non-inertial frames.

The inhomogeneous term is the change in proper acceleration of the frame.

Symbolically X = GCT which physically is the transformation between two COINCIDENT LNIFs each with proper acceleration encoded in their corresponding LCs.

LC --> LC' = XXXLC + XdX

I must have the patience of a saint.

More accurately of a demon!

The proper acceleration of the test particle is the tensor

DV(test particle)/ds = dV(test particle)/ds - LC(LNIF)V(test particle)V(test particle)

The proper acceleration of the LNIF in its rest frame is

DV(LNIF)/ds = dV(LNIF)/ds - LC(LNIF)V(LNIF)V(LNIF)

but in the rest frame, for the 3-vector parts

V(LNIF) = 0

dV(LNIF)/ds = 0

Therefore

DV^i(LNIF)/ds = - LC(LNIF)^i00V^0(LNIF)V^0(LNIF)

V^0 = 1 in the REST FRAME always

therefore,

DV^i(LNIF)/ds = - LC(LNIF)^i00

Under the GCT in the REST FRAME of LNIF'

DV^i'(LNIF)/ds = - LC(LNIF')^i'0'0'V^0'(LNIF')V^0'(LNIF') = - LC(LNIF')^i'0'0'

Because, just like in the U(1) EM gauge transformations

dV(LNIF)/ds -> dV(LNIF')/ds = dV(LNIF)/ds + XdX

Whilst

- LC(LNIF)^i00 -> - LC(LNIF)^i'0'0 - XdX

Therefore the XdX inhomogeneous terms cancel out for the transformation LNIF -> LNIF' when we calculate the proper acceleration change of the center of mass /origin of the LNIF.