Saturday, December 18, 2004

"The Question is: What is The Question?" J.A. Wheeler

The {LC} connection field in GR curved spacetime is analogous to the EM vector potential A connection in internal space.

The tidal stretch-squeeze GCT tensor field = {LC} curl of itself is analogous to the Maxwell EM Fuv field tensor.

{LC} and A are both Cartan 1-forms. Curvature and Fuv are both Cartan 2-forms.

Z's hullaballo bruhaha has been a search to find the GR analog to the Maxwell decomposition

F = -GradU + CurlA

Note Cartan identities d^2 = 0

CurlGradU = 0

DivCurlA = 0

GradU is a Cartan 1-form = dS

Curl 1-form = d(1-form)

dS is an exact 1-form

CurlA =/= 0 if the 1-form A is NOT exact.

CurlA =/= 0 is an exact 2-form = dA

DivCurlA is d^2A = 0 vanishing 3-form

All vector fields in 3D are UNIQUELY defined if their circulation densities (curl) and source densities (divergence) are given functions of the coordinates at all points in space, and if the totality of the sources, as well as the source density, is zero at infinity. Panofsky & Phillips "Classical Electricity and Magnetism" p. 2

Note an infinite flat plate (vacuum domain wall) violates the required asymptotic flatness in the GR application.

Therefore, The Question is, is there an analogous decomposition for GR at the level, not of the tidal curvature, but of the {LC} itself?

That is, thinking of {LC} as a "vector field" does it have a COORDINATE INDUCED (INERTIAL) divergence part from accelerating LNIF non-geodesic observers + INTRINSIC GEOMETRY curl part?

Z says yes. However, Z has failed to produce explicit formulae in even the simplest case of the Schwarzschild vacuum solution.

Now in the above analogy

{LC} the connection of 1916 GR is a 1-form.

Therefore, The Question is, is there a decomposition

LC 1-form = d(Zero Form) + 1-form = exact 1-form + non-exact 1-form

The exact 1-form is the INERTIAL FIELD "coordinate-dependent" part.

The non-exact 1-form is the INTRINSIC GEOMETRY part.

Note that the CURVATURE 2-form is

d{LC} = d(non-exact 1-form)

However, in non-Abelian GR

d{LC} = {LC} covariant curl of itself!

Back to EM

U = Integral over a Green's function of a scalar density d = I(Gs)

A = Integral over the same Green's function of a vector density B = I(GB)

Therefore if

{LC} = GradU + CurlA

Curl{LC} = CurlCurlA

But self-consistency requires that

B = Curl{LC}

A = I(Gcurl{LC})

Therefore, the CONJECTURE is, in analog form:

{LC} = -GradI[Gs] + CurlI(GCurl{LC}) ?

The GLOBAL boundary conditions are in the Green's function G.

INERTIAL FORCE coordinate part of {LC} = -GradI(Gs)


The Curl and Grad are of course {LC} covariant operations so that this is all highly nonlinear. There is also the problem of the definition of the Green's functions as well as the Integrals over the curved manifold.

It is not clear if Z's idea can be carried out.

There is also the side issue of whether a stationary homogeneous g-field i.e.

mc^2{LC}^ztt + External Non-Gravity Force^z = 0

In the LNIF rest frame of a HOVERING non-geodesic observer at FIXED DISTANCE z from the FLAT PLATE source Tuv =/= 0, where

{LC}^ztt = g/c^2

{LC} Curl of {LC} = 0 where Tuv = 0

Can be an EXACT SOLUTION of Einstein's Guv = (8piG/c^4)Tuv ?

Z claims Vilenken's vacuum domain wall is an example.

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