Levi-Civita Connection Decomposition in GR?

From: Jack Sarfatti (sarfatti_at_pacbell.net)
Date: 12/19/04 

Date: Sun, 19 Dec 2004 13:18:41 +0000 (UTC)

This problem arose from a student in the General Relativity course I am
teaching. Comments would be helpful.

"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.

Students have asked me if there is 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?

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)

INTRINSIC GEOMETRY PART of {LC} = CurlI(GCurl{LC})

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 this 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 ?

One student in my relativity class claims Vilenken's vacuum domain wall
is an example.

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