Sunday, August 13, 2006

Einstein's gravity field emerges from the coherent Goldstone phases of the vacuum Higgs inflation field.

Given a p-form P and an q-form Q

P/\Q is a p + q form

P/\Q = (-1)^(pq) Q/\P

d(P/\Q) = (dP)/\Q + (-1)^pP/\(dQ)

Let O be a 0-form and I be a 1-form

O/\I = (-1)^(0x1)I/\O = I/\0

d(O/\I) = dO/\I + (-1)^0O/\dI = dO/\I + O/\dI

d^2 = 0

Let Theta & Phi be Goldstone phase 0-forms of the vacuum ODLRO inflation field.

Define the 1-form

A = Theta/\dPhi - dTheta/\Phi

d(Theta/\dPhi) = dTheta/\dPhi + Theta/\d^2Phi = dTheta/\dPhi

d(dTheta/\Phi) = d^2(Theta)/\Phi - dTheta/\dPhi = -dTheta/\dPhi

Therefore, the 2-form is

dA = +2dTheta/\dPhi

Let eu^a be a set of Einstein-Cartan sub-geometrodynamic tetrad field components. The O(1,3) indices a,b,c... are raised and lowered with the globally flat constant Minkowski metric nab. The GCT Diff(4) indices are raised and lowered with the locally curvilinear GCT metric guv. Einstein's local principle of equivalence is formally here for a point event E

guv(E) = eu^a(E)nabev^b(E)

The 4 GCT 1-form tetrad fields are

e^a = eu^adx^u

Form the contracted O(1,3) scalar

e = e^a&a

&a is a basis of vector fields in flat tangent space.


e = 1 + A

1 is the constant unit scalar

When there is no real gravity curvature field

A = 0 identically in the region of events {E}.

A = Theta/\dPhi - dTheta/\Phi

Au^a = Lp(Theta/\dPhi - dTheta/\Phi)u^a

Lp = (hG/c^3)^1/2

dA = 2(dTheta)/\(dPhi) = (dA)uv^adx^u/\dx^v&a

(dA)uv^a = 2Lp^2(dTheta/\dPhi)uv^a

A topological 3D space point defect (world line in 4D spacetime) in the macro-quantum "More is different" vacuum ODLRO local order parameter inflation field from a spontaneously broken continuous symmetry has Theta & Phi undefined.

The integral of dA around a closed surface that surrounds that point defect is quantized in order to keep the local order parameter single valued.


Integral of dA on a closed surface with no boundary that is itself not a boundary ~ (Integer)Lp^2, i.e. Hawking-Bekenstein black hole entropy rule of 1 c-bit per Planck quantum of area and t'Hooft-Susskind world hologram rolled into one formula.

Note, if there was only one Goldstone phase Theta instead of 2 the situation would be like
1. Superfluid helium
2. Superconductor
3. Self-trapped laser filaments in nonlinear optics.

The topological defects would be string vortex lines, i.e. world sheets in 4D spacetime.
The surrounding surface would be a non-bounding 1D loop without boundary enclosing a vortex line, and the corresponding loop integral would be quantized as integer h/m for a neutral superfluid, or h/2e magnetic flux quantum for a superconductor.

No comments: