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World Hologram in Emergent Gravity



 
 
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Old August 14th 06 posted to sci.math,sci.physics.relativity,sci.philosophy.tech,sci.space,sci.astro
Jack Sarfatti
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Default World Hologram in Emergent Gravity

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.

Ansatz

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.

Therefore

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.











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