From: Jack Sarfatti
Date: November 10, 2005 3:47:02 PM PST
To: ROBERT BECKER
Subject: A curious observation about ds^2
In ordinary tensor notation
ds^2 = guvdx^udx^v
But in Cartan tetrad notation we can write
ds^2 = eae^a
ds = e = 1 + B 1-form = 'd' (Goldstone Phase of Higgs Vacuum Field)
B is the local compensating gauge potential from global T4 ---> Diff(4).
Therefore, the Hodge dual *ds is a 3-form
From Stoke's theorem - ignoring topological defects for the moment
Loop integral of e = surface integral of dB
But *ds is a 3-form, whose volume integral is equal to the surface integral of a 2-form F where
*ds = dF
Heuristically "s" is the 0-form Goldstone phase in the standard model's internal symmetry breaking for origin of inertia of leptons & quarks (here W-1,W0,W+1, massless photon, & 1 Higgs boson in n = 3 order parameter.
G = U(1)hyperchargeSU(2)weak --3 adjoint irrep--> H = U(1)em
with degenerate vacuum manifold G/H = U(1)hyperchargeSU(2)weak / U(1)em ~ S2
The second homotopy group PI2(S2) = Z
Z is the set of wrapping numbers (i.e. 2D version of 1-D winding numbers) such that when one covers the S2 sphere in 3D space that surrounds the point defect, there is a wrapping N times around the vacuum manifold G/H (+ or -).
This comes about because of the single-valuedness of the order parameters in G/H ~ S2.
Therefore, the Hawking-Bekenstein quantization of area + World Holography seems to pop out trivially, i.e. from Gauss's theorem for quantized surface integral about the point defect where Higgs amplitude vanishes and Goldstone phase is undefined. Take for space the WHOLE UNIVERSE at some cosmic epoch with the "surface" as the "screen" and we need only assume ONE geometrodynamic point defect somewhere, same idea as Dirac's string to get quantized electric charge if there is only ONE magnetic monopole per universe.
To review, consider ds as a 1-form with s the 0-form as the Goldstone vacuum coherent world hologram phase.
Because of U(1)SU(2) Higgs mass generation in standard model + equivalence principle, jump from 1-form ds to 3-form *ds and apply the singular "flux-without-flux" version of Gauss's theorem for a surface in physical space that surrounds the point defect demanded by the standard model.