Thanks to Robert Israel for correcting a math error. However I think I
see how to formulate the same basic physical idea, which I think is
correct, with the proper math.

First for the record the math error I made:

On Dec 9, 2005, at 12:31 PM, Jack Sarfatti wrote:
OK, let's see

B = d(Theta Phi) = (dTheta)Phi + (Theta)dPhi

is true

So next is

d[(dTheta)Phi] = (d^2(Theta))Phi + (-1)^deg(dTheta)dTheta/\dPhi = -
dTheta/\dPhi

and

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

So you were right. Thanks. :-)

d^2(ThetaPhi) = 0 still

So I was wrong about that.

On Dec 9, 2005, at 8:45 AM, Robert Israel wrote:

I'm not going to argue. This is the way the exterior derivative was set
up by Elie Cartan. See e.g.
http://en.wikipedia.org/wiki/Exterior_derivative
or Spivak, "Calculus on Manifolds".

Cheers,
Robert Israel

On Thu, 8 Dec 2005, Jack Sarfatti wrote:


On Dec 8, 2005, at 10:26 PM, Robert Israel wrote:
Note, by the way, that if B = d(Theta Phi) = Theta dPhi + Phi dTheta,
then dB = dTheta /\ dPhi + dPhi /\ dTheta = 0, not 2 dTheta /\ dPhi.

Thanks for pointing that out.


Corrected Draft 2

On Dec 9, 2005, at 12:30 AM, Jack Sarfatti wrote:

Imagine that the first Planck scale vacuum symmetry breaking at ~ 10^-44
seconds has the stable point monopole topological defect with vacuum
manifold (this is not a magnetic monopole of the EM field, but, rather,
a geometrodynamic monopole.

V(Planck) = G(false unordered vacuum)/H(ordered vacuum) = S2

i.e. Second Homotopy Group PI(S2) = Z

the integers are 2D "wrapping numbers" around a sphere surrounding the
point defect in physical based space rather than 1D winding numbers
around a circle surrounding the line defect in physical base space when
First Homotopy Group PI(S1) = Z.

The discrete wrapping numbers come from the single-valuedness of the
local macro-quantum vacuum coherent order parameter of the inflating
emergent curved space-time universe shown in the picture below from
Richard Gott III. Gott ist Gut! ;-)

These are obviously the Hawking-Beckenstein BITS that is a trivial
result from the generic Higgs field vacuum coherent order parameter if
the order parameter manifold is S2.

Note that these surrounding surfaces in physical 3D space are cycles
that do not bound of dim 2 & 1 respectively.

The vacuum manifold S2 in fiber space, coincidently has an S2
surrounding surface of the point defect in physical 3D space.

The macro-quantum vacuum coherence order parameter is LOCAL and this
explains how local curved space-time emerges from the nonlocal quantum
substratum. There is also an entropy drop here. However, Richard Gott
and Li-Xin-Li explain the low entropy of early self-creating universe as
a Planck scale time-loop CTC in which only retarded photons having
redshifts in the inflating universe can give a stable globally
self-consistent curved vacuum.



The red arrows are the retarded zero rest mass photons getting
redshifted. The inflation starts in the white neck part where G - H.

http://www.sns.ias.edu/~dejan/CCS/wo...t.III.eng.html

OK the Second Law of Thermodynamics is explained by globally
self-consistent time travel to the past in the sense of Igor Novikov's
idea (that I got independently years before I knew of Novikov's view).

How about the 't Hooft-Susskind hologram?

Again assuming V(Planck 10^-44 sec) = S2

from below

Next consider an S2 fiber. This has TWO functionally-independent
"orthogonal" Goldstone phases Theta(x) and Phi(x), i.e. 3 real scalar
fields where

tan(Theta) = phi(x)1/phi(x)2

tan(Phi) = phi(x)2/phi(x)3

Note the tangent of the angle from phi(x)1/phi(x)3 is not independent of
the first 2 because sum of squares of direction cosines add to 1.

Physically the two angles polar and azimuthal describe linearly
independent displacements in 3D fiber space.

The exterior derivative d on a 0-form is like a gradient operator in
base (x) space. The sphere S^2 is a fiber at each point x in base space.

Consider exact 1-form is

B = Lpd(ThetaPhi) = Lp(dTheta)Phi + Theta(dPhi)

The vanishing 2-form is

dB = 0

However,


Because d(Theta) is not "parallel" to d(Phi)

We simply define the non-vanishing 2-form

C =Lp^2 dTheta/\dPhi

= area flux density


This is an angular area element in vacuum manifold fiber space not
physical space.

Of course dC = 0

The closed exact 1-form B is essentially the curved part of the
Einstein-Cartan tetrad field that is roughly the square root of
Einstein's metric tensor field.

The non-vanishing 2-form C is the geometrodynamic flux density analogous
to the magnetic flux density.

B is also the compensating potential in the local gauging of the
translation T4 group as well as emergent from the S2 internal symmetry
breaking at the initial Planck phase transition. B, however, is not a
potential for the area flux density 2-form C.

Simply use the Bohm-Aharonov singular extension of Gauss's theorem

The integral of the 2-form area flux density C around the closed
nonbounding surface is equal to the integral of dC in the interior
volume. Even though dC = 0 locally and near the surface nevertheless the
global integral is quantized. This is basically the World Hologram that
all the physics of the interior is coded on the surrounding surface of
the point defect that defines the Planck scale vacuum phase transition.

That the 3-form dC = 0 locally is analogous to the Bohm-Aharonov effect
where the 2-form dA = 0 on the path of the electrons where A is the EM
1-form potential. The electrodynamic case is for an S1 fiber. In
contrast the geometrodynamic case is for an S2 vacuum manifold fiber. In
other words the World Hologram is the Bohm-Aharonov effect for the
geometrodynamic field with a point (neutral) gravity monopole defect in
the initial Planck-scale Higgs field.

On Dec 8, 2005, at 11:15 PM, Jack Sarfatti wrote:

On Dec 8, 2005, at 10:05 PM, Jack Sarfatti wrote:

For spontaneous symmetry breakdown of the ground state of a system
described by one real scalar field order parameter, phi(x)1 the vacuum
manifold G/H in a renormalizable quartic potential has the topology S0
with only 2 possible discrete values of the Goldstone phase 0 or pi.

The topological defect there where the order parameter vanishes is a 2D
domain wall in physical 3-space.

Dim(Vacuum Manifold) + Dim( Stable Topological Defect in Physical Space)
+ 1 = Dim of Physical space

0 + 2 + 1 = 3

For two real scalar fields the single Goldstone phase describes a vacuum
manifold with the topology of the circle S1. Dim of the topological
defect in physical space is the 1-D string.

1 + 1 + 1 = 3


For three real scalar fields with two independent Goldstone phases, the
vacuum manifold has dim 2, the topological defect has dim 0, i.e. a
point monopole.


2 + 0 + 1 = 3


The effective quartic "Mexican Hat" potential here is

V = a(phi1^2 + phi2^2 + phi3^2) + b(phi1^2 + phi2^2 + phi3^2)^2

The degenerate minima of V is a sphere S2 of radius eta, where

eta^2 = phi1^2 + phi2^2 + phi3^2

The direction cosines in the vacuum manifold fiber space are, at fixed x
in base space

phii/eta

dV/dphij = 0

d^2V/dphij^2 0

One can generalize this to include anisotropies in fiber space. What
happens then?

Obviously S2 - S1 - S0 in fiber vacuum manifold order parameter space.

i.e. topological defects in physical space go from point to line to
surface. That is from monopole to vortex to domain wall.



OK, consider a circle fiber S1 over a base space in a fiber bundle.

S1 has a Goldstone phase Theta that is the 0-form Theta(x) where x is in
the base space.

You can think instead of 2 real scalar fields phi(x)j where

tan(Theta) = phi(x)1/phi(x)2

In this case all exact forms are closed as everyone takes for granted.

That is

A = Theta

B = dTheta

dB = d^2Theta = 0 locally

Next consider an S2 fiber. This has TWO functionally-independent
"orthogonal" Goldstone phases Theta(x) and Phi(x), i.e. 3 real scalar
fields where

tan(Theta) = phi(x)1/phi(x)2

tan(Phi) = phi(x)2/phi(x)3

Note the tangent of the angle from phi(x)1/phi(x)3 is not independent of
the first 2 because sum of squares of direction cosines add to 1.

Physically the two angles polar and azimuthal describe linearly
independent displacements in 3D fiber space.

Define the 0-form Theta(x)Phi(x)

The exterior derivative d on a 0-form is like a gradient operator in
base (x) space. The sphere S^2 is a fiber at each point x in base space.

The exact 1-form is

B/Lp = d(ThetaPhi) = (dTheta)Phi + Theta(dPhi)

The non-vanishing 2-form is

C/Lp^2 = 2d(Theta)/\d(Phi) =/= dB/Lp

Because d(Theta) is not "parallel" to d(Phi)

This is an angular area element in vacuum manifold fiber space not
physical space.

Of course dC = 0