Space Forum / Astronomy / September 2005

Lecture 2

Ex 4
Mechanical Model of a Phase Singularity

Simplest case.

Imagine a plane. Pick an origin O. Use polar coordinates, (r,theta) for
arbitrary moving point P.

Pick a point 0' with fixed coordinates (a, chi).

Draw a circle of radius b < a centered at O' with coordinates (b,phi)

Let point P move around this circle whose center O' is displaced from
origin O.

Obviously when a =/= 0 the  total theta angle integral of the 1-form dtheta
swept out in one complete circuit round the circle is ZERO. Basically
theta oscillates.

Note that the angle theta depends on the angles chi and phi.

Half of the movement is clockwise and then counter-clockwise for dtheta
on successive semicircles as P winds around the circumference of the
displaced circle. This is most easily seen intuitively all at once when
O' is vertical compared to O (on y-axis ordinate).

Note what happens when you move the circle to different locations on the
plane.

Draw tangents from O to the circle in different locations.

In contrast, when a = 0, or alternatively, b > a the total angle
integral of dtheta is 2pi.

Homework Problem

Use trigonometry to make an algebraic proof.

-------------------

For 3D flat metric, the Hodge * is with the right-hand rule convention

*dx/\dy = dz
*dy/\dz = dx
*dz/\dx = dy

Left-hand rule is

*dy/\dx = dz
*dz/\dy = dx
*dx/\dz = dy

Parity transformation interchanges left and right hand rules in 3D.

(x,y,z) -> (-x,-y,-z)

SU(2)hypercharge breaks parity symmetry and it also may be the origin of
inertia and gravity.

p-forms  |p) = (p!)^-1Fuv ... dx^u/\dx^v ...

p-factors, p =< n = dim of manifold.

Important formula

The exterior product /\ of forms is a parallelepiped in the co-tangent
n-dim space of constant phase wave fronts in contrast to the tangent
space of particle paths normal to the wave fronts.

For R^3

A = Axdx + Aydy + Azdz

F =  dA = ( Az,y - Ay,z)dy/\dz + (Az,x - Ax,z)dz/\dx + (Ax,y - Ay,x)dx/\dy

2-form independent of metric

*F = *dA = ( Az,y - Ay,z)dx + (Az,x - Ax,z)dy + (Ax,y - Ay,x)dz

* dual 1-form in 3D manifold with a metric specified.

Note, if

A = df

F = dA = d^2f = 0

Therefore

 ( Az,y - Ay,z) = 0 etc

, is ordinary partial derivative

i.e. mixed second order partial derivatives of the 0-form f commute in
that case.

However, in the case of a phase-singularity, there is some kind of
region in the manifold where the mixed partials of the 0-form Goldstone
phase of the local macro-quantum coherent vacuum order parameter Higgs
field in our primary application to physics of this formalism do not
commute. This is a topological defect in the vacuum manifold G/H, where
I write

A = 'd'f

d'd' =/= d^2 = 0

due to multiply-connected manifolds

F = dA =/= 0

e.g. non-integrable anholonomic multi-valued gauge transformation of
Hagen Kleinert

AKA

Flux without flux

see also the related idea of the nonlocal Bohm-Aharonov effect using
Feynman amplitude Wilson loop operators.

In 3 space

d|0) is gradient of a function, i.e. scalar field

d|1) is curl of a vector field

d|2) is divergence of a vector field

B = Bxydx/\dy + Byzdy/\dz + Bzxdz/\dx

dB = (Bxy,z + Byz,x + Bzx,y)dx/\dy/\dz

Static 4D Metrics without gravimagnetism (non-rotating spacetimes) &
without gravity waves (c = 1 convention) here

(curved metric) = g = -dt^2 + 3^g

The toy model wormhole is of this form.

We need positive dark zero point energy density with negative pressure
to keep the wormhole open. There is no event horizon in this wormhole.
It's not a black hole!

A metric allows the symmetric inner product { , }.

Classical energy density of the EM field in the absence of sources is

(1/2)[{E,E} + {B,B}]

The Lagrangian density is

(1/2)[{E,E} - {B,B}

E = (Ftx, Fty, Ftz)  electric field

B = (Fyz, Fzx, Fxy) magnetic field

F = B + E/\dt

F & B are 2-forms

E is a 1-form

We need a classical EM stress-energy density tensor T to compute

T ~ &(Dynamical Action)/&(metric)

& is functional derivative of classical Lagrangian field theory (not
particle mechanics).

w = (pressure/energy density)

Note, the above is classical without any quantum zero point fluctuations.

w = +1/3 for classical far-field radiation with only 2 transverse
polarizations.

For example, the cosmic black body radiation has w = +1/3

http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_05_jpg.htm

It's wrong to use w = +1/3 for vacuum zero point energy that bends
spacetime absolutely.

This is an error in SED used by HRP. The Casimir force is not important
for metric engineering.

Equivalence principle + local Lorentz invariance imply w = -1 for all
kinds of zero point energy (isotropically distributed).

That is the ZPF stress-energy diagonal is for pressure P

(P,-P,-P,-P) i.e. Trace is -2P

If we stick in plates or somehow break the rotational symmetry (rotating
superconducting disks that phase lock to the vacuum Goldstone phase?)

Then

(-P, +aP, + bP, + cP), the trace is now (1 - a - b - c)P

This is quintessence and it can perhaps be done with the Shipov torsion
field.

On Sep 28, 2005, at 4:20 PM, Jack Sarfatti wrote:

Lecture 1 on Cartan Forms

I am using John Baez's Ch 4 of "Gauge Fields, Knots and Gravity" for the
standard ideas.

All the local physical observables in classical gauge force field
theories are examples of Eli Cartan's "differential forms", e.g., Au,
Fuv, ju.

The integrals of forms over manifolds are premetrical until we define a
Hodge * operation taking a p-form to a N-p form for N-dim manifolds.

The p-forms are very much like Bishop Berkeley’s “ghosts of departed
quantities.” They “are neither finite, nor … infinitely small, nor yet
nothing.”

The 1-forms are dual to tangent vector fields on the manifold. A vector
field is like a bundle of particle paths in Bohm’s hidden variable
picture of quantum theory. The 1-form (AKA “cotangent vector”) is like a
stack of wave fronts (AKA “little hyperplanes”) of small extent as in
Fig. 1 p. 45 (Baez) also in MTW’s “Gravitation.” “The bigger df is, the
more tightly packed the hyperplanes are.” Given a Cartesian coordinate
basis of tangent vector fields {,u} and a dual basis of 1-forms {dx^u},
then duality here is

dx^u,v = 1v^u = Kronecker delta NxN identity matrix.

Given a 1-form df and any vector field v, the directional real number
df(v) “counts how many little hyperplanes in the stack df the vector v
crosses.”  Linearity is built in as a postulate. The Cartan forms are
invariants of local coordinate LNIF transformations Diff(N). Diff(N) is
what you get when you locally gauge the global ND translation group. In
1915 GR the Cartan forms are also invariant under the local LIF Lorentz
transformations O(1,3). In general this would be O(N) pre-signature.
That is TNxO(N).

/\ is the exterior product. Obviously we have a kind of quasi-algebra
equivalent to dissecting an N-1 simplex or “brane” also giving partially
ordered (non-Boolean?) lattices with the 0-form on bottom and the N-form
on the top.

N = 1 (dx), i.e. 1

N = 2 (dx, dy, dx/\dy = -dy/\dx), i.e. 3

N = 3 (dx,dx,dz, dx/\dy, dx/\dz, dy/\dz, dx/\dy/\dz), i.e  7

N = 4 (dx,dy,dz,dt, dx/\dy, dx/\dz, dy/\dx, dt/\dx, dt/\dy,dt/\dz,
dx/\dy/\dz, dt/\dx/\dy, dt/\dx/\dz, dt/\dy/\dz, dt/\dx/\dy/\dz), i.e. 15

If we include the 0-form we have 2, 4, 8, 16, i.e. 2^N elements in the
quasi-algebra that suggests the Clifford Algebras. There are obviously
N!/p!(N-p)! p-forms in N space. This is also like an information space
of N c-Bit Shannon Boolean strings. Obviously there will be some kind of
matrix representation. For example N = 2 should correspond to the 3 Paul
2x2 spin matrices with the unit matrix. Therefore, there is a connection
to U(1)xSU(2) here. N = 3 should have something to do with the 8 SU(3)
matrices, and N = 4 obviously connects with the Dirac algebra and
possibly U(4) especially when we complexify each real number space-time
dimension and even go beyond that to quaternions & octonians.

Classical gauge force theories include Maxwell's U(1) electromagnetic
theory, Yang-Mills theories of the SU(2) weak and SU(3) strong forces of
the leptons and quarks in the standard model and Einstein's theory of
gravity (General Relativity, 1915 AKA GR) provided you do not work at
the symmetric metric tensor level guv(x), but work at the "square root"
1-form tetrad "e" level. Note that Einstein's local equivalence
principle is simply

(curved metric ) = e(flat metric)e

where e is the Einstein-Cartan 1-form tetrad field.

You can write

e = 1 + B

B = curvature tetrad field

Since the forms are local frame invariant this decomposition is objective.

Global Special Relativity 1905 AKA SR is when B = 0 everywhere-when.

Note that the (curved) metric has linear in B "elastic" terms and
nonlinear quadratic in B "plastic" terms (H. Kleinert), i.e.,

(curved metric) = (flat metric) + 1(flat metric)B + B(flat metric)1 +
B(flat metric)B

The B^2 terms show that the gravity field is self-interacting like the
SU(2) & SU(3) gauge fields, but unlike the U(1) Maxwell EM field.

The Cartan exterior derivative operator d on forms generalizes the
gradient, curl and divergence. Together with its dual boundary operator
& on co-forms, there is a generalization of Stokes & Gauss's theorems to
N-dimensional manifold integrations with multiple-connectivity (e.g.
wormholes).

The p-dim form |p)  is the thing integrated. The dual co-form (p| is the
manifold on which the integral is done. I use a variation on the Dirac
bra-ket notation.

The basic integration theorem, is like the adjoint operation in quantum
theory, i.e.

(&(p+1)|p) = (p+1|d|p)

The two identities

d^2 = 0

&^2 = 0

are analogous to the antisymmetric Pauli exclusion principle in quantum
field theory where

a^2 = 0

a*^2 = 0

a* creates a fermion, a destroys a fermion.

However, we use the notations ‘d’ and ‘&” partially introduced by John
Baez on p. 130 of his book, where he writes:

Ex. 1:

‘dtheta’ = (xdy – ydx)/(x^2 + y^2)

for the polar angle “theta” where

dr = (xdy + ydx)/(x^2 + y^2)

x = rcos(theta)

y = rsin(theta)

The 1-form ‘dtheta’ above is closed, but not exact. In effect this means

d’d’ =/= 0

ONLY when the integral is over a non-bounding co-form (AKA non-bounding
cycle).

Therefore, for this particular example, there is phase (theta) ambiguity
at the origin r = 0. When the closed loop integral

(&’2|’d’theta’) = (2’|d’dtheta’) =/= 0

encircles the origin r = 0 it does not vanish. Note that if the closed
loop integral of ‘dtheta’ does not encircle the branch point r = 0, it
will vanish. In this sense, ‘dtheta’ is closed, but not exact and &2’ is
not a true boundary because of the “hole” at r = 0. Note that the
co-form (2’| is the area enclosed by the loop &’2 minus the “hole” at r
= 0.  If we extend this to cylindrical coordinates, then we have a
vortex core string provided we have a local U(1) complex order parameter
PSI(r,theta,z) such that

PSI(0,theta,z) = 0

PSI(r,theta,z) = PSI(r, theta + 2pi, z)

for equilibrium “stationary states” when the closed system relaxes
expelling excess flux in the Meissner effect.

In that case,

(&’2|’d’theta’) = (2’|d’dtheta’) = 2piN

N = +-1, +-2 ….

N = winding number around the string vortex core on the z-axis.

To review, the rigorous theorem is

(&(p+1)|p) = (p+1|dp)

Where

(&(p+1)| is a true boundary, which means

(&^2(p+1)| = 0

When |p) is exact, that means

and

However, when the topology of the co-form manifold is multiply connected
we can have closed p-manifolds, AKA “non-bounding p-cycles”, (&’(p+1)|
that are not true boundaries together with non-exact p-forms |d’(p-1))
such that

(&’(p+1)|d’(p-1) = (p+1’|dd’(p-1)) =/= 0

The non-bounding p-cycles are p-dim wormhole mouths or “Star Gate
Portals” that are “Through The Looking Glass” Darkly as it were, down
the Rabbit Hole in Hyperspace.

Ex. 2:

Consider the 3D space-like metric of a static spherically symmetric
non-rotating uncharged wormhole Star Gate is

(3-metric) = dr^2 + f(r)^2(dtheta^2 + sin^2thetadphi^2)

Where f(r) is the wormhole shape function. Each wormhole mouth looks
like a closed spherical surface of radius R where

R = f(r*)

df(r*)/dr = 0

d^2f(r*)/dr^2 > 0

This closed S2 surface is not a complete boundary (&3| enclosing a
3-space because it has a twin wormhole mouth somewhere-when else perhaps
in a parallel universe next door in hyperspace. Therefore, all wormhole
mouths for actual time travel to distant places in negligible proper
time for the traveler are really (&’3| not (&3|. Furthermore, the curved
tetrad field B = (hG/c^3)^1/2’d’(Goldstone Phase) is not exact, i.e. in
a 1-D loop around the wormhole mouth S2 surface with the
multiply-connected quasi Stoke’s theorem

(1’|B) = (&’3|dB) =/= 0

This is the curved tetrad flux through the closed loop that “cuts” the
spherical wormhole mouth.

Ex. 3:

Given in cylindrical coordinates the vortex string along the z-axis

 ‘d’theta = (xdy – ydx)/(x^2 + y^2)

For any closed loop

(1’| =  (&’2|

around the z-axis

(1’|’d’theta) = (2’|d’d’theta) =/= 0

i.e. Nonlocal Bohm-Aharonov “Flux without flux”

Given the above wormhole 3-metric define a Hodge * operation, with the
non-exact 2-form

3*’d’theta = (xdy/\dz + ydz/\dx + zdx/\dy)/f(r)^3

Where now we have a multiply-connected quasi-divergence Gauss theorem

(3’|d3*’d’theta) = (&’3|3*’d’theta)=/= 0

when (&’3| is a wormhole portal. There is now a radial 3*’d’theta flux
through the wormhole closed surface in addition to the ‘d’theta
circulation around a closed loop that cuts the wormhole closed surface
that is not a complete boundary.