Metric Engineering Star Gates

Lectures 2 & 1

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

|p)/\|q) = (-1)^(pq)|q)/\|p)

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



|p) = |d(p-1))



and



|dp) = |d^2(p-1)) = 0



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.