| Topic: |
Science > Physics |
| User: |
"Jack Sarfatti" |
| Date: |
03 Oct 2005 05:03:16 PM |
| Object: |
Warp, Wormhole & Time Travel Lecture 5 |
Star Gate
Warp Drive, Wormholes & Time Travel
(a technical book under construction)
Lectures in 21st Century Physics
Quotes below in Lecture 5 are by David Thouless unless otherwise specified.
On Sep 30, 2005, at 2:00 PM, Jack Sarfatti wrote:
Super Cosmos is now at http://amazon.com
These rough notes below to appear (refined) in book Star Gate: Metric
Engineering Warp Drive for Wormhole Space & Time Travel.
Lecture 5 Topological Concepts in Physics
(ref: Topological Quantum Numbers in Nonrelativistic Physics by David
Thouless)
The "More is different" (P.W. Anderson) Emergence of Order. This
includes both gravity and consciousness in different physical systems
that have key formal similarities. Roger Penrose has also suggested a
deep connection between gravity and consciousness. Roger's ideas based
on quantum gravity collapse of tiny differences in the geometrodynamic
field are different from what I am proposing. There is no "collapse" in
Bohmian quantum theory nor in the emergent macro-quantum theory of local
order parameters in both the "off-shell" physical vacuum (for gravity)
and in the "on-shell" brain of higher organisms. Indeed the
Landau-Ginzburg "phase rigidity" of the local non-unitary order
parameter is an effective defense against the environmental decoherence
degradation that Max Tegmark used against the Penrose-Hameroff model of
micro-tubule consciousness in open non-equilibrium living systems. The
latter must have "signal nonlocality" in violation of unitary nonlocal
micro-quantum theory. Penrose does not use the concept of a giant
macro-quantum order parameter.
"The space in which an order parameter resides is ... the quotient set
of the ... symmetry group of the disordered phase and of the symmetry
group of the ordered phase. This quotient means ... the set of all
different residual symmetry groups. For example the quotient set
SO(3)/SO(2), where SO(n) is the group of proper rotations in n
dimensions, is the set of all different possible symmetry axes ... and
is equivalent to the sphere S^2 which we have used to describe the
directions of magnetization of the isotropic ferromagnet. The
topological properties ... are related both to the topology of the space
in which the material that possesses an ordered phase is confined,
whether it is simply-connected or has holes in it, for example, and to
the nature of the order parameter."
Ex. 5 The Hedgehog of the NASA Pioneer Anomaly
"but keeping constant magnitude" (Thouless below)
a_g = - cH(t) ~ 1 nanometer per second per second
pointing back to Sun on scale of 10 to 10^2 AU
H(t) = a(t)^-1da(t)/dt
in FRW metric (unexpected small scale residue of large-scale cosmology)
There is a directional order parameter inside the physical vacuum as well.
"For an isotropic magnetic material contained in the space between an
inner and outer spherical surface the magnetization might point always
outwards from the inner surface, taking up the direction (but keeping
constant magnitude) of the electric field which would be produced by an
electric charge inside the inner surface ... No continuous deformation
of the magnetization, keeping the magnitude constant" (decoherence-proof
generalized phase rigidity, e.g. space-time stiffness c^4/8piG ~
10^19Gev per 10^-33 cm G-string tension) "can turn this into a state of
uniform magnetization. This is a topological configuration of the
magnetization known as a hedgehog." p. 7
Hedgehog defect in the order parameter is not possible in a planar 2D
thin film.
"Bose-Einstein condensation ... responsible for superfluidity ...
manifested in the one-particle Dirac density matrix ... in the normal"
unordered "state its eigenvalues are all of most of order unity."
This is necessary for micro-quantum unitarity with signal locality in
what Antony Valentini calls "sub-quantal heat death". The eigenvalues of
the density matrix are the Born ensemble probabilities that only obtain
in sub-quantal heat death. As soon as one of the eigenvalues exceed
unity the orthodox quantum probability Ansatz breaks down!
The single-particle density matrix at zero temperature for a pure state
coherent superposition decomposed in simultaneous eigenstates |j>
|psi> = Sum over j |j><j|psi>
of a set of commuting observables for some total experimental arrangement is
rho = |psi><psi| = Sums over j & j'|j><j|psi><psi|j'><j'|
The Born micro-quantum probabilities are
Trace{rho|j><j|}= <psi|j><j|psi>
These are the eigenvalues of the density matrix rho. Note that the Trace
of a matrix is the sum of the eigenvalues of the matrix. Each projection
operator filters out one of the eigenvalues. The determinant is the
product of the eigenvalues. Each coefficient in the secular polynomial
of the matrix is a sum of distinct products of the eigenvalues in
increasing order from Trace to Determinant.
Obviously, when one of the eigenvalues exceeds unity the sub-quantal
heat death approximation breaks down completely and signal nonlocality
is no longer forbidden.
This is the Achilles Heel of Lenny Susskind's argument for the recovery
of quantum information from evaporating black holes even if there was
world enough and time to wait that long - there ain't.
"In the superfluid state there is a macroscopic eigenvalue no of order
N, and the corresponding eigenvector |PSIo> can be regarded as the
condensate wave function. The value of no/N for superfluid helium at low
temperatures seems to be of the order of 10% and shrinks continuously to
zero as the temperature approaches the superfluid transition
temperature. This order parameter should not be confused with the the
superfluid density of the two-fluid model, which is equal to the total
fluid density in the zero temperature limit."
In the case of the universe this suggests that the vacuum condensate
density is ~ 4%, i.e. no/N ~ 0.04 because both the dark energy and the
dark matter constitute 96% of the stuff of the world (negative and
positive zero point pressures respectively) outside of the condensate
coherent vacuum wave.
Slides from Rocky Kolb
http://www-conf.slac.stanford.edu/ssi/2005/program.htm
Think "vacuum condensate" when you see "superfluid" below. The
superfluid is a giant coherent "macro-quantum" wave in ordinary 3D space
made out of real on-shell boson particles outside the vacuum. The vacuum
condensate is a mathematically similar giant local wave made out of
virtual off-shell bosons (e.g. virtual electron-positron pairs) inside
the vacuum.
Multiple-Connectivity
"If the superfluid is confined to a region which is not simply
connected, such as the interior of a torus, there may be metastable
states in which the phase of the condensate wave function changes in a
non-trivial way around a path that cannot be shrunk continuously to
zero. It is this sort of consideration that ... led Onsager to the idea
of quantized circulation" [vorticity quanta] "in superfluid 4He. This
method of defining the order parameter was generalized by Yang and given
the name off-diagonal-long-range-order (ODLRO). In the case of fermions
.... the one-particle density matrix cannot have a macroscopic eigenvalue
.... but the two-particle Dirac density matrix can. In the BCS
equilibrium state of a superconductor the eigenvector corresponding to
to the macroscopic eigenvalue is an S-state spin singlet function of the
relative coordinates, and is independent of the center of mass
coordinate except near the boundaries: this represents the Bose
condensation of a ^1S0 electron pair. For 3He the pair is in a triplet
P-state, with the spin and orbital angular momentum coupled together in
different ways in the A and B phases ... Again for the singlet S
superfluid fermion system there can be states in non simply connected
geometries in which the phase of the center of mass dependence of the
order parameter changes by a multiple of 2pi around the system.
.... In the quantum Hall effect .. the concept of an order parameter is
not so clear ... Nevertheless, the ideas of topological quantum numbers
run through discussions of the subject. Not only is the Hall
conductance" e^2/h quanta of physical dimension speed "itself a
topological quantum number, but the charge carriers themselves ...
fractionally charged quasiparticles, behave ... like topological defects.
Is the aether an ODLRO 4D Diff(4) covariant supersolid? Remember local
Diff(4) is simply the rigid global 4D translation group T4 locally
gauged. The compensating gauge potential here is
B = (hG/c^3)^1/2'd'(Goldstone Phase of Vacuum Higgs Field)
Here comes Gennady Shipov's torsion field theory extension of Einstein's
1915 GR.
"There are two important order parameters in a solid, which are the
position of the actual unit cell with respect to an ideal unit cell." p. 10
This is B above, i.e. the curvature field 1-form part of the
Einstein-Cartan tetrad field.
"and the orientation of the unit cell."
This is S the compensating torsion field potential 1-form from locally
gauging the 4D rotation Lorentz group O(1,3) that generalizes O(3) for
the 3D solid.
T = dS + W/\S + S/\(1 + B + S) = torsion field 2-form (dislocations)
W = spin connection 1-form determined by B from
dB + W/\(1 + B) = 0
R = dW + W/\W is disclination geodesic deviation tidal curvature 2-form.
Homotopy Classes
Simplest space is contractible, i.e. continuously shrinkable to a point.
The space between the two spheres in the hedgehog is simply connected
for closed 1-D loops that can all be shrunk to a point, but is not so
for all closed surfaces. The 2D torus has closed paths that cannot be
shrunk to a point and that wind around the torus.
Assume "a continuous order parameter associated with each point in
space. Around each closed path there is a continuous change of the order
parameter. Homotopy classes classify such continuous changes - mapping
of the loops onto the order parameter space - according to whether they
can be continuously deformed into each other or not. ... The winding
numbers w are integers, and are formally defined by"
w = (1/2pi)('&'2|'d'Theta) = (1/2pi)(2'|d'd'Theta)
Theta = Im ln(Order Parameter)
"for a complex scalar order parameter or a planar vector field. These
must be the same for all loops that can be continuously deformed into
each other, and are additive when loops are strung together. The winding
numbers round paths that go once around the system are ... all we need
to know. This process defines the homotopy group PI1 for the order
parameter" p. 13
Note that the winding number is a property of the order parameter space
G/H as one closed 2pi path is taken around some obstruction in physical
3D space that contains a zero of the order parameter. Order parameters
are LOCAL in physical space. This is why emergent "More is different"
macro-physics is local even though the micro-quantum substratum is
nonlocal with a complex web of Einstein-Podolsky-Rosen entanglements in
higher dimensional configuration space. Indeed, this is the real
solution of the Schrodinger Cat paradox.
For the U(1) order parameter, PI1 is the group of integers. The A phase
of superfluid 3He & liquid crystals have finite groups for PI1. One can
also define the 2D homotopy group PI2 "in terms of the behavior of the
order parameter on a simple closed surface. This is trivial for the U(1)
order parameter (complex scalar field). The quantum Hall effect is more
complicated, "use the mapping of the surface of a two-torus onto a
complex projective space ... The topological invariant that describes
such a mapping defines its Chern class, and this turns out to be the
Hall conductance in units of e^2/h.
The order parameter can have discontinuities in physical space.
Curvature and torsion are examples of such discontinuities in the World
Crystal Super-Solid.
"If the singularity cannot be removed by smoothing the order parameter,
then ... it is topologically stable. The topological stability of these
defects depends on homotopy class." p.14
In the case of the U(1) order parameter, the topological stability of a
string vortex requires a non-trivial winding number = number of
circulation-vorticity quanta.
"The winding number ... gives the number of complete turns the phase ...
makes on a circuit around" a closed path in physical space that
encircles the defect.
"The ac Josephson effect involves matching rotations of the relative
phases of superconductor wave functions with the temporal oscillations
of an ac circuit." p. 15
Low-power metric engineering the local curvature field B =
(hG/c^3)^1/2'd'(Phase of Vacuum) to steer the timelike geodesic glide
path of the ship from the ship with weightless warp drive (WWD) requires
a non-vanishing torsion field potential S and the above Josephson effect
in which the stiff macro-quantum "carrier" phase of the vacuum is
modulated by a high Tc "signal phase" (probably using an anyon
condensate in a thin 2D film on the fuselage of the saucer). The signal
phase, in turn, is controlled by the EM potential A ~ J using the
Bohm-Aharonov effect in tiny nano-solenoids in the 2D layer.
The stability of the Bohm hidden variable extended electron (& quark)
micro-geon with strong short-range gravity induced directly by the dark
zero point energy density core needs a point defect with non-trivial PI2
whose quantized flux quanta is the electric U(1) charge itself, +-e/3 &
+-2e/3 for quarks & -e for electrons.
Lecture 4 Fractional Quantum Hall Effect for 2D Anyons
Note that the Hodge* as used in most elementary applications apparently
only works for static curved metrics and needs to be modified for
rotating Kerr metrics with gravimagnetism H = (g01,g02,g03) as in
frame-dragging in standard 1915 GR with zero torsion.
What about (anti)self-dual instantons in Euclidean metric? (vacuum
tunneling)
Given a 2-form, e.g. curvature, EM field, torsion, then relative to some
metric define a *
The 2-form is self-dual if
*F = +F
it is anti self-dual if
*F = - F
*^2 = +1 for Euclidean signature ++++
*^2 = - 1 for Lorentzian signature - +++
where with causal light cones
F = F+ + F-
*F+- = +-iF+-
Self-interacting nonlinear Yang-Mills eqs in ++++ have (anti)self-dual
"instanton" solutions.
Can we generalize this to the N roots of unity?
Closed and exact forms:
If dB = 0, B is a closed form.
If B = dA
B is an exact form.
All exact forms are closed, but not all closed forms are exact.
If we have a p-form A
B = 'd'A
('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) =/= 0
Where '&'(p+1) is a non-bounding p-cycle in a multiply-connected p+1
manifold.
For example, if p = 2, then &'3 is a closed 2D surface AKA wormhole
mouth (portal) that is not a complete boundary of the interior 3D space.
There is at least one other '&'3 closed surface some where-when in our
universe or in a parallel universe next door in hyperspace. There can be
many such '&'3 in a NONLOCAL multi-pronged network of Star Gate wormhole
tunnels held open by negative pressure positive zero point dark energy
density distributions.
The general theorem is
(&(p+1)|A) = Sum over all ('&'(p+1)|A) = Sum over all (p+1|'d'A) = (p+1|dA)
In the special case that dA = 0 then the final term on RHS = 0.
Of course there is no reason why dA = 0 is necessary. You can have dA
=/= 0.
Restricting ourselves to a LOCAL wormhole mouth where
('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) =/= 0
Let p = 1
('&'2|A) = (2|'d'A)
Let
A = 'd'(Theta)
Theta = 0-form phase of a LOCAL macro-quantum order parameter valued in
the complex plane.
Single-valuedness of the local order parameter around the closed loop
('&'2| that is not the complete boundary of the inner area (2| then
gives us quantized "Flux without flux".
Consider winding around the closed non-bounding loop ('&'2| once.
Imagine that the order parameter Goldstone phase changes by 2pi/n (n an
integer) in order parameter space G/H for 1 winding of 2pi around the
non-bounding closed loop in physical 3D space. In that case, the order
parameter changes by e^i2pi/n in G/H vacuum manifold space for each 2pi
circuit around the Theta Goldstone phase singularity (e.g. vortex core
string) in 3D space.
In the simplest case n = 1, Then for N windings in physical 3D space we
have the nonlocal Bohm-Aharonov effect
('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) = 2piN
of flux quanta through the closed loop in the "normal core" in a
stationary state. We can actually pump non-integer flux through, but the
state will not be stationary, it will shake off the excess flux in the
form of radiating quasi-particles if not externally pumped in a Meissner
effect.
Suppose n = 2, then for N windings
('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) = piN
This is a macro-quantum spinor condensate.
For n = 3
('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) = 2piN/3
suggestive of the fractional Quantum Hall Effect for high Tc 2D films of
anyon condensate with para-statistics of the normal fluctuations?
Lecture 3 Gravity energy is nonlocal and Yilmaz et-al are wrong.
The Bianchi identities generalize d^2 = 0 to D^2 = 0 when there are
gauge field connections A, W etc. present.
For example:
D = d + A/\
for SU(2) & SU(3) internal symmetry non-Abelian Yang-Mills gauge force
models of the parity-violating weak hypercharge and parity-conserving
strong gluon forces respectively in the standard model. It is the weak
hypercharge group SU(2) that is spontaneously broken to produce the
small inertia of the leptons and quarks and to produce emergent gravity
via my original formula
B = (hG/c^3)^1/2d(Goldstone Phase)
Here Goldstone Phase is the "mean" SU(2) hypercharge phase from using
the Trace operation (details later on).
The large mass of the hadrons comes from the kinetic energy of the
confined quarks in dark energy bags.
In 1915 General Relativity, locally gauging T4 -> Diff(4):
D = d + W/\
Where the spin connection W obeys
T = De = d(1 + B) + W/\(1 + B) = 0
Therefore
dB + W/\(1 + B) = 0
B is the curvature tetrad field that corresponds to disclination
topological defects in the Vacuum ODLRO Manifold G/H and in Hagen
Kleinert's "World Crystal Lattice".
Gennady Shipov's torsion field is when the Lorentz group is locally
gauged in addition to T4. This adds 6 new scalar fields that act like
extra space dimensions, i.e. 10D space-time. Just as locally gauging T4
brings in the compensating 1-form curvature disclination connection B,
provided SU(2) hypercharge spontaneously breaks in the inflation, so
does locally gauging the Lorentz group O(1,3) bring in the compensating
1-form torsion dislocation connection S where the non-vanishing torsion
2-form T is now
T = dS + W/\S + S/\(1 + B + S)
Note the torsion-curvature coupling terms W/\S and S/\B.
The 1915 GR curvature 2-form is
R = DW = dW + W/\W
The Einstein-Hilbert dark energy vacuum action is the 4-form
R/\(1 + B)/\(1 + B)+ /\zpf(1 + B)/\(1 + B)/\(1 + B)/\(1 + B)
The 1915 Bianchi identity is, for /\zpf = 0
DR = 0
i.e.
(d + W/\)R = dR + W/\R = 0
The Einstein field equation is
D*R(Geometry) = *J(Matter)
D^2R = D*J(Matter) = 0
is the local conservation stress-energy current density.
Note
D*J(Matter) = d*J(matter) + W/\*J(Matter)
W/\*J(matter) = d*j(vacuum-matter)
*j(vacuum-matter) is a 3-form that is the pseudo-tensor of the
(matter-vacuum coupling)
Nonlocality of total gravity energy is multiply-connected "Flux without
flux", i.e.
(3|D*R) =/= 0
When *J(matter) = 0 everywhere-when
i.e.
*R = *"D"W
D^2 = 0
but
D"D" =/= 0
i.e. Flux without flux from multiple connectivity of 3-Manifold slices
of space-time.
If no S, then /\zpf is really a constant, i.e. "Cosmological Constant"
made small by vacuum ODLRO. When S =/= 0, then /\zpf is a local scalar
field, indeed the one we need for metric engineering warp and wormhole
with the Josephson homodyne detection method.
This is in complete analogy to Maxwell's U(1) equations
dF = 0
d*F = *J
d^2*F = d*J = 0
And the Yang-Mills SU(2) & SU(3) equations
DF = 0
D = d + A/\
D*F = *J
D^2F = D*J = 0
The complete theory with torsion and U(1)xSU(2)xSU(3) is clearly the
same template where now
D' = d + W/\ + S/\ + A/\ + C/\
A is U(1)xSU(2)xSU(3) or whatever internal G symmetry group works.
W & S are from locally gauging the 10-parameter space-time symmetry
Poincare group.
If we go to the 15 parameter conformal group we will get an additional
conformal or Penrose twistor connection. If we go to Kaluza-Klein
extra-dimensions and Grassmann fermion dimensions of supersymmetry (not
observed) then everything generalizes in the same template. Now however
R' = D'(W + S + C + A ...)
D'R' = 0
D'*R' = *J'
As well as the Yang-Mills equations
F' = D'A
DF' = 0
D*F' = *j
etc.
Similarly for S, B & C?
On Sep 29, 2005, at 2:37 PM, Jack Sarfatti wrote:
Lecture 2 (Revised Draft #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
|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
<kolb1new_Page_05.jpg>
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 Weightless Warp Drive and Wormhole Time Travel to
The Past (using old wormholes made at the beginning of our local
universe and even connecting to the parallel universes of Super Cosmos).
Equivalence principle + local Lorentz invariance imply w = -1 for all
kinds of zero point energy (isotropically distributed).
That is the Zero Point Energy Stress-Energy Current Density Tensor
tuv(ZPF) diagonal is for Energy Density dE/dV & Pressure P
w(ZPF) = P/(dE/dV) = -1 from EEP & LI of GR
(dE/dV,-dE/dV,-dE/dV,dE/dV),
Therefore, since w = -1
Trace is -2(dE/dV) = (1 + 3w)(dE/dV)
Compare to EM radiation where
w = P/(dE/dV) = + 1/3
( dE/dV, +(1/3)dE/dV, +(1/3)dE/dV, +(1/3)dE/dV), i.e. Trace
= +2dE/dV = (1 + 3w)(dE/dV)
Note that dark energy is P < 0 & dE/dV > 0.
Dark Matter is P > 0 & dE/dV < 0
Assuming here above, of course, compact dark matter sources like the
Galactic Halos when P > 0 and measurements by external observers.
If we stick in Casimir plates to break translational symmetry or somehow
break the rotational symmetry (rotating superconducting disks that phase
lock to the vacuum Goldstone phase?)
There is analogy here to homodyne detection of quantum information with
continuous variables where the local oscillator in a beam splitter is
like the vacuum ODLRO field. (Rev Mod Phys, p. 513, April 2005) Squeezed
vacuum states in quantum optics is when one quadrature of the zero point
virtual photons has less vacuum noise than does the other quadrature
which has excess noise to compensate.
http://www.cco.caltech.edu/~qoptics/squeeze.html
Then
( dE/dV, +a(dE/dV), + b(dE/dV), + c(dE/dV)), the trace is now (1 + a + b
+ c)dE/dV
1 + 3w = 1 + a + b + c
w = (a + b + c)/3
This is quintessence when w < -1/3 and it can perhaps be done with the
Shipov torsion field. Phantom energy with the Big Rip tearing the fabric
of the Universe apart is when w < -1.
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.
.
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