The Theory of Everything for Everyone
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On Apr 15, 2005, at 3:13 PM, iksnileiz@earthlink.net wrote:
If you read the Arcos & Pereira paper, how do you understand the authors' position regarding the extraction of a localizable tensor vacuum stress-energy density
from the net gravitational-inertial stress-energy of the Einstein field in (1) their own teleparallel formalism; and (2) in the conventional curved-manifold formalism?
Jack, this question was specifically addressed to Art.
J: Yes, I know.
What A&P do, very nicely is to show the B field!
Z: In the context of a *teleparallel* theory.
J: They do not have my Vacuum ODLRO SBS idea, i.e. they do not write
B = (Lp/2pi)"dargVacuumODLRO"
argVacuumODLRO = Goldstone Phase of Higgs Field (post-inflation).
Z: Right. Of course not.
J: If B is closed non-exact 1-form that's A&P torsion "flux without flux".
Z: So you are saying that your "tetrad" B can be modeled as a torsion field?
J: Yes, it's the "potential" for the A&P torsion field not the Shipov torsion field. B is like A in EM, F = dB is like the EM field tensor, but here it is the A&P tetrad substratum torsion field tensor.
As R. Kiehn points out the most general p-form has 3 possible independent components
form = exact + closed not-exact + non-closed
If B has a non-closed piece I call that "flux with flux" i.e.
F = dB
If B is only closed and not-exact
Stokes theorem says |B = 0 on a bounding cycle but need not be zero on a non-bounding cycle that is only a piece of the total boundary that consists of several cycles when the manifold of integration of dB is not simply-connected.
If, in addition, there is macro-quantum vacuum ODLRO (a new independent physical idea)
B = (Lp/2pi)"dtheta" in sense described in John Baez's book on Gauge Fields, Knots & Gravity, then the "period integral" of 1-form over the boundary fragment, which is nevertheless a closed loop (nonbounding 1-cycle) is, from single-valuedness of the ODLRO order parameter
|B = NLp
N is integer winding number
This leads me to define another "Ghost of a departed quantity" (Bishop Berkeley)
NLp = (Area)dB'
Mean value theorem for integrals.
I put the ' to denote "flux without flux". The "Area" is inside the non-bounding cycle and all the other inner non-bounding cycles isolating phase singularities where |ODLRO| = 0 (like vortex cores in Type II superconductor of real EM flux with flux) shrink to zero (I mean small). Actually HeII quantized circulations may be "flux without flux" in this sense.
Thus in cylindrical coordinates for |ODLRO| = 0 along z-axis
dB' = NLp/pir^2
Even though technically dB = 0.
NLp is like a finite integral of a zero integrand only because we have ignored the inner cycles that add up to -NLp of course. But as physicists we can do that since we only actually measure the outer non-bounding cycle in the case of quantized vortices in superfluid HeII.
So this is an archetype analogy for the complete nonlocality of gravity energy in ordinary vacuum /\zpf = 0 i.e. gravity vacuum energy without gravity vacuum energy.
Integrating a local "zero" to get a "finite" total is not so crazy when the manifolds of integration are not simply-connected and the "holes" correspond to phase singularities, i.e. zeros in the ODLRO intensity.
The situation is like that of infinitesimals in calculus that are both zero and not zero in the heuristic sense. As physicists we don't care about what Feynman called "rigor mortis". For example the Feynman path integrals are mathematically non-rigorous yet we get good results from them.
If B is not closed that's torsion "flux with flux"
i.e.
F = dB is the A&P torsion field (not same as Shipov's).
I assume here G/H = S1 like Abelian Maxwell E&M, this is only a toy model
dF = 0
d*F = *J
The local stress-energy tensor of this torsion field is
~ F/\*F i.e. a 4-form
But NOTE if B is closed and non-exact, then strictly speaking LOCALLY
F = dB = 0
Yet, from "flux without flux" the global integral is not zero even though the local density is!
WHERE HAVE WE SEEN THAT BEFORE? :-)
Z: OK, I think I get it.
J: Yes, this is an interesting strategy. Penrose gives a good heuristic why total gravity ordinary vacuum energy really must be nonlocal so that Yilmaz's theory is asking the wrong question and so is Alex's. Valentini's new papers on the creative tension between locality & nonlocality add fuel to that fire. There are other arguments as well.
"Flux without flux" means
|B = ||dB = NLp, N an integer
Strictly of course |B = 0 over a boundary, but here we take the incomplete integral over a non-bounding cycle and ignore the inner cycle that gives -NLp (Stoke's theorem) that shrinks to the topological defect Goldstone phase singularity where |Vacuum ODLRO| = 0 in the G/H order parameter space. Note, unlike micro-quantum theory, in macro-quantum theory the radius of S1 has independent physical meaning. Also see A. Valentini's latest papers on sub-quantum non-equilibrium signal nonlocality. Let P be the non-equilibrium distribution, the non-local signal is an integral with kernel P - |psi|^2, in macro-quantum theory, I think this becomes ~ |ODLRO|^2, which of course is zero in nonlocal micro-quantum theory with "presponse" signal locality. That is, any system with ODLRO (in vacuum for virtual quanta E^2 =/= (pc)^2 + (mc^2)^2, in ground states E^2 = (pc)^2 + (mc^2)^2 for real quanta) is "sub-quantal non-equilibrium". Call this the "Sarfatti Conjecture" since our conscious minds are pumped open giant quantum robust ODLRO systems immune to warm wet environmental decoherence of the Zurek-Tegmark kind.
Z: OK, looks good. Very interesting idea.
J: Valentini has a formula in one of his new papers
Nonlocal signal ~ Integral of stuff [P(sub-quantum non-equil) - P(sub-quantal equil)]
I think I can argue as a sufficient, though maybe not necessary, condition for signal nonlocality in macro-quantum theory
|OLDRO|^2 = [P(sub-quantum non-equil) - P(sub-quantal equil)]
I get this from theory of reduced density matrices as given by L. Onsager & O. Penrose.
P(sub-quantal equil) = "normal fluid" single-particle reduced density matrix in diagonal limit = Born probability density! (pair density matrix in case of fermions)
The macroscopic eigenvalue of first-order reduced density matrix is "sub-quantal non-equilibrium" in A. Valentini's sense and it is immune to warm wet decoherence. This is WHY Dick Bierman sees "presponse" in his data.
Z: I'm also curious about how you understand A&P's claim that that the teleparallel formalism is "physically equivalent" to the curved manifold formalism?
Again, this was addressed to Wagner.
In answer to R.Kiehn today
On Apr 15, 2005, at 5:00 PM, Jack Sarfatti wrote:
Clarification of notation
When I write
g(curved) =[(I + B)^T][(flat)]([I + B)]
That is just my lazy notation. I do not mean that as an equation in Cartan forms. All I mean by that is the well known
guv = eu^a(Minkowski)abev^b
where
eu^a = Iu^a + Bu^a
Iu^a is Kronecker Delta
B = dx^uBu^a(Pa(Goldstone Phase)/ih)
{Pa} is Lie Algebra of T4
It means simply one version of EEP
g(curved) = (I + B)(flat)(I + B)
Z: So you think that by "physical equivalence" A&P simply mean that they recover standard EEP? You think they mean "equivalence" as in "equivalence principle"? In other words -- "pun intended"?
J: Pun intended sure. What they mean is simply that starting from the zero curvature torsion tetrad substratum, one can use the formal EEP to derive standard vanilla 1916 GR curved torsionless model.
There is a kind of curvature/torsion duality here between the local gauge force and geometrodynamical pictures.
The torsion flat gauge force picture is the "square root" of the curved torsionless geometrodynamic picture.
So, the torsion picture is a covering theory of the geometrodynamics. It is closed to the Ashtekar picture!
That is, in the torsion tetrad substratum it is obvious how gravity is really simply the local gauge theory of T4 -> Diff(4.
In 1905 special relativity
x -> x' = x + xo -> conservation of total linear momentum px
t -> t' = t + to -> conservation of total energy E
These are GLOBAL displacements in Minkowski space-time making up the RIGID group T4 (with x & y).
It is completely trivially obvious that Diff(4) GCT's
x^u(P) -> x^u'(x^u(P))
at a fixed PHYSICAL EVENT P = equivalence class of manifold points p ~ p', p =/= p'
is by definition the local gauging of T4 to Diff(4) with compensating gauge force potential Bu^a where
guv(curved) = eu^a(Minkowski)abev^b
Z: I think they mean something much broader -- more like matrix mechanics vs. wave mechanics.
J: Frankly I don't care what they mean. What I care about here is what I mean. However, I suspect that what I mean and they mean are not that far apart.
This is exactly how curved space-time is Vacuum ODLRO emergent bottom -> up.
Back-of-the-envelope heuristics
The quantized Goldstone vibrations in argVacuum ODLRO are not spin 2 gravitons.
OK.
Are they like phonons?
More like phonons yes. They are not spin 2 at least not when G/H = S1 where
Vacuum ODLRO = PSI = Re^iS (Bohm Pilot Wave notation)
If Bu^a = X^a(PuS/ih) = X^aS,u
X^a is "position operator" conjugate to Pa in Minkowski space
i.e. X^a makes displacement in momentum Minkowksi space
Z = (X + Lp^2P/ih) & Z* =(X - Lp^2P/ih)
Give Wigner Phase Space Density Representation of Glauber & Squeezed macro-quantum coherent states
Z|z> = z|z>
z = re^itheta
Use the formalism of quantum optics
D(z) = e^(zA* - z*A) Glauber coherent states
S(w) = e^(wA*A* - w*AA) Bogoliubov pairing transformation for squeezed states.
w vacuum ODLRO squeezing parameter ~ BCS Nambu/Jona-Lasino gap function, i.e. rest masses of lepto-quarks?
No gravitons. No quantum foam. No quantum gravity.
Z: Well, you do have a true QFT layer beneath your BEC macro-condensate, don't you? That describes the micro-behavior of "off-mass-shell" electrons and positrons?
Yes, it's their pairing that is the "chaotic" inflation phase transition. The pairing binding energy powers the hot phase of the Big Bang. It's all pretty obvious intuitively. It fits Lenny Susskind's "landscape" with WEP and also Max Tegmark's Level I & Level II (see his homepage for pretty pictures)
Who ordered that?
What about Duff's argument for quantum gravity?
What happens when you reach rock bottom?
Z: Yes, of course this is the Sakharov emergent-gravity thesis. Same in Hu's papers, for example.
J: Same in general philosophy not in detail. This stuff is in the air. Mine is the simplest of em all. Are all the rest over-complicated or is my model simpler than is possible like Puthoff's PV and Haisch's EM SED ZPE "origin of inertia".
Is a theory of everything for everyone impossible? Do you have to be a fancy mathematician to grok it? Calabi-Yau and all that really needed? Too soon to tell.
You return to the false globally flat massless conformal vacuum of spin1/2 & spin 1 QFT with no gravity and no inertia, i.e. all rest m = 0
Z: But is it kinematically false?
J: I do not know what that even means. The center of the cyclone where |ODLRO| = R = 0 thats FALSE VACUUM. So it's physical as real as anything is.
Z: Or just unstable if construed as a physical vacuum?
J: It's real and unstable. It's as real as riding a unicycle.
http://stardrive.org/cartoon/Saturn.html
Z: I still sense Einsteinian ambiguity and even confusion here.
J: The fault dear Z is in your mind and not beyond the stars.
Z: What did Sakharov say about this "emergence"?
J: Not much. Like Bohr he mumbled some things about "metric elasticity" and if you read Adler Rev Mod Phys 1982 you will see more very complicated stuff. It's not clear which comes first however. It's Chicken & Egg. Been up so long it looks like down to them. Adler seems to get Einstein-Hilbert action from SBS & estimates of G & /\, but it's far from obvious what he is doing. Also I don't see no tetrads there.
Z: What's wrong with a bi-metric approach at the macro-level? I can imagine that a bi-metric approach would fit your BEC model quite well if you were open to it.
J: Occam's razor. Parsimony. More with less. Excess Baggage. If it ain't broke don't fix it. Who ordered a second metric. That's an unimaginative act of desperation - not at all like Max Planck's.
Z.
E = pc for all real quanta
This is what you get for |Vacuum ODLRO| = 0 in the core of a topological defect in G/H order parameter space.
Note that the non-perturbative lepto-quark mass scale is
m ~ (hwD/c^2)e^1/rho(Ef)<0|V|0>Theta(-V)
i.e.
1 Mev ~ 10^19Geve^-1/|x|
e^-1/|x| ~ 10^-22
Theta(x) = step function
V is the attractive interaction potential between the virtual electrons and positrons at the edge of the negative energy Dirac Sea with Fermi energy Ef = 0, but pf ~ h/L
Let V -> 0 from negative values where it is attractive i.e. V ~ -e^2/r, this gives m = 0 at V = 0-, m stays zero for repulsive V > 0.
rho is density of states of massless negative energy virtual electrons Fermi liquid at Ef per unit energy.
Subject: Z's confounding of tetrads with Diff(4) GCTs
Date: Thu, 14 Apr 2005 14:06:26 -0700
Admittedly Paul I suspect you are not the only one confused by this issue of how to separate intrinsic geometry from non-inertial local frame inertial g-forces especially since one cannot tell the difference without making local tidal geodesic deviation measurements.
On Apr 14, 2005, at 12:12 PM, iksnileiz@earthlink.net wrote:
[JS] What I am telling you is that you cannot use a GCT X when B =/= 0 to make (LC)' = 0.
[Z] Then there is something wrong with your B. Of course you can always cancel net g-forces with a GCT.
[J] There is nothing wrong with B. In fact it is what you were looking for, but you looked in the wrong place - at the wrong level. Math is math. Diff(4) GCT non-dynamical Xu'^u connects non-geodesic LNIFu with non-geodesic LNIFu'. Indeed, you can always locally cancel the g inertial force of an LNIF by jumping to a coincident LIF, but you do that with a tetrad eu^a not with a GCT Xu'^u. The globally flat Minkowski space-time is degenerate i.e. eu^a = Kronecker delta Iu^a so that the u/a distinction vanishes and, indeed, in that case only, you can use GCT to get from an inertial frame to a non-inertial frame. Your intuition on Minkowski space-time does not carry over to curved space-time. Can I teach this Lazy Dog a new trick?
e.g. x -> x' - (1/2)gt^2
t -> t' = t
in Newtonian limit.
In the B = 0 degenerate case where a = u so to speak
(Minkowski)uv = Iu^a(Minkowski)abIv^b
Now, take any Diff(4) GCT Xu'^u at all to get the PHONY 100% inertial force zero tidal warp pseudo-curved metric
gu'v'(Minkowski) = Xu'^u(Minkowski)uvXv'^v
B = 0
Ruvwl = Ru'v'w'l' = 0 everywhere-when in Minkowski
These Diff(4) GCT Xu'^u describe infinite sets of pairs of coincident possible local observers u & u' in arbitrary relative motion at event P. Mathematically Xu'^u is the overlap transition function of two local coordinate charts x^u & x^u' in same neighborhood of event P. P is an equivalence ~ class of bare manifold points p i.e. P is the coset of ~ p's. The space of events P is a coset quotient space under gauge equivalent active Diff(4) p -> p'.
There is a qualitative phase transition from the bottom -> up non-perturbative inflationary emergence of
B = (Lp/2pi)d(argVacuum ODLRO)
closed non-exact 1-form means period integrals of 1-form B on non-bounding 1-chain cycle need not vanish, but is NLp hence "flux without flux"? The boundary round a |Vacuum ODLRO| = 0 "hole" is two concentric opposite running cycles and we ignore the inner one to get "flux without flux" that is, in fact, measured by observers on the outer non-bounding 1-cycle. Goldstone phase = argVacuum ODLRO is not a unique 0-form (it is a set of partially overlapping functions) because of |Vacuum ODLRO| = 0 topological defects in the G/H quotient order parameter space of the inflationary vacuum phase transition. The defects give non-trivial cohomology/homology/homotopies for all the p-forms, p = 0,1,2,3,4 in 4D space-time. The macro-quantum order parameters out of which curved space-time emerges must be single-valued. It's not enough that |Vacuum ODLRO| match. The phases must match mod 2piN round a closed circuit in G/H that corresponds to some path in ordinary 3D-space. The number of circuits in each space need not be identical.
[JS] that Taylor series procedure
[Z]: What "Taylor series procedure"?
[JS]: guv(P') = guv(P) + guv,w(P)(P'-P)^w + (1/2)guv,w,l(P)(P'-P)^w(P'-P)^l
Use the non-trivial tetrads e = (I + B)
Finding the local dynamical tetrad eu^a(P) requires the critical point
guv,w = (LC)uv,w - (LC)vu,w = 0 ? (from memory - error?)
This seems to require zero torsion to get orthodox EEP?
So what we really want is
gab,c = ea^ueb^vec^wguv,w = 0
P'is now in the tangent space
gab(P') ~ (Minkowski)ab + (1/2)gab,c,d(P)(P'-P)^c(P'-P)^d + ...
where second term on RHS is the actual tidal geodesic deviation (curvature) in the LIF.
Note also
guv(P) = (Iu^a + Bu^a(P))(Minkowski)ab(Iv^b + Bv^b(P))
and inversely
(Minkowski)ab = (Ia^u + Ba^u(P))guv(P)(Ib^v + Bb^v(P))
local all at same P
Therefore, the Taylor series can be reduced to expansions of the dynamical B field.
Friday, April 15, 2005
Sunday, April 10, 2005
The Origin of Everything
Keep up with discussion forum at http://stardrive.org/
for fast-breaking news!
TOPOS is prior to Geometrodynamica!
What I cannot create on the spot spontaneously ab-initio I certainly do not understand. It's the "non-mechanical" Bohmian unfolding of a compressed implicate algorithm that we call "intuition" with a dash of "presponse".
The idea of John Archibald Wheeler in the late 1950's & early 60's was to fulfil "Einstein's Vision" that only made sense in terms of David Bohm's pilot wave/ hidden variable theory because Einstein envisioned elementary particles as pure vacuum tiny wormholes with quantized trapped gauge force fluxes. The idea, like Andrei Sakharov's vision of emergent gravity from the "metric elasticity" of zero point quantum vacuum fluctuations, was very incomplete.
1. Gravity is too weak by 40 powers of ten unless it gets stronger on the scale of 1 fermi. Abdus Salam suggested that in early 70's "f-meson" and I showed it explained the Regge trajectories of hadronic resonances J ~ alpha'E^2 + ... where alpha' = (1Gev)^-2. Salam invited me to ICTP Trieste in 1973-4 because of that.
2. Wheeler had no explanation of quantized flux because he did not have idea of vacuum ODLRO. The Penrose-Onsager idea of ODLRO was new. The Higgs mechanism was not understood. (Bernie Haisch et-al still doesn't get it that it falsified his random ZPF approach to the "origin of inertia". :-)) Yang-Mills local gauging of global to local symmetries was new. The Bohm-Aharonov effect was not well understood. The time was not ripe. Also no one even conceived of anti-gravity "dark energy" except Hermann Bondi & Stalin's physics spymaster, Y. Terletskii in a very vague way as "negative mass propulsion". No one understood the exotic vacuum equation
Guv + /\zpfguv = 0
with /\zpf a local scalar field FROM vacuum ODLRO that in Newtonian limit is the Poisson equation for isotropic source
Grad^2V(source) = 4piG(mass density)(1 + 3w)
Ordinary matter w = 0
Radiation w = +1/3
All random zero point fluctuations from all quantum fields w = -1
"mass density" of ZPF can be both > 0 and < 0!
Virtual electron-positron pairs have negative effective mass density.
Virtual photons have positive effective mass density.
Virtual QED quanta cannot directly make QED counters click, but they DO directly warp space-time as "dark energy" and "dark matter" are BOTH telling us!
The post-inflation field is essentially the vacuum ODLRO field.
3. Laughlin-Chapline and myself independently have been working on the idea that it is in the ODLRO cohering of the ZPF that gravity emerges bottom->up and that all top->down versions of "quantum gravity" are the "wrong question". No gravitons, no quantum foam is the message from top->down quantization of guv is unrenormalizable. [Ashtekar's loop variables seem to be a version of my B warp tetrad torsion field?]
4. The idea that black holes may be prevented by repulsive dark energy cores (dark energy stars) is also immediately obvious, though it is too early to tell if it is correct. Laughlin & Chapline do argue from actual evidence however. I thought of the general idea in 2002, but did not do anything with it. Ken Shoulders "EVOs" may be a mesoscopic charged analog to the "dark stars" in that exotic vacuum cores can stabilize the N electrons, indeed the single electron solving the Lorentz problem of the "stresses" from 100 years ago, i.e. what prevents a shell of charge from exploding. Simplest example is
V = +(Ne^2)/mr + c^2/\zpfr^2
dV/dr = -(Ne^2)/mr^2 + 2c^2/\zpfr = 0
Therefore a UNIFORM DARK ENERGY core obviously stabilizes the shell of electric charge. The problem becomes trivial!
d^2V/dr^2 = +2(Ne^2)/mr^3 + 2c^2/\zpf
Static stability is d^2V/dr^2 > 0 at the critical point dV/dr = 0
OBVIOUSLY TRUE when /\zpf > 0, i.e. dark energy w = -1 negative pressure
Message: 2
Date: Sat, 09 Apr 2005 12:40:52 -0700
From: iksnileiz@earthlink.net
Subject: Re: Flux without flux
Jack Sarfatti wrote:
A funny thing happened to me on my way around the singularity.
It is intuitively obvious to me that the reason the A&P vacuum
"torsion" stress-energy tensor is allegedly "local" in the usual
sense you describe above is that in my theory it is only second
order in derivatives of the vacuum ODLRO Goldstone Phase of the
post-inflationary "Higgs Ocean" (Brian Greene's term). Similarly,
the reason why Yilmaz theory is fundamentally wrong, and that the
geometrodynamic gravity energy is FUNDAMENTALLY NONLOCAL, so what
Alex P does should not be done, is that using EEP ("Grad" in 4D sense)
g(Einstein Curved ST) = [I + B(torsion)Grad(Phase)](Flat False
Vacuum)[I + B(torsion)Grad(Phase)]
The Einstein 4th rank tidal geodesic deviation curvature tensor is
third order in the derivatives of the Goldstone phase, and even when
you contract it to second order Guv = Ruv - (1/2)Rguv you have those
3rd order "jerk" field theory terms very much like the "radiation
reaction" in which you need nonlocal presponse from the future to
avoid runaway solutions in the charged particle mechanics case. Same
thing going on here. Of course Roger Penrose is unaware of this new
way of looking at the problem, which is why he is vague about it -
at least in Road to Reality. But MTW are essentially correct EEP
demands nonlocal gravity energy. They did not give the correct
argument however. Z has a point there.
Z: Two points.
(1) The reason why the vacuum stress-energy is local in A&P is that
they use a "physically equivalent" teleparallel formalism that
dispenses with manifold curvature, allowing an objective local
decomposition of the inertial and gravitational parts; and
J: Perhap. But that says nothing about the problem in the geometrodynamic
picture.
Z: Remember that A&P also insist that their teleparallel formalism is
"physically equivalent" to standard curved-manifold GR.
J: See my remarks on implicate/explicate Fourier transform hologram relation between the two pictures rather than 1-1 mechanical mapping of parts to parts, it's parts to wholes & vice versa.
I am willing to grant a local "torsion field" stress-energy tensor in
the teleparallel SUBSTRATUM to the Einstein curved geometrodynamic field.
Z: OK.
J: Intuitively this tetrad substratum is the "square root" of the
geometry. The tetrad field is essentially the distortion field of the
vacuum ODLRO 4D "supersolid" (Diff(4) covariant "aether") world
crystal Planck lattice.
Z: OK, but am I correct in thinking that your "B" transformation always
represents an actual deformation of the manifold that results in a change in the intrinsic geometry, while your "I" transformation represents *only* a change of local coordinates (transformation of coordinate bases) that can occur even while the
geometry of the manifold is fixed?
J: YES! That's my idea here.
Z: The point here is that a mere change of coordinates in itself has no
effect on intrinsic geometry.
J: Obviously. Also the Cartan forms are manifestly coordinate-independent.
The non-trivial warp part of the Einstein-Cartan tetrad field is the
Cartan 1-form
B ~ (Lp/2pi)d(Goldstone Phase of vacuum ODLRO Higgs Field)
where d is the Cartan exterior derivative.
Lp^2 = hG/c^3
Complete Einstein-Cartan tetrad is e = I + B, where I = Identity
Z: If I = Identity, exactly what does I transform?
J: Paul, you really should understand this by now! It's obvious. When B = 0 the intrinsic geometry is globally flat Minkowski. Therefore the tangent bundle is degenerate i.e. the tangent fiber space is IDENTICAL to the BASE SPACE
(Minkowski)uv = Iu^a(Minkowski)abIv^b
a,b are tangent space indices
u,v are base space indices
Nevertheless you still can IMPOSE GCT's i.e. Xu^u' arbitrary LNIF observer fields. When you do that EVERYTHING that results is PHONY GLOBAL GRAVITY in Landau & Lifshitz sense, even though LOCALLY in sense of "correspondance" if you like you cannot tell if you do not try too hard with tidal geodesic deviation measurements.
That is
gu'v' = Xu'u(Minkowski)uvXv^v is a GLOBAL PHONY GRAVITY g-FIELD, i.e. pure inertial force field from the LNIF observers firing their rockets in space.
As Kiehn points out one must be careful to distinguish exact forms
from closed forms. All exact forms are closed but not vice versa.
Integrating a closed form around a hole can give a non-vanishing
result even when there is no dynamical local gauge field present.
Z: OK.
J: In more detail, the most general decomposition for a p-form that is the DeRahm "integrand"
p-form = exact p-form + closed non-exact p-form + non-closed p-form
The cohomology factor group is
closed means d(p-form) = 0
Note the gravity energy nonlocality is connected with inability to generalize Stoke's theorem:
Integral over a bounding set of p-cycles of any p-form = Integral over the bounded p+1 manifold of d(p-form)
to replacing d by the gauge covariant D = d + B
where B is a connection 1-form.
H^p = (All closed p-forms/exact p-forms) = p-th cohomology group
dimH^p = integer number of p-hole topological obstructions
dimH^p = 0 means simply-connected NO p-holes, i.e. every closed p-form is exact.
Similarly
Hp = (cycles/bounding cycles) = Dual pth homology group for the domain manifold of integration.
Obviously dimHp = dimH^p
On a simply-connected manifold where
Exact p-form = d(p-1 form)
The "potential" p-1 form is a state function, the integrals of exact p-form over p-co-forms are path-independent.
Examples: 1. Conservative force fields in Newton's particle mechanics.
2. Reversible thermodynamics of iso-entropic Carnot Heat Engines.
3. Non-closed forms mean irreversibility & TURBULENCE (R. Kiehn)
So God DID answer von Neumann at the Pearly Gate after all!
4. Classical Action Principle of Quantum Field Theory uses closed 1-form.
Conservative force fields are exact action Lagrangian 1-forms in configuration space.
Hamiltonian is in symplectic phase space where we need to use Wigner phase space functions for the quantum histories!
Non-closed Action 1-form must be the vacuum/ground state zero point fluctuations of paths away from classical limit of constructive interference of Feynman histories.
In contrast take the archetypal 1-form that is closed, but not exact. Given any abstract flat x-y plane with polar coordinates
1-form = (xdy - ydx)/(x^2 + y^2) = "d(Theta)"
I will put quotes to indicate that, despite the d, this closed 1-form is not exact. That is, the "0-form" Theta is NOT UNIQUE (R. Kiehn), but is at least 2 overlapping functions, same as covering the 2-sphere where polar axis is a topological obstruction.
Think of this as a local coordinate chart with origin at x^2 + y^2 = 0, this is a topological obstruction. So, in this abstract plane
Imagine points A = (x = -1,0) and B = (x = +1,0) integrate the 1-form "dTheta" CW from A to B around upper semi-unit circle x^2 + y^2 = 1 to get +pi.
Similarly, integrate the same 1-form from CW A to B along lower semi-unit circle to get -pi. Obviously, integrating around the NON-BOUNDING closed circle gives 2pi, i.e. winding number = + 1. So Stoke's theorem is suspended for it!
In this simple world dimH^1 = 1
However that unit circle is not a boundary! So the period integral of "dTheta" does not vanish! Imagine an inner circle of radius epsilon -> 0 that we traverse CCW to get -2pi. The boundary is BOTH circles and for the interior of the boundary d^2Theta = 0 shows that the sum of the two integrals over the actual boundary isolating the topological Theta phase singularity is ZERO.
But now notice we never really need to imagine some kind of second source flux at the hole! We can shrink epsilon to zero!
This is "flux without flux".
No quantization yet because no single-valued ODLRO as yet!
Also we get dimH^p = N by sewing N copies of above together in a quilt of overlapping coordinate patches similar to making N-tori S^1xS^1 x ... N times.
In general "x,y" above are abstract. In ODLRO they live in G/H SBS order parameter space, where physical space x^u or x^a are CONTROL PARAMETERS (Catastrophe theory & V.I. Arnold?)
In Bohm's language for a single component complex order parameter G/H = U(1)
x = RcosS
y = RsinS
PSI = Re^iS
R(x^u) S(x^u).
Therefore, the topological obstructions (S phase singularities) are the loci of x^u where R(x^u) = 0. For G/H = U(1) these loci are "strings" or vortex lines where R -> 0. Each contiguous string or vortex is a single obstruction. That is
DimH^1 = number of vortices in the system in the emergent gravity problem
B = (Lp/2pi)"dTheta" for the non-trivial intrinsic gravity warp piece of the Einstein-Cartan tetrad, where "Theta" is the set of 0-form branches of the Goldstone Phase of the vacuum ODLRO that cover the entire G/H order parameter space for a given space-time region x^u.
e = (I + B)
The EEP is
g(Einstein real gravity) = (I + B)(Minkowski)(I + B)
This splits into 3 qualitatively different pieces
I(Minkowski)I globally flat component
B(Minkowski)I + I(Minkowski)B = weak field linear elastic Diff(4) covariant ODLRO supersolid
B(Minkowski)B = strong field "geon" nonlinear plastic component
Note that B is the compensating connection 1-form from locally gauging T4 to Diff(4),
but IF FLUX WITHOUT FLUX is true, that comes entirely from the Wheeler "wormholes" i.e. non-trivial cohomology
DimH^p =/= 0 in the torsion tetrad SUBSTRATUM of curved geometrodynamics!
Therefore,
I. Flux without flux.
->
II Charge without charge
III Mass without mass.
Also
IV Spin without spin?
e.g. Spinor ODLRO from boson condensate?
i.e. make a closed non-bounding loop around singularity in ordinary space (e.g. vortex line) that induces in order parameter space U(1) in this case
Theta = argODLRO
Theta -> Theta' = Theta + pi
|ODLRO|^2 -> |ODLRO|'^2 = -i(ODLRO)*i(ODLRO) = +|ODLRO|^2
away from the singularity |ODLRO|^2 = 0 globally along a given string in ordinary space.
This is "Flux without flux" - Ghost of a departed quantity.
Not only "Mass without mass", "Spin without spin", "Charge without
charge" (Real Flux), BUT EVEN "Flux without flux"? This is really
getting something from nothing. Actually it's cohomology on
multiply-connected manifolds.
Z: Name of the game. :-)
J: The A&P torsion field F uses the gauge covariant exterior derivative
D = d + B
F = DB = dB + B/\B a 2-form
The issue now is dB
dB = Lpd^2B = 0
since d^2 = 0
That is B is a closed 1-form.
However, because of "Flux without flux" we cannot simply conclude that
F = B/\B under all conditions.
Why? Because the physical quantities are always global loop integrals
of the closed forms even in the limit that the loops shrink to get
"local observables". Anandan and others have shown this. There is a
nice article by Wheeler explaining this point. Basically everything is
related to the Bohm-Aharonov effect and it's "inverse".
"Ghosts of departed quantities"
Z: "Ghosts of the departed aether".
It seems now, if "Flux without flux" is a genuine discovery and not a delusion, not a false attractor on my mental PSI ODLRO landscape like your "(LC) = Tensor + Non-Tensor & Tensor =/= 0" delusion Paul! ;-), then the REASON for gravity as T4 locally gauged to Diff(4) is completely TOPOLOGICAL in the sense of non-trivial cohomology of the Cartan p-forms that ultimately express ALL of physics.
That is TOPOS is prior to Geometrodynamica!
Consider a tightly wound thin solenoid with an actual electric current
spiraling around the coil and the uniform magnetic flux along the axis
of the coil. A single electron passes a beam splitter with two paths
around the coil. The electron feels a local A-field where B = curlA =
0. Nevertheless, changing the magnetic flux through the coil will
shift the probability "fringe" pattern of where the electron is likely
to be detected. The relative phase between the Feynman amplitudes for
the two indistinguishable alternative single-electron paths is the
loop integral of p/h - (e/hc)A where dA = 0 everywhere along all
possible paths for the electron. Nevertheless, the nonlocal influence
of the isolated dA =/= 0 region where there is an actual B field from
the spiraling currents is felt quantum mechanically. But this is
nonlocal action at a distance of "Flux with flux" and there is no
macro-quantum ODLRO in this micro-quantum nonlocality of the
Bohm-Aharonov effect, which generalizes.
Now consider ODLRO. We have, in simplest case a complex local order
parameter Psi(r,z,phi,t), i.e a GIANT quantum wave function. This is
different from MIDGET quantum wave functions. Orthodox quantum theory
is only about the Dwarves and Midgets, the Little People as it were. :-)
I use cylindrical coordinates. Imagine now a closed loop in ordinary
space around a line defect (z-axis) where the ODLRO parameter vanishes.
That is
|Psi(0,z,phi,t)| = 0
Psi = |Psi|e^iTheta
Therefore, although Theta(0,z,phi,t) =/= 0 it is ill-defined in the
complex Psi plane where the "origin" is at |Psi| = 0.
Psi must be single-valued. At least |Psi|^2 must return to itself in
any closed loop in ordinary space. This does not preclude a giant
ODLRO SPINOR (made from bosons). For now exclude the GIANT SPINOR
emergent from boson ODLRO condensate and only consider the case
Theta(r,z,0,t) = Theta(r,z,2pi) + N2pi = Goldstone phase change round
a closed loop in ordinary space
N = +- 1, +- 2, .... around the phase singularity where |Psi| = 0
B = (Lp/2pi)dTheta
Therefore, integral of closed 1-form B round the closed ordinary space
loop is
|B = NLp = ||dB =/= 0 .
This is "Flux without flux"
The effective mean ghostly torsion field in the teleparallel
substratum is then ~ NLp/4pir^2
Therefore, around a Goldstone phase singularity in the substratum
F = DB = NLp/4pir^2 + B/\B
What this portends in curved geometrodynamics needs to be studied.
This is TOTALLY NEW!
g(Curved) = (I + B)(Minkowksi)(I + B)
The LOCAL stress-energy tensor in the substratum ~ (1/8pi)F/\*F
(something like that), but this says nothing directly about the
corresponding problem in curved geometrodynamics.
Z: OK. Not directly. Agreed it is not yet clear exactly how this relates mathematically to the corresponding problem in curved-manifold theory.
(2) The presence of cross terms in the formal tetrad expression for
the LC connection does not necessarily prevent a local linear
decomposition of the Christoffel symbols into tensor and non-tensor
parts.
J: Yes, it does. Your idea there is clearly wrong. The Christoffel symbol
is a non-tensor period (except for linear GCT's where it is a tensor
under that restriction only)
Z: Of course I agree that the Christoffel symbol is a non-tensor under GCTs. You haven't explained exactly why it is a non-tensor under non-linear
transformations, while it is a tensor under linear transformations.
J: Paul that IS trivial!
GCT = Group{X}
GCT:(LC) = (LC)' = XXX(LC) + XY
where from the construction
Y = 0 for ALL linear GCT's.
Y =/= 0 in essence defines a necessary property for nonlinear GCT.
Z: What is it exactly about *non-linear* coordinate transformations that
spoil the tensor property of the Christoffel symbols?
J: Uh Oh Paul it's Alzheimers! Do you remember your name? What planet is this? Look's like The Grays are working on you at night when you think you are in bed.:-)
Keep up with discussion forum at http://stardrive.org/
for fast-breaking news!
TOPOS is prior to Geometrodynamica!
What I cannot create on the spot spontaneously ab-initio I certainly do not understand. It's the "non-mechanical" Bohmian unfolding of a compressed implicate algorithm that we call "intuition" with a dash of "presponse".
The idea of John Archibald Wheeler in the late 1950's & early 60's was to fulfil "Einstein's Vision" that only made sense in terms of David Bohm's pilot wave/ hidden variable theory because Einstein envisioned elementary particles as pure vacuum tiny wormholes with quantized trapped gauge force fluxes. The idea, like Andrei Sakharov's vision of emergent gravity from the "metric elasticity" of zero point quantum vacuum fluctuations, was very incomplete.
1. Gravity is too weak by 40 powers of ten unless it gets stronger on the scale of 1 fermi. Abdus Salam suggested that in early 70's "f-meson" and I showed it explained the Regge trajectories of hadronic resonances J ~ alpha'E^2 + ... where alpha' = (1Gev)^-2. Salam invited me to ICTP Trieste in 1973-4 because of that.
2. Wheeler had no explanation of quantized flux because he did not have idea of vacuum ODLRO. The Penrose-Onsager idea of ODLRO was new. The Higgs mechanism was not understood. (Bernie Haisch et-al still doesn't get it that it falsified his random ZPF approach to the "origin of inertia". :-)) Yang-Mills local gauging of global to local symmetries was new. The Bohm-Aharonov effect was not well understood. The time was not ripe. Also no one even conceived of anti-gravity "dark energy" except Hermann Bondi & Stalin's physics spymaster, Y. Terletskii in a very vague way as "negative mass propulsion". No one understood the exotic vacuum equation
Guv + /\zpfguv = 0
with /\zpf a local scalar field FROM vacuum ODLRO that in Newtonian limit is the Poisson equation for isotropic source
Grad^2V(source) = 4piG(mass density)(1 + 3w)
Ordinary matter w = 0
Radiation w = +1/3
All random zero point fluctuations from all quantum fields w = -1
"mass density" of ZPF can be both > 0 and < 0!
Virtual electron-positron pairs have negative effective mass density.
Virtual photons have positive effective mass density.
Virtual QED quanta cannot directly make QED counters click, but they DO directly warp space-time as "dark energy" and "dark matter" are BOTH telling us!
The post-inflation field is essentially the vacuum ODLRO field.
3. Laughlin-Chapline and myself independently have been working on the idea that it is in the ODLRO cohering of the ZPF that gravity emerges bottom->up and that all top->down versions of "quantum gravity" are the "wrong question". No gravitons, no quantum foam is the message from top->down quantization of guv is unrenormalizable. [Ashtekar's loop variables seem to be a version of my B warp tetrad torsion field?]
4. The idea that black holes may be prevented by repulsive dark energy cores (dark energy stars) is also immediately obvious, though it is too early to tell if it is correct. Laughlin & Chapline do argue from actual evidence however. I thought of the general idea in 2002, but did not do anything with it. Ken Shoulders "EVOs" may be a mesoscopic charged analog to the "dark stars" in that exotic vacuum cores can stabilize the N electrons, indeed the single electron solving the Lorentz problem of the "stresses" from 100 years ago, i.e. what prevents a shell of charge from exploding. Simplest example is
V = +(Ne^2)/mr + c^2/\zpfr^2
dV/dr = -(Ne^2)/mr^2 + 2c^2/\zpfr = 0
Therefore a UNIFORM DARK ENERGY core obviously stabilizes the shell of electric charge. The problem becomes trivial!
d^2V/dr^2 = +2(Ne^2)/mr^3 + 2c^2/\zpf
Static stability is d^2V/dr^2 > 0 at the critical point dV/dr = 0
OBVIOUSLY TRUE when /\zpf > 0, i.e. dark energy w = -1 negative pressure
Message: 2
Date: Sat, 09 Apr 2005 12:40:52 -0700
From: iksnileiz@earthlink.net
Subject: Re: Flux without flux
Jack Sarfatti wrote:
A funny thing happened to me on my way around the singularity.
It is intuitively obvious to me that the reason the A&P vacuum
"torsion" stress-energy tensor is allegedly "local" in the usual
sense you describe above is that in my theory it is only second
order in derivatives of the vacuum ODLRO Goldstone Phase of the
post-inflationary "Higgs Ocean" (Brian Greene's term). Similarly,
the reason why Yilmaz theory is fundamentally wrong, and that the
geometrodynamic gravity energy is FUNDAMENTALLY NONLOCAL, so what
Alex P does should not be done, is that using EEP ("Grad" in 4D sense)
g(Einstein Curved ST) = [I + B(torsion)Grad(Phase)](Flat False
Vacuum)[I + B(torsion)Grad(Phase)]
The Einstein 4th rank tidal geodesic deviation curvature tensor is
third order in the derivatives of the Goldstone phase, and even when
you contract it to second order Guv = Ruv - (1/2)Rguv you have those
3rd order "jerk" field theory terms very much like the "radiation
reaction" in which you need nonlocal presponse from the future to
avoid runaway solutions in the charged particle mechanics case. Same
thing going on here. Of course Roger Penrose is unaware of this new
way of looking at the problem, which is why he is vague about it -
at least in Road to Reality. But MTW are essentially correct EEP
demands nonlocal gravity energy. They did not give the correct
argument however. Z has a point there.
Z: Two points.
(1) The reason why the vacuum stress-energy is local in A&P is that
they use a "physically equivalent" teleparallel formalism that
dispenses with manifold curvature, allowing an objective local
decomposition of the inertial and gravitational parts; and
J: Perhap. But that says nothing about the problem in the geometrodynamic
picture.
Z: Remember that A&P also insist that their teleparallel formalism is
"physically equivalent" to standard curved-manifold GR.
J: See my remarks on implicate/explicate Fourier transform hologram relation between the two pictures rather than 1-1 mechanical mapping of parts to parts, it's parts to wholes & vice versa.
I am willing to grant a local "torsion field" stress-energy tensor in
the teleparallel SUBSTRATUM to the Einstein curved geometrodynamic field.
Z: OK.
J: Intuitively this tetrad substratum is the "square root" of the
geometry. The tetrad field is essentially the distortion field of the
vacuum ODLRO 4D "supersolid" (Diff(4) covariant "aether") world
crystal Planck lattice.
Z: OK, but am I correct in thinking that your "B" transformation always
represents an actual deformation of the manifold that results in a change in the intrinsic geometry, while your "I" transformation represents *only* a change of local coordinates (transformation of coordinate bases) that can occur even while the
geometry of the manifold is fixed?
J: YES! That's my idea here.
Z: The point here is that a mere change of coordinates in itself has no
effect on intrinsic geometry.
J: Obviously. Also the Cartan forms are manifestly coordinate-independent.
The non-trivial warp part of the Einstein-Cartan tetrad field is the
Cartan 1-form
B ~ (Lp/2pi)d(Goldstone Phase of vacuum ODLRO Higgs Field)
where d is the Cartan exterior derivative.
Lp^2 = hG/c^3
Complete Einstein-Cartan tetrad is e = I + B, where I = Identity
Z: If I = Identity, exactly what does I transform?
J: Paul, you really should understand this by now! It's obvious. When B = 0 the intrinsic geometry is globally flat Minkowski. Therefore the tangent bundle is degenerate i.e. the tangent fiber space is IDENTICAL to the BASE SPACE
(Minkowski)uv = Iu^a(Minkowski)abIv^b
a,b are tangent space indices
u,v are base space indices
Nevertheless you still can IMPOSE GCT's i.e. Xu^u' arbitrary LNIF observer fields. When you do that EVERYTHING that results is PHONY GLOBAL GRAVITY in Landau & Lifshitz sense, even though LOCALLY in sense of "correspondance" if you like you cannot tell if you do not try too hard with tidal geodesic deviation measurements.
That is
gu'v' = Xu'u(Minkowski)uvXv^v is a GLOBAL PHONY GRAVITY g-FIELD, i.e. pure inertial force field from the LNIF observers firing their rockets in space.
As Kiehn points out one must be careful to distinguish exact forms
from closed forms. All exact forms are closed but not vice versa.
Integrating a closed form around a hole can give a non-vanishing
result even when there is no dynamical local gauge field present.
Z: OK.
J: In more detail, the most general decomposition for a p-form that is the DeRahm "integrand"
p-form = exact p-form + closed non-exact p-form + non-closed p-form
The cohomology factor group is
closed means d(p-form) = 0
Note the gravity energy nonlocality is connected with inability to generalize Stoke's theorem:
Integral over a bounding set of p-cycles of any p-form = Integral over the bounded p+1 manifold of d(p-form)
to replacing d by the gauge covariant D = d + B
where B is a connection 1-form.
H^p = (All closed p-forms/exact p-forms) = p-th cohomology group
dimH^p = integer number of p-hole topological obstructions
dimH^p = 0 means simply-connected NO p-holes, i.e. every closed p-form is exact.
Similarly
Hp = (cycles/bounding cycles) = Dual pth homology group for the domain manifold of integration.
Obviously dimHp = dimH^p
On a simply-connected manifold where
Exact p-form = d(p-1 form)
The "potential" p-1 form is a state function, the integrals of exact p-form over p-co-forms are path-independent.
Examples: 1. Conservative force fields in Newton's particle mechanics.
2. Reversible thermodynamics of iso-entropic Carnot Heat Engines.
3. Non-closed forms mean irreversibility & TURBULENCE (R. Kiehn)
So God DID answer von Neumann at the Pearly Gate after all!
4. Classical Action Principle of Quantum Field Theory uses closed 1-form.
Conservative force fields are exact action Lagrangian 1-forms in configuration space.
Hamiltonian is in symplectic phase space where we need to use Wigner phase space functions for the quantum histories!
Non-closed Action 1-form must be the vacuum/ground state zero point fluctuations of paths away from classical limit of constructive interference of Feynman histories.
In contrast take the archetypal 1-form that is closed, but not exact. Given any abstract flat x-y plane with polar coordinates
1-form = (xdy - ydx)/(x^2 + y^2) = "d(Theta)"
I will put quotes to indicate that, despite the d, this closed 1-form is not exact. That is, the "0-form" Theta is NOT UNIQUE (R. Kiehn), but is at least 2 overlapping functions, same as covering the 2-sphere where polar axis is a topological obstruction.
Think of this as a local coordinate chart with origin at x^2 + y^2 = 0, this is a topological obstruction. So, in this abstract plane
Imagine points A = (x = -1,0) and B = (x = +1,0) integrate the 1-form "dTheta" CW from A to B around upper semi-unit circle x^2 + y^2 = 1 to get +pi.
Similarly, integrate the same 1-form from CW A to B along lower semi-unit circle to get -pi. Obviously, integrating around the NON-BOUNDING closed circle gives 2pi, i.e. winding number = + 1. So Stoke's theorem is suspended for it!
In this simple world dimH^1 = 1
However that unit circle is not a boundary! So the period integral of "dTheta" does not vanish! Imagine an inner circle of radius epsilon -> 0 that we traverse CCW to get -2pi. The boundary is BOTH circles and for the interior of the boundary d^2Theta = 0 shows that the sum of the two integrals over the actual boundary isolating the topological Theta phase singularity is ZERO.
But now notice we never really need to imagine some kind of second source flux at the hole! We can shrink epsilon to zero!
This is "flux without flux".
No quantization yet because no single-valued ODLRO as yet!
Also we get dimH^p = N by sewing N copies of above together in a quilt of overlapping coordinate patches similar to making N-tori S^1xS^1 x ... N times.
In general "x,y" above are abstract. In ODLRO they live in G/H SBS order parameter space, where physical space x^u or x^a are CONTROL PARAMETERS (Catastrophe theory & V.I. Arnold?)
In Bohm's language for a single component complex order parameter G/H = U(1)
x = RcosS
y = RsinS
PSI = Re^iS
R(x^u) S(x^u).
Therefore, the topological obstructions (S phase singularities) are the loci of x^u where R(x^u) = 0. For G/H = U(1) these loci are "strings" or vortex lines where R -> 0. Each contiguous string or vortex is a single obstruction. That is
DimH^1 = number of vortices in the system in the emergent gravity problem
B = (Lp/2pi)"dTheta" for the non-trivial intrinsic gravity warp piece of the Einstein-Cartan tetrad, where "Theta" is the set of 0-form branches of the Goldstone Phase of the vacuum ODLRO that cover the entire G/H order parameter space for a given space-time region x^u.
e = (I + B)
The EEP is
g(Einstein real gravity) = (I + B)(Minkowski)(I + B)
This splits into 3 qualitatively different pieces
I(Minkowski)I globally flat component
B(Minkowski)I + I(Minkowski)B = weak field linear elastic Diff(4) covariant ODLRO supersolid
B(Minkowski)B = strong field "geon" nonlinear plastic component
Note that B is the compensating connection 1-form from locally gauging T4 to Diff(4),
but IF FLUX WITHOUT FLUX is true, that comes entirely from the Wheeler "wormholes" i.e. non-trivial cohomology
DimH^p =/= 0 in the torsion tetrad SUBSTRATUM of curved geometrodynamics!
Therefore,
I. Flux without flux.
->
II Charge without charge
III Mass without mass.
Also
IV Spin without spin?
e.g. Spinor ODLRO from boson condensate?
i.e. make a closed non-bounding loop around singularity in ordinary space (e.g. vortex line) that induces in order parameter space U(1) in this case
Theta = argODLRO
Theta -> Theta' = Theta + pi
|ODLRO|^2 -> |ODLRO|'^2 = -i(ODLRO)*i(ODLRO) = +|ODLRO|^2
away from the singularity |ODLRO|^2 = 0 globally along a given string in ordinary space.
This is "Flux without flux" - Ghost of a departed quantity.
Not only "Mass without mass", "Spin without spin", "Charge without
charge" (Real Flux), BUT EVEN "Flux without flux"? This is really
getting something from nothing. Actually it's cohomology on
multiply-connected manifolds.
Z: Name of the game. :-)
J: The A&P torsion field F uses the gauge covariant exterior derivative
D = d + B
F = DB = dB + B/\B a 2-form
The issue now is dB
dB = Lpd^2B = 0
since d^2 = 0
That is B is a closed 1-form.
However, because of "Flux without flux" we cannot simply conclude that
F = B/\B under all conditions.
Why? Because the physical quantities are always global loop integrals
of the closed forms even in the limit that the loops shrink to get
"local observables". Anandan and others have shown this. There is a
nice article by Wheeler explaining this point. Basically everything is
related to the Bohm-Aharonov effect and it's "inverse".
"Ghosts of departed quantities"
Z: "Ghosts of the departed aether".
It seems now, if "Flux without flux" is a genuine discovery and not a delusion, not a false attractor on my mental PSI ODLRO landscape like your "(LC) = Tensor + Non-Tensor & Tensor =/= 0" delusion Paul! ;-), then the REASON for gravity as T4 locally gauged to Diff(4) is completely TOPOLOGICAL in the sense of non-trivial cohomology of the Cartan p-forms that ultimately express ALL of physics.
That is TOPOS is prior to Geometrodynamica!
Consider a tightly wound thin solenoid with an actual electric current
spiraling around the coil and the uniform magnetic flux along the axis
of the coil. A single electron passes a beam splitter with two paths
around the coil. The electron feels a local A-field where B = curlA =
0. Nevertheless, changing the magnetic flux through the coil will
shift the probability "fringe" pattern of where the electron is likely
to be detected. The relative phase between the Feynman amplitudes for
the two indistinguishable alternative single-electron paths is the
loop integral of p/h - (e/hc)A where dA = 0 everywhere along all
possible paths for the electron. Nevertheless, the nonlocal influence
of the isolated dA =/= 0 region where there is an actual B field from
the spiraling currents is felt quantum mechanically. But this is
nonlocal action at a distance of "Flux with flux" and there is no
macro-quantum ODLRO in this micro-quantum nonlocality of the
Bohm-Aharonov effect, which generalizes.
Now consider ODLRO. We have, in simplest case a complex local order
parameter Psi(r,z,phi,t), i.e a GIANT quantum wave function. This is
different from MIDGET quantum wave functions. Orthodox quantum theory
is only about the Dwarves and Midgets, the Little People as it were. :-)
I use cylindrical coordinates. Imagine now a closed loop in ordinary
space around a line defect (z-axis) where the ODLRO parameter vanishes.
That is
|Psi(0,z,phi,t)| = 0
Psi = |Psi|e^iTheta
Therefore, although Theta(0,z,phi,t) =/= 0 it is ill-defined in the
complex Psi plane where the "origin" is at |Psi| = 0.
Psi must be single-valued. At least |Psi|^2 must return to itself in
any closed loop in ordinary space. This does not preclude a giant
ODLRO SPINOR (made from bosons). For now exclude the GIANT SPINOR
emergent from boson ODLRO condensate and only consider the case
Theta(r,z,0,t) = Theta(r,z,2pi) + N2pi = Goldstone phase change round
a closed loop in ordinary space
N = +- 1, +- 2, .... around the phase singularity where |Psi| = 0
B = (Lp/2pi)dTheta
Therefore, integral of closed 1-form B round the closed ordinary space
loop is
|B = NLp = ||dB =/= 0 .
This is "Flux without flux"
The effective
substratum is then
Therefore, around a Goldstone phase singularity in the substratum
F = DB = NLp/4pir^2 + B/\B
What this portends in curved geometrodynamics needs to be studied.
This is TOTALLY NEW!
g(Curved) = (I + B)(Minkowksi)(I + B)
The LOCAL stress-energy tensor in the substratum ~ (1/8pi)F/\*F
(something like that), but this says nothing directly about the
corresponding problem in curved geometrodynamics.
Z: OK. Not directly. Agreed it is not yet clear exactly how this relates mathematically to the corresponding problem in curved-manifold theory.
(2) The presence of cross terms in the formal tetrad expression for
the LC connection does not necessarily prevent a local linear
decomposition of the Christoffel symbols into tensor and non-tensor
parts.
J: Yes, it does. Your idea there is clearly wrong. The Christoffel symbol
is a non-tensor period (except for linear GCT's where it is a tensor
under that restriction only)
Z: Of course I agree that the Christoffel symbol is a non-tensor under GCTs. You haven't explained exactly why it is a non-tensor under non-linear
transformations, while it is a tensor under linear transformations.
J: Paul that IS trivial!
GCT = Group{X}
GCT:(LC) = (LC)' = XXX(LC) + XY
where from the construction
Y = 0 for ALL linear GCT's.
Y =/= 0 in essence defines a necessary property for nonlinear GCT.
Z: What is it exactly about *non-linear* coordinate transformations that
spoil the tensor property of the Christoffel symbols?
J: Uh Oh Paul it's Alzheimers! Do you remember your name? What planet is this? Look's like The Grays are working on you at night when you think you are in bed.:-)
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