On Nov 4, 2005, at 6:28 PM, Jack Sarfatti wrote:
Derivation of the World Hologram?
I am not proposing the usual magnetic monopole here. I am proposing a geometrodynamic monopole in which the global translation group T(4) generated by total 4-Momentum Pa in 1905 Special Relativity is locally gauged to Diff(4) Group of Local Coordinate Transformations.
For example, if we were doing quantum theory in flat spacetime, the unitary operator
U(La) = e^iPaL^a
Makes the globally rigid space-time displacement L^a on the simple quantum wave psi(x).
To make it simpler lets consider only 1 space dimension x and P^a is the linear momentum in the x direction. Then by Taylor series since P = (h/i)d/dx
U(L)psi(x) = psi(x + L)
everywhere in space the same L.
Periodicity (i.e. S1 topology) and single-valuedness of the quantum wave imply quantization of linear momentum, i.e. discrete momentum eigenvalues hn/2L, n in discrete group Z from homotopy
PI1(S1) = Z I suppose?
In any case, off that momentary tangent to GR.
T(1) --> Diff(1) means that L is not a constant but is an arbitrary function L(x).
This means we want a gauge covariant derivative with a compensating potential.
I leave this as a homework problem for this 1 Dim space. Actually do it in 1+1 Space-Time.
OK doing it in 4D gives the curved tetrad field
B = (hG/c^3)^1/2'd'THETA
THETA is the Goldstone Phase of the Vacuum Coherence given below. THETA is a MULTI-VALUED 0-form from branch cuts something akin to a Riemann surface!
Now in 1915 GR without torsion - start simple.
B determines the spin connection W via the
ZERO TORSION CONSTRAINT, i.e.,
De = dB + W/\(1 + B) = DB + W/\I = 0
The exterior Diff(4) covariant derivative is, therefore,
D = d + W/
If we treat B as a Yang-Mills field then
F = DB = - W/\I
note that DB = dB + W/\B
means
DB^a = dB^a + W^ab/\B^b
But vanishing torsion implies
DB^a = - W^ab/\I^b
DF = 0 Bianchi identity, i.e. like Faraday induction & no magnetic monopoles
D*F = *J(curved tetrad sources) i.e. like Ampere's law & Gauss's law
D*J = 0 local current density conservation
In any case the VACUUM MONOPOLE will be in this curved tetrad B field not the EM A field.
The GEOMETRODYNAMIC FLUX is the two-form in 4D space-time
*F = *DB
Therefore, the surface integral of 2-form *F is the volume integral of the 3-Form *J and they are QUANTIZED obviously in accord with the Bekenstein-Hawking thermodynamics of blackholes
The 't Hooft-Polyakov point defect of the Higgs field when surrounded by the surface integral of *F is ~ NArea/Lp^2 where N is in Z
Since *F is normalized to 1/Lp^2 ~ c^3/hG.
Seems logical? Pretty simple from my new POV.
Is this "the organizing idea" the M-theorists are looking for?
On Nov 4, 2005, at 11:24 AM, Jack Sarfatti wrote:
I here UNIFY the Electro-Weak Force with Einstein's Gravity Geometrodynamics.
I think this is the first time anyone has even thought of doing this?
Why did I think of this?
Simple!
Einstein's Equivalence Principle
That's why!
If leptons and quarks get their rest inertia from the electro-weak force vacuum symmetry breaking Higgs mechanism then so must gravity (curved space-time) emerge from that because:
Any physical theory that does not have gravity and inertia emergent together is a wrong theory.
Theorem of 'tHooft & Polyakov
"If the manifold of degenerate vacua contains non-contractible two-surfaces like the sphere S2. This corresponds to the vacuum manifold V = G/H having a non-trivial second homotopy group PI2(G/H) =/= I. If the unbroken VEV invariant normal subgroup G of H has a non-trivial FUNDAMENTAL GROUP PI1(H)" HENCE A CONJUGATE GRAVITY-GENERATING GOLDSTONE PHASE IS GUARANTEED! "then there are point defect 'monopole' solutions from
PI1(U(1)) = Z"
IN FACT IN GENERAL
"Grand unified models based on simple groups G which break to leave H = U(1)EM must produce MONOPOLES.
The simplest monopole is the 't Hooft-Polyakov ... which appears when SU(2) is broken to U(1) ... the 3-component Higgs field phi far from the core of the monopole takes a HEDGEHOG configuration
VEV = (0|phi|0)3r^
where r^ is the unit radial vector in physical 3D spacelike slice of 4D space-time."
p. 64 "Cosmic Strings & Topological Defects"
In this case however we do not have a magnetic monopole, but a charge neutral geometrodynamic monopole because the gauge potential is not Aem, but the curved tetrad
B = (hG/c^3)^1/2'd'(Goldstone Phase) Translation group to Diff(4)
The effective vacuum order parameter here in the 3-irrep of SU(2)weak is
a_g = -dV(Dark Energy)/dr = cH(t)r^
From Einstein's eq using w = -1
G00 ~ c^2/\zpfg00
/\zpf ~ H(t)/cr
g00 ~ 1
See below.
NOTE TO HISTORIANS:
I thought of the HEDGEHOG from the NASA Pioneer data alone BEFORE I knew of the 't Hooft-Polyakov "theorem".
I claim that Einstein's gravity emerges from the single Goldstone phase in
e^-i(Goldstone Phase)T3
that leaves the only non-vanishing VEV (0|Higgs Field|0)3 invariant in the 3-adjoint irrep of SU(2)weak spontaneous symmetry breaking of the vacuum
G = U(1)hypercharge SU(2)weak --3--> H = U(1)EM (photon)
i.e.
The degenerate vacuum manifold for constant (0|Higgs|0)3 is S2.
n' = 3 is the number of real scalar Higgs fields in the 3x3 irrep of SU(2) in which the VEV is taken along the 3-direction in fiber group parameter space.
However, U(1)xSU(2) has n = 4 real scalar fields in d = 4 space-time corresponding to group U(2). That is one mode for each generator in the Lie algebra.
e^-i(Goldstone Phase)T3(0|Higgs|0)3 = (0|Higgs|0)3
Einstein's geometrodynamic field, therefore is the curved tetrad 1-form
B = (hG/c^3)^1/2'd'(Goldstone Phase of U(1)xSU(2) --3--> U(1))
'd' means topological singularities in the vacuum manifold
V = G/H = U(1)hyperchargexSU(2)weak/U(1)em = S2
N = 4 = number of generators of G = U(1)xSU(2).
N - K = 1 is number of generators of H = U(1)
Therefore,
K = 3
is the number of massive SU(2)weak force gauge bosons
i.e.
W+, W0, W-
On the other hand, the number of real Higgs quanta is
n - K = 4 - 3 = 1
This provides our UNIQUE Goldstone phase for emergent Einstein-Gravity as a c-number low-energy long range effective field theory.
The Higgs quanta on mass shell are small vibrations of the single real Higgs VEV along the 3-direction in SU(2) group parameter space in the 3x3 matrix adjoint irrep. The physical Goldstone phase leaving that (0|Higgs Field|0) invariant mod small oscillations is the S1 circle of renormalizable Mexican Hat minima in the 1-2 plane of internal weak force 3x3 group representation fiber space.
The number of massless gauge quanta is
N - K = 1, i.e. H = U(1)em.
OK, think of U(1)hypercharge SU(2)weak as a 4D group parameter space.
One real dimension for each Lie algebra generator. We have 4 with 4 conjugate phase parameter.
2 orthogonal TRANSVERSE directions are W+ & W- they are T1 & T2.
T3 is electrically charged neutral.
T4 the hypercharge is also electrically charged neutral.
The T4-T1 plane (like longitudinal-timelike virtual photon polarizations)
has a rotated frame in which one axis is W0 the massive weak electrically neutral weak current boson
and the orthogonal axis is the MASSLESS electrically neutral PHOTON!
So that's ELECTRO-WEAK THEORY in a NUT SHELL without the PARITY VIOLATION - we stick that in later. Now for the emergence of gravity:
Suppressing the tangent fiber indices, in compact symbolic form
ds^2 = guvdx^udx^v = (1 + B)(Minkowski)(1 + B)
D = d + W/
De = dB + W/\B = 0
R = DW = curvature 2-form
Einstein-Hilbert Lagrangian density is for /\zpf = 0
L = R/\e/\e
DR = 0 is Bianchi identity
D*R = *J(Matter) is Einstein's field equation.
* = Hodge dual
D*J(Matter) = 0 is local conservation of matter stress-energy current densities.
TYPO-CORRECTED Draft 2
On Nov 3, 2005, at 7:35 PM, Jack Sarfatti wrote:
On Nov 3, 2005, at 2:42 PM, art wagner wrote:
Are you totally anti-Chaplygin...? Is Lobo loco...?
No.
(http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0511/0511003.pdf)
http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_05_jpg.htm
Note that the energy density in the expanding universe scales as
energy density ~ (space expansion factor)^-3(1 + w) = a^-3(1 + w)
OK the skewon field has w = -1/3, therefore
skewon energy density ~ a^-3(2/3) = a^-2
Phantom energy has w < - 1 therefore its energy density increases as universe expands. This causes the BIG RIP totally destroying the universe.
However, Lorentz invariance and equivalence principle DEMAND UNIVERSALLY for ALL vacuum ZPF
w = - 1 EXACTLY (at least for isotropic fields).
Einstein's equation for ZPF i.e.
D*R = *J(ZPF)
R = curvature 2-form
translates to the w = -1 exotic vacuum equation
Guv + /\zpfguv = 0
In general in the Newtonian weak curvature Galilean v/c << 1 limit
Guv + kTuv(sources) = 0
in the 00 component
G00 ~ Grad^2V ~ (G/c^2)(Energy Density)(1 + 3w)
For w = 0 we have Newton's gravity theory in Poisson eq. form.
for radiation w = +1/3 we have the famous Einstein factor of 2 in gravity bending of light. Note that fast matter looks like light in this regard.
For Skewons we have w = -1/3 so that the source term on RHS VANISHES.
Skewons DO NOT BEND SPACE-TIME (at least on large scale isotropic case - that may be the loophole).
ZPF BENDS SPACE-TIME ABSOLUTELY, i.e. NO CASIMIR FORCE SUBTRACT NO BOUNDARY FROM BOUNDARY as in QED. QED thinking FAILS for gravity. That's one of Hal Puthoff's BIG BLUNDERS in his attempt at metric engineering. Another is his naive use of a single global coordinate system - another story.
For zero point energy w = -1 ON THE NOSE and in isotropic case
G00(ZPF) ~ Grad^2V(ZPF) ~ -2(G/c^2)(Energy Density)
Therefore,
POSITIVE ZPF ENERGY DENSITY ANTIGRAVITATES with a UNIVERSAL REPULSION
NEGATIVE ZPF ENERGY DENSITY GRAVITATES with a UNIVERSAL ATTRACTION
Put them together for NEGATIVE MATTER PROPULSION of the GEODESIC KIND - no g-forces!
The basic story is simple!
You need torsion to make /\zpf inhomogeneous. Maybe you need a Skewon as well as an auxilliary field? I don't know yet.
Also small scale anistropies may be important. Above is only for FRW cosmology very large scale homogeneous isotropic "aether".
Further
Total Exotic Vacuum Virtual Particle "4D Super-Solid" Density = Higgs Vacuum Locally Coherent Condensate Density + Nonlocally Coherent Locally Incoherent EPR ZPF density
The second term on RHS can be locally negative as can the LHS.
Remember in the Einstein-Podolsky-Rosen-Bohm effect, the photon pair state is nonlocally coherent but locally incoherent. Same for momentum pair correlated interferometry. The Higgs field however is also LOCALLY COHERENT and it is that LOCAL COHERENCE that Einstein's curved space-time gravity emerges from!
B(curved) ~ 'd'(Goldstone Phase of U(1)yxSU(2)weak -> U(1)em)
i.e. point defects like Pioneer anomaly since
n = 4 real scalar fields
N = 4 generators of U(1)hyperchargexSU(2)weak
N - K = 1 = number of generators of U(1)em
K = 3
n - K = 1 is the single Higgs boson amplitude oscillation whose conjugate phase is the Goldstone phase out of which Einstein's 1915 GR field equations emerge quite simply.
K = 3 massive weak force bosons W+, W0, W-
N - K = 1 is the massless photon of H = U(1)em
The NASA PIONEER gravity anomaly data show this effect clearly on macro-scale of tens of AU with the point defect d' = 0 at center of our Sun
i.e.
This is a point defect with "wrapping number" = -1.
Instead of magnetization we have
a_g = -cH(t) ~ 1 nanometer per sec^2 anomalous dark energy blue shift between the two S2 spherical surfaces. First one at ~ 20AU. Don't know where 2nd one ends.
That is anomalous part of the radial geodesic equation is
dr^2/dt^2 - a_g = 0
a_g as a vector is constant
/\zpf = H(t)/cr
V(ZPF) = c^2/\zpfr^2
= cH(t)r
a_g = - dV(ZPF)/dr = - cH(t)
as observed.
Most of this is in my book Super Cosmos.
http://amazon.com
But the U(1)hyperchargeSU(2)weak ---> U(1)em
will be in book #4 of
Space-Time and Beyond: The Series
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