Review flat space EM
Locally gauge 1-parameter U(1) to get A
The EM field is
F = dA
on a simply-connected manifold (no Dirac strings etc)
d^2 = 0
dF = 0 (Bianchi identities)
are Faraday's law & no magnetic monopoles
d*F = *J (Source eq)
are Gauss's law & Ampere's law.
d^2*F = d*J = 0 current density conservation
Next Yang-Mills theory (weak, strong forces without Higgs-Goldstone
Vacuum ODLRO SSB fields)
Locally gauge N-parameter internal Lie group G to get A'
Define covariant exterior derivative
D = d + A'/\
F' = DA' = dA' + A'/\A'
Bianchi identities
DF' = 0
Source eq
D*F' = *J'
Now go to General Relativity
Locally gauge 10-parameter Poincare group T4xO(1,3) to get B from T4 & C
from O(1,3).
The tetrad field is (LOCAL FRAME INVARIANT notation)
e = 1 + B + C
B = C = 0 is CONFORMAL Special Relativity (GLOBALLY FLAT S-T NO GRAVITY
NO INERTIA FALSE VACUUM)
Define Einstein's metric field g(CURVED) using ONLY B from T4 -> Diff(4)
i.e. EEP is
g(CURVED) = (1 + B)(Flat)(1 + B)
Torsion 2-Form is
T = De
where we need to introduce the SPIN CONNECTION W
D = d + W/\
T = de + W/\e
In 1915 GR i.e. ONLY T4 -> Diff(4)
T = 0 & C = 0
Therefore B determines W completely.
Tidal Curvature 2-Form is
R = DW = dW + W/\W
Bianchi identities
DR = 0
So we expect
D*R = *J
to map to
Guv(Geometry) = kTuv(Matter)
with
DD*R = D*J = 0
for "simply-connected" manifold.
Einstein-Hilbert Action Density is the 4-form
R/\e/\e + /\zpfe/\e/\e/\e
Energy momentum tensor is functional derivative with respect to SUB-tetrad
e' = 1 + B (ignoring torsion C)
But we also have OTHER EQUATIONS
When T =/= 0
DT = D^2e = 0
D*T = *J'
D^2*T = D*J' = 0
NOTE
C = J'
i.e. *J' is a 3-form. Therefore J' is a 1-form
The source torsion current comes from locally gauging the Lorentz group
O(1,3)
i.e. the torsion connection is its own source.
We also have the SUBSPACE equations from
D' = d + B/\ + C/\
to investigate
F' = D'(B + C)
D'F' = 0
D'*F' = *J"
D'^2F' = D'*J" = 0
On Sep 3, 2005, at 7:18 PM, Jack Sarfatti wrote:
Note
S(x) = *dT(x) + *W(x)/\T(x) = *DT(x)
d^2 = 0 if NO multiply-connected branch cut phase jumps
T(x) = de(x) + W(x)/\e(x)
S(x) = *d^2e(x) + *d(W(x)/\e(x)) + *[W(x)/\de(x)] + *[W(x)/\W(x)/\e(x)]
In 1915 GR
S(x) = 0
d^2e(x) = d^2B(x) =/= 0 when there are argphi branch cuts (Dirac
strings/Abrikosov vortex lattices), i.e. multiply-connected gauge
transformations (Hagen Kleinert).
On Sep 3, 2005, at 4:31 PM, Jack Sarfatti wrote:
Indeed
http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb2/Kolb2_Page_12_jpg
http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_21_jpg.htm
The percentages are probably contingent, i.e. WAP "Landscape" (Susskind).
Remember my theory describes the 23% dark matter as negative zero point
energy density with positive pressure /\zpf < 0, and the 73% dark energy
as positive zero point energy density of negative pressure /\zpf > 0.
i.e. we are a little to the RIGHT of the "0" on the above plot at
current epoch.
Since the negative ZPE density CLUMPS from positive pressure, it looks
like w = 0 CDM to us distant observers scaling as a(t)^-3.
Note Lenny Susskind's "complementarity" between distant LNIF "rest
observers" ("FIDOS") and LIF geodesic observers ("FREDOS") free falling
through an event horizon. That idea will pop up here in a slightly
different guise - later on.
http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_05_jpg.htm
Both obey Einstein's exotic vacuum equation
Guv + /\zpfguv = 0
but at different length scales, i.e. do a scale-dependent wave number
power spectrum instead of only temporal frequency. To be more precise
use the whole k-w "space" and plot how much zero point energy density
there is - but you need to use Wigner phase space density, i.e. 8-dim
domain.
But we do need Gennady Shipov's torsion field extension of 1915 GR to
get variation of /\zpf away from a uniform constant value demanded by
Bianchi identities
Guv^;v = 0
breaks down with torsion field from locally gauging entire 10-parameter
Poincare group of 1905 SR rather than ONLY its 4-parameter T4
translation group.
Something like /\zpf(x,p) both positive and negative in different "flat"
tangent fibers - the extent of the tangent fiber at x must be small
compared to ~ Guv(x)^-1/2.
B = (hG/c^3)d(Vacuum ODLRO Phase) = Budx^u
Vacuum ODLRO Phase = argphi
http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_27_jpg.htm
Bu = Bu^a&a
B = (hG/c^3)d(Vacuum Phase) = Bu^adx^u&a
{dx^u} = basis of forms in base space
{&a} = basis of co-forms in tangent fiber
Einstein's geometrodynamic field of 1915 GR is
guv(x) = [Iu^a + Bu^a(x)](flat metric)ab[Iv^b + Bv^b(x)]
Iu^a is 4x4 unit matrix tetrad (see Rovelli's "Quantum Gravity" Ch 2)
This relation holds even when we add the torsion potential 1-form S =/=
0 where the Einstein-Cartan tetrad is
e(x) = 1 + B(x) + S(x)
The torsion "dislocation" 2-form is
T(x) = de(x) + W(x)/\e(x)
T(x) =/= 0 only when S(x) =/= 0
W(x) is the SPIN CONNECTION 1-FORM
The tidal geodesic deviation "disclination" curvature 2-form is
R(x) = dW(x) + W(x)/\W(x)
In general W(x) is determined from both B(x) and S(x).
B(x) from locally gauging T4
S(x) from locally gauging O(1,3).
Note that the one-form S(x) is the Hodge-dual of DT(x) where D... = d...
+ W(x)/\ ...
S(x) = *dT(x) + *W(x)/\T(x) = *DT(x)
T(x) = T(x)^a^b&a&b
T(x)^a^b = (hG/c^3){(argphi)^,a^,b - (argphi)^,b^,a }
On Sep 3, 2005, at 3:19 PM, Saul-Paul & Mary-Minn Sirag wrote:
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