|Oct12-06, 04:14 AM||#1|
Cartan Forms & General Relativity
Review flat space-time 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'
DF' = 0
D*F' = *J'
Now go to General Relativity
Locally gauge 10-parameter Poincare group T4xO(1,3) to get B from T4 & C
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
DR = 0
So we expect
D*R = *J
to map to
Guv(Geometry) = kTuv(Matter)
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
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
i.e. the torsion connection is its own source.
We also have the SUBSPACE equations from
D' = d + B/\ + C/\
F' = D'(B + C)
D'F' = 0
D'*F' = *J"
D'^2F' = D'*J" = 0
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