| Topic: |
Science > Physics |
| User: |
"Jack Sarfatti" |
| Date: |
06 Sep 2005 10:34:32 AM |
| Object: |
Quantized "vortices" in Emergent Gravity |
All of these formulae are local frame invariant
e = 1 + B Einstein-Cartan tetrad field 1-form
T = De = de + W/\e
T = torsion 2-form
W = spin connection 1-form
d = exterior derivative
T - W/\e = de
T - W/\(1 + B) = dB
since d1 = 0
dT = d^2e + d(W/\e) = d(W/\e) = d(W/\(1 + B)) is a 3-form
B = (hG/c^3)^1/2"d(argphi)" = warp gravity field
B is the compensating gauge potential from T4 -> Diff(4) and it is also
the exterior derivative of the Goldstone phase 0-form of the Higgs
Vacuum ODLRO field. Branch cut singularities (Dirac strings in space)
associated with non-trivial cohomology make Goldstone phase jumps so that
d"d(argphi)" =/= 0 i.e. Hagen Kleinert's "multiply-valued gauge
transformations"
i.e. dB =/= 0 i.e. d^2 = 0 only in trivial cohomology (simply-connected
manifolds)
R = DW = dW + W/\W
The "Bianchi identity" is
DR = 0
We also have
D*R = *J
* = Hodge dual
*D*R = J
J is a 1-form in 4D space-time
Covariant conservation of stress-energy current density would be
DJ = 0
i.e.
D*D*R = 0
dT = d^2e + d(W/\e) = d(W/\e) = d(W/\(1 + B)) 3-form
*dT is a 1-form torsion connection addition to the tetrad
e' = e + *dT
allows /\zpf^,v =/= 0
,v is ordinary partial derivative
Note
<2|dB> = <&2|B>
for a finite area <2| with closed loop boundary <&2|
Single-valuedness of the LOCAL Higgs VEV phi field gives quantized
"circulations"
<&2|B> ~ integers
Similarly,
R = DW
dW = R - W/\W
<2|dW> = <&2|W> ~ integer
.
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