Register to reply

Cartan Forms & General Relativity

by Jack Sarfatti
Tags: cartan, forms, relativity
Share this thread:
Jack Sarfatti
#1
Oct12-06, 04:14 AM
P: n/a
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'

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

Phys.Org News Partner Physics news on Phys.org
Engineers develop new sensor to detect tiny individual nanoparticles
Tiny particles have big potential in debate over nuclear proliferation
Ray tracing and beyond

Register to reply