Cartan Forms & General Relativity

Jack Sarfatti

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