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Cartan Forms & General Relativity 
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Oct1206, 04:14 AM

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Review flat spacetime EM
Locally gauge 1parameter U(1) to get A The EM field is F = dA on a simplyconnected 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 YangMills theory (weak, strong forces without HiggsGoldstone Vacuum ODLRO SSB fields) Locally gauge Nparameter 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 10parameter 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 ST 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 2Form 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 2Form 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 "simplyconnected" manifold. EinsteinHilbert Action Density is the 4form R/\e/\e + /\zpfe/\e/\e/\e Energy momentum tensor is functional derivative with respect to SUBtetrad 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 3form. Therefore J' is a 1form 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 


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