Ivancevic_Applied-Diff-Geom

950 Applied Differential Geometry: A Modern IntroductionLet K be a world space–time connection andΓ Kαβνλ = 1 2 [kα ενk ε βλ − k α ελk ε βν + ∂ α K α βν + ∂ ν K α βλ−2K ε (νλ)(K α βε − k α βε) + K ε βλk α εν + K ε βνk α ελ − K α ελk ε βν − K α ενk ε βλ]be the corresponding connection on the bundle C (5.438). Let K ′ be anothersymmetric world connection. Building on these connections, we setup the following connection on the bundle C × Σ,Γ αβ α = −K ′ αελg εβ − K ′ βελg αε ,Γ α βνλ = Γ Kαβνλ − 1 2 Rα βνλ , (5.442)where R α βνλ is the Riemann curvature tensor of K.For all connections (5.442), the following Hamiltonian forms are associatedwith the Lagrangian density L HE and constitute a complete family:H HE = (p αβ α dg αβ + p α βνλ dk α βν) ∧ ω α − H HE ω,H HE = −p αβ α (K ′ αελg εβ + K ′ βελg αε )+ p α βνλ Γ Kαβνλ − 1 2 Rα βνλ(p α βνλ − π α βνλ )= −p αβ α (K ′ αελg εβ + K ′ βελg αε ) + p α βνλ Γ α βνλ + ˜H HE ,˜H HE = 12κ R√ −g. (5.443)Given the Hamiltonian form H HE (5.443) plus a Hamiltonian form H Mfor matter, we have the corresponding Hamiltonian equations∂ α g αβ + K ′ αελg εβ + K ′ βελg αε = 0, (5.444)∂ α k α βν = Γ Kαβνλ − 1 2 Rα βνλ, (5.445)∂ α p α αβ = p σ εβ K ′ εασ + p σ εα K ′ εβσ (5.446)− 12κ (R αβ − 1 2 g αβR) √ −g − ∂H M∂g αβ ,∂ α p α βνλ = −p α ε[νγ] k β εγ + p ε β[νγ] k ε αγ − p α βεγ K ν (εγ)−p α ε(νγ) K β εγ + p ε β(νγ) K ε αγ , (5.447)plus the motion equations of matter. The Hamiltonian equations (5.444)and (5.445) are independent of canonical momenta and so, reduce to thegauge–type conditions. The equation (5.445) breaks into the following two