Ivancevic_Applied-Diff-Geom

952 Applied Differential Geometry: A Modern Introduction(5.442) where K and K ′ are given by the expressions (5.450). It readsT sαµ = δ α µ ˜H HE = 12κ δα µR √ −gand the identity (5.398) takes the form(∂ µ + Γ αβ µ∂ αβ + Γ i µ∂ i − ∂ i Γ j µp α j ∂ i λ)( ˜H HE + ˜H M )≈ddx α (T s α µ + T Mαµ ) + p α βνλ R α βνλµ + p α i R i λµ (5.454)where T M is the SEM–tensor for matter.One can verify that the SEM–tensor T s meets the condition(∂ µ + Γ αβ µ∂ αβ ) ˜H HE = ddx α T s α µ, (5.455)so that on solutions (5.450), the curvature of the connection (5.442) vanishes.Hence, the identity (5.454) is reduced to the conservation law (5.421)of matter in the presence of a background metric. The gravitation SEM–tensor is eliminated from the conservation law because the Hamiltonianform H HE is affine in all canonical momenta. Note that only gauge–typeconditions (5.444), (5.445) and the motion equations of matter have beenused.At the same time, since the canonical momenta p αβ α of the world metricare equal to zero, the Hamiltonian equation (5.446) on the Lagrangianconstraint space becomes∂ αβ ( ˜H HE + ˜H M ) = 0.Hence, the equality (5.454) takes the formπ βνλ α ∂ µ R α βνλ+(∂ µ +Γ i µ∂ i −∂ i Γ j µp α j ∂λ) i ˜H M ≈ddx α (T s α λµ+T M µ )+p α i Rλµ.i(5.456)This is the form of the energy–momentum conservation law which we observealso in case of quadratic Lagrangian densities of affine–metric gravity.Substituting the equality (5.455) into (5.456), we get the above result.As a test case of quadratic Lagrangian densities of affine–metric gravity,let us examine the sumL = (− 12κ gβλ F α βαλ + 1 4ε g αγg βσ g νµ g λε F α βνλF γ σµε) √ −gω (5.457)