Conservation laws and variational principles in metric theories of gravity

Conservation of energy, momentum, and angular momentum in metric theories of gravity is studied extensively both in Lagrangian formulations (using generalized Bianchi identities) and in the post-Newtonian limit of general metric theories. Our most important results are the following: (i) The matter response equations T_^μν_;ν^=0 of any Lagrangian-based, generally covariant metric theory (LBGCM theory) are a consequence of the gravitational-field equations if and only if the theory contains no absolute variables. (ii) Almost all LBGCM theories possess conservation laws of the form θ_μ^_^ν_,ν^_^=0 (where θ_μ^_^ν reduces to T_μ^_^ν in the absence of gravity). (iii) θ_μ^_^ν is always expressible in terms of a superpotential, θ_μ^_^ν=Λ_μ^_^[να]_,α^_^, If the superpotential Λ_μ^_^[να] can be expressed in terms of asymptotic values of field quantities, then the conserved integral P_μ=∫θ_μ^_^νd³Σ_ν can be measured by experiments confined to the asymptotically flat region outside the source. (iv) In the Will-Nordtvedt ten-parameter post-Newtonian (PPN) formalism there exists a conserved P_μ if and only if the parameters obey five specific constraints; two additional constraints are needed for the existence of a conserved angular momentum J_μν (This modifies and extends a previous result due to Will.) (v) We conjecture that for metric theories of gravity, the conservation of energy-momentum is equivalent to the existence of a Lagrangian formulation; and using the PPN formalism, we prove the post-Newtonian limit of this conjecture. (vi) We present "stress-energy-momentum complexes" θ_μ^_^ν for all currently viable metric theories known to us.