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 $T_{}^{\mu \nu }{}_{;\nu }{}^{}=0\mathrm{}$ 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 $\theta _{\mu }^{}{}_{}{}^{\nu }{}_{,\nu }{}^{}{}_{}{}^{}=0\mathrm{}$ (where $\theta _{\mu }^{}{}_{}{}^{\nu }$ reduces to $T_{\mu }^{}{}_{}{}^{\nu }$ in the absence of gravity). (iii) $\theta _{\mu }^{}{}_{}{}^{\nu }$ is always expressible in terms of a superpotential, $\theta _{\mu }^{}{}_{}{}^{\nu }=\Lambda _{\mu }^{}{}_{}{}^{\left[\nu \alpha \right]}{}_{,\alpha }{}^{}{}_{}{}^{}$, If the superpotential $\Lambda _{\mu }^{}{}_{}{}^{\left[\nu \alpha \right]}$ can be expressed in terms of asymptotic values of field quantities, then the conserved integral $P_{\mu }=\int \theta _{\mu }^{}{}_{}{}^{\nu }{d}^3{\Sigma }_{\nu }$ 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 $P_{\mu }$ if and only if the parameters obey five specific constraints; two additional constraints are needed for the existence of a conserved angular momentum $J_{\mu \nu }$ (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" $\theta _{\mu }^{}{}_{}{}^{\nu }$ for all currently viable metric theories known to us.

• Received 30 January 1974

DOI:https://doi.org/10.1103/PhysRevD.10.1685