Conservation laws and variational principles in metric theories of gravity

Abstract

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.