Conservation laws and variational principles in metric theories of gravity

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

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