This paper studies whether familiar global conservation laws — for total mass, momentum and angular momentum — can still be defined when compact astrophysical bodies (like neutron stars) have effective masses that depend on the local gravitational environment. The authors work inside the parameterized post-Newtonian (PPN) framework, which is the standard, theory‑independent toolkit for describing weak gravitational fields and for testing deviations from Einstein’s general relativity. They show that conserved quantities can still exist in the presence of such “sensitivities,” and they give explicit formulas and conditions for when this happens.

A bit of background helps to explain the problem. The PPN approach expands the metric of spacetime as a weak perturbation on flat space and expresses results in terms of a set of gravitational potentials (for example the Newtonian potential U and several post‑Newtonian potentials labeled Φi). In some alternative gravity theories, the strong equivalence principle (SEP) is violated, and a compact object’s effective mass can depend on the local values of those gravitational fields. This dependence is encoded by sensitivity parameters such as sU = ∂ ln ρ / ∂U and similar derivatives with respect to other potentials. When such sensitivities are present, the effective energy‑momentum tensor used to model the distant weak field is not covariantly conserved in the usual way.

What the researchers did was extend the usual classification of gravitational theories — fully conservative, semi‑conservative, or non‑conservative — to include these theory‑independent sensitivities. They derive conservation equations at leading and higher post‑Newtonian orders, identify the specific conditions under which a conserved tensor τµν (with ∂ντµν = 0 in ordinary coordinates) can be constructed, and write explicit expressions for the resulting global conserved integrals Pµ and Jµν. They also show new ways to organize the coefficients that appear in the PPN metric when conserved quantities exist, and they check their formalism against known results from scalar‑tensor gravity to demonstrate consistency.