Flat-background scalar-force theory of gravity ($d{p}^i={\partial }^i\Phi p_{j}d{x}^j-(p^j{\partial }_j\Phi )d{x}^i$)

This covariant scalar-force model uses the Minkowski metric and the displayed momentum law, with $\square \Phi =-4\pi G{T}^i{}_i$. For a massive test particle, write $p^i=m{u}^i$, $u^i{u}_i=1$. Contracting the force law with $p_i$ gives constant rest mass; the acceleration is $d{u}^i/d\tau ={\partial }^i\Phi -u^i(u^j{\partial }_j\Phi )$, independent of the mass. A static weak field obeys the Poisson equation and yields Newtonian acceleration $-\nabla \Phi$. For a photon, $p\cdot dx=0$ and $dx$ is parallel to $p$, so $d{p}^i=-p^{i}d\Phi$: photon energies change but their unparametrized paths remain straight. The motion law has the same unparametrized curves as the conformal metric $e^{2\Phi }\eta$, but the field equation must not be silently identified with a different scalar-gravity theory.