Limits of the Rastall--Einstein Equivalence: Matter-Action Compatibility, FLRW Dynamics, and Exceptional Sectors

Abstract

A regular Rastall metric equation can be written exactly as an Einstein equation with a conserved algebraic source. We examine whether this source redefinition also preserves a specified local matter-action class. For an isentropic barotropic fluid, keeping the particle-density variable and its conserved current fixed gives the compatibility condition $3n\rho_{,nn}-\rho_{,n}=0$, whose non-Einstein solutions are $\rho(n)=\rho_\Lambda+C n^{4/3}$. For a minimally coupled first-derivative scalar field with the same field and kinetic variable, preservation of the local $\mathcal L_\phi(X,\phi)$ class gives $X\mathcal L_{\phi,XX}-\mathcal L_{\phi,X}=0$, and hence $\mathcal L_\phi=\frac12\mathcal A(\phi)X^2+\mathcal B(\phi)$. These are off-shell functional compatibility tests within restricted continuum action classes; they do not by themselves establish equivalence of full solution spaces or observables. Throughout the analysis, $T_{\mu\nu}$ denotes the physical energy--momentum tensor of the specified matter model. Its Einstein-form algebraic image is $\Theta_{\mu\nu}[T]=T_{\mu\nu}-\alpha g_{\mu\nu}T$. We also complete the dictionary between two Ricci--trace parametrizations, including the coupling, and state a limited on-shell variational obstruction with its exceptions. The analysis includes Poisson-level weak-field matching for an operational rest-mass density, the singular FLRW branch $D(w)=0$, a classification of exceptional parameter sectors, and a comparison of Rastall gravity with standard unimodular gravity and nonconservative trace-free completions. The regular Rastall metric equation is therefore algebraically equivalent to an Einstein equation with a redefined source; equivalence of completed matter--gravity models is a separate, stronger requirement.