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 ${\mathcal{L}}_{\varphi }(X,\varphi )$ class gives $X{\mathcal{L}}_{\varphi ,XX}-{\mathcal{L}}_{\varphi ,X}=0$, and hence ${\mathcal{L}}_{\varphi }=\frac{1}{2}\mathcal{A}(\varphi )X^2+\mathcal{B}(\varphi )$. 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.

arXiv comment29 pages, 1 figure