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

Limits of the Rastall–Einstein Equivalence: Matter-Action Compatibility, FLRW Dynamics, and Exceptional Sectors
Jos’e A. C. Nogales, Karen-Luz Burgoa Rosso, Marcelo H. Alvarenga
arXiv:2606.09819v3 Announce Type: replace-cross
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 $3nrho_{,nn}-rho_{,n}=0$, 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 $Xmathcal L_{phi,XX}-mathcal L_{phi,X}=0$, and hence $mathcal L_phi=frac12mathcal A(phi)X^2+mathcal B(phi)$. 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, $T_{munu}$ denotes the physical energy–momentum tensor of the specified matter model. Its Einstein-form algebraic image is $Theta_{munu}[T]=T_{munu}-alpha g_{munu}T$. 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:2606.09819v3 Announce Type: replace-cross
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 $3nrho_{,nn}-rho_{,n}=0$, 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 $Xmathcal L_{phi,XX}-mathcal L_{phi,X}=0$, and hence $mathcal L_phi=frac12mathcal A(phi)X^2+mathcal B(phi)$. 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, $T_{munu}$ denotes the physical energy–momentum tensor of the specified matter model. Its Einstein-form algebraic image is $Theta_{munu}[T]=T_{munu}-alpha g_{munu}T$. 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.