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 $$3n\rho _{,nn}-\rho _{,n}=0$$ 3 n ρ , n n - ρ , n = 0 , whose non-Einstein solutions are $$\rho (n)=\rho _\Lambda +C n^{4/3}$$ ρ ( n ) = ρ Λ + C n 4 / 3 . For a minimally coupled first-derivative scalar field with the same field and kinetic variable, preservation of the local $$\mathcal {L}_\phi (X,\phi )$$ L ϕ ( X , ϕ ) class gives $$X\mathcal {L}_{\phi ,XX}-\mathcal {L}_{\phi ,X}=0$$ X L ϕ , X X - L ϕ , X = 0 , and hence $$\mathcal {L}_\phi =\frac{1}{2}\mathcal {A}(\phi )X^2+\mathcal {B}(\phi )$$ L ϕ = 1 2 A ( ϕ ) X 2 + B ( ϕ ) . 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, $$T_{\mu \nu }$$ T μ ν denotes the physical energy–momentum tensor of the specified matter model. Its Einstein-form algebraic image is $$\Theta _{\mu \nu }[T]=T_{\mu \nu }-\alpha g_{\mu \nu }T$$ Θ μ ν [ T ] = T μ ν - α g