Deriving Entangled Relativity

Entangled Relativity is a non-linear reformulation of Einstein's theory that cannot be defined in the absence of matter fields. It recovers General Relativity without a cosmological constant in the weak matter density limit or whenever $\Lm = T$ on-shell, and it is also more parsimonious in terms of fundamental constants and units. In this paper, we show that Entangled Relativity can be derived from a general $f(R,\Lm)$ theory by imposing a single requirement: the theory must admit all solutions of General Relativity without a cosmological constant whenever $\Lm = T \neq 0$ on-shell, though not necessarily only those solutions. An important consequence is that all vacuum solutions of General Relativity without a cosmological constant are limits of solutions of Entangled Relativity when the matter fields tend to zero. In addition, we introduce a broader class of theories featuring an \textit{intrinsic decoupling}, which, however, do not generally admit the solutions of General Relativity.