A Manifestly Covariant Divergence Law for Energy--Momentum Current Density in an Open System: On the Modified Strong Equivalence Principle, Active Gravitational Mass Density, and Dark Energy

In standard general relativity, the identity $\nabla_{\mu}(T^{\mu\nu}U_{\nu}) = \mathcal{F}^\nu U_\nu + T^{\mu\nu}\left(\nabla_{\mu}U_{\nu}\right)$ is a tautology. As a resolution to this issue, in this paper we have developed a first-order manifestly covariant divergence law for energy--momentum current density in an open system using the Herglotz, Euler framework, a modified Cauchy momentum balance law in Eulerian form and invariant scalars. It models heat and viscous effects through an external four-force density instead of an internal properties of the fluid. It retains the system's hyperbolicity and causality for any arbitrarily chosen frame in spacetime. It reveals a \emph{fundamental symmetry}: the observer field determines how the energy--momentum current density is measured, while the energy--momentum density exchange determines how that measured current density diverges over spacetime. This same divergence is found to exhibit a material-dependent thermodynamic signature, and therefore we modify the strong equivalence principle to account for this missing signature. This formulation redefines the active gravitational mass density as a dynamic susceptibility that allows ordinary matter to effectively produce a dark-energy-like gravitational regime, obviating the need for modified Einstein field equations or ad-hoc dark energy vacuum fluid to model it. This paper predicts four testable phenomenons.