A Generally 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 this paper, we develop a first-order generally covariant divergence law for energy--momentum current density in an open system without modifying the Einstein field equations. It models heat and viscous effects through an external four-force density instead of an internal properties of the fluid. It does not require an equation of state, and retains the system's hyperbolicity and causality for any arbitrarily chosen frame in spacetime. Consequently, this approach obviates both the second-order dissipative relativistic fluid theory's thermodynamic relaxation times and density frames. It reveals a 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 exotic matter or modified Einstein field equations to model it. This paper provides four testable predictions: (i) externally supplied power density exhibits a nonlinear observer dependence in General Relativity, (ii) a generally nonzero externally supplied power density arises in calorimetry for non-geodesic observer fields, even within an otherwise locally isolated system, (iii) a fluid's nonzero internal pressure inherently induces a nonzero divergence of energy--momentum current density, and (iv) counter-divergent energy--momentum current density induces a dark-energy-like gravitational regime.