A Herglotz-based Manifestly Covariant Divergence Law: Emergent Gravity and Kinematic Criteria for Cosmic Phases

From first principles, we develop a first-order, manifestly covariant divergence law for the ideal fluid stress–energy tensor contracted with the four-velocity through a bottom-up approach using the Herglotz formulation, the Eulerian framework, and four standard coordinate-invariant ten