Advanced Analytical Formalism and Energy-Momentum Tensor Bridge for Effective Gravitational Fields in the Dynamic Substrate Theory

Abstract General Relativity treats the stress-energy tensor T^{\mu\nu} as the geometric source of spacetime curvature via Einstein's field equations. However, within the Dynamic Substrate Theory (DST), spacetime is not a fundamental pseudo-Riemannian manifold, but an emergent macroscopic hydrodynamic manifestation of a Maxwell-Oldroyd-B viscoelastic vacuum. To establish rigorous predictive power, we construct an explicit analytical bridge mapping internal stress, strain rates, and momentum fluxes of the cosmic substrate to effective gravitational potentials and curvature analogues. Through variational formulation and tensor decomposition, we derive the effective field equations where viscoelastic extra-stress terms act as active physical source terms, successfully grounding gravitational sources in explicit continuum mechanics. Keywords: Energy-Momentum Tensor Bridge, Maxwell-Oldroyd-B Vacuum, Viscoelastic Continuum, Effective Gravitational Fields, Dynamic Substrate Theory.