A Minimal Scalar-Tensor Gravity Model with Time-Density Scalar Field and Mass-Time Equivalence: Variational Formulation, Cosmological Implications and Emergent Spacetime

We propose a minimal scalar-tensor gravity model in which a time-density scalar field ρ_t is introduced, leading to a natural mass-time equivalence relation m = λ t (where λ is a scaling constant with units kg·s⁻¹). The core equivalence m = t = F/a = w/g = E/c² = F·s/c² is derived variationally and shown to be fully consistent with the Einstein equivalence principle in the lowdensity limit. The effective action yields modified Einstein equations and a Brans-Dicke-like scalar-tensor theory with effective gravitational constant G_eff = G₀ ρ_t^α (α = 1 for the minimal case). High-precision numerical simulations confirm internal consistency with error < 10^{-12}. Cosmological solutions in the FLRW metric demonstrate that the time-density field can mimic late-time acceleration (dark energy analog) and provide an emergent-time description of spacetime. The model is compatible with recent equivalences between scalar-tensor and scale-dependent gravity theories and offers falsifiable predictions for gravitationalwave propagation and cosmological redshift. This framework extends classical general relativity into a new class of effective theories with potential applications in quantum gravity and emergent spacetime studies.