Time-Density Field Theory: On the ν Field and a Testable Model of Gravity with Regular Black Hole Cores

We propose a unified framework in which time is described as a physical field associated with the expansion of space. The fundamental quantity is the time-density field ν, representing the local rate of spatial expansion and defining the local flow of time through dt = 1/ν We further relate ν to a gravitational potential Φ through ν = ν₀ exp(Φ / c²) where ν₀ is the vacuum reference value. By introducing the auxiliary scalar field φ = ln ν, we construct a Lagrangian formulation that reproduces Newtonian gravity in the weak-field limit. Gravity arises from spatial gradients of the time-density field, while time dilation naturally emerges from variations in ν.We derive a relation between energy density and the time-density field and introduce an expansion-induced potential together with a density-dependent overlap scale, leading to a nonlinear field equation.The model predicts that gravitational collapse results in a finite core rather than a singularity. In addition, gravitational wave echoes provide a direct observational probe of the internal structure of compact objects. This work provides a physically testable framework connecting time, gravity, and vacuum structure through a single scalar field.