Emergent Time: A Causal-Machian Time Field Coupled to Matter

A causal-Machian model of time is formulated in which physical proper time is defined by the matter and energy in the causally relevant past of an event. The retarded horizon momentum of this matter selects a preferred rest frame $U_{H}^{\mu}(x)$. In that frame the maximum proper time is accumulated. A bounded scalar field Θ(x)∈[0,1] describes the degree to which temporal structure is realised: Θ = 1 gives the General Relativity (GR) limit, 0 < Θ <1 gives reduced temporal structure, and $\Theta=0$ corresponds to a completely matter-free causal domain in which no physical proper time is defined. The physical clock metric modifies only the time direction selected by $U_{H}^{\mu}$; it is not a conformal rescaling of space and time. Overdense regions therefore cannot make clocks run faster than the GR limit. All physical fields, including massless ones, propagate on the clock metric $\hat{g}_{\mu\nu}$; the resulting speed bound $v_{H}\leq c\sqrt{\Theta}$ is a prediction of the theory. The model provides a preferred frame for compact spatial topologies and thereby fixes the inertial twin comparison in such spaces. GR is recovered as a limiting case whenever $\Theta=1$ and $\nabla\Theta=0$; in that sector the standard energy-momentum conservation law $\nabla^{\mu}T_{\mu\nu}=0$ is recovered. The present paper establishes the kinematic and constraint structure of the theory; a complete variational closure and the regularisation of sharp causal boundaries are identified as open tasks.