Geometrizing the Speed of Light via a Temporal Scalar Field

This work presents a theoretical derivation of the speed of light squared, , directly from a fundamental temporal field , without relying on empirical assumptions or external postulates. The derivation starts from a generalized action principle in a 4-dimensional spacetime, where the Lagrangian density is constructed solely from the gradient of the temporal field. The kinetic term naturally introduces a Lorentzian structure, and through variation of the action, a wave equation for emerges. By analyzing the energy-momentum tensor associated with this field, and requiring consistency with the relativistic invariant interval, the temporal field formulation yields a geometric constraint equivalent to setting the proper time flow to unity. This leads to the emergence of a natural coefficient with units of , identified as , purely from the structure of the field. This derivation avoids circular reasoning by not assuming any prior knowledge of light propagation or Maxwellian dynamics. It instead reconstructs as a fundamental consequence of the causal structure imposed by the temporal field. This result opens the path to reconstructing relativity and quantum field theory from a common time-based foundation.