Emergent Gravity from Temporal Field Geometry: Deriving Einstein's Equations from Scalar Time Dynamics

This paper presents a derivation of Einstein’s field equations from a scalar temporal field , without assuming spacetime curvature a priori. By constructing a covariant Lagrangian based on the gradient and dynamics of the temporal field, we recover the Einstein tensor and show that gravity arises naturally from the geometry of time itself. The resulting equations match general relativity in all tested regimes, including planetary precession, light deflection, gravitational time dilation, and black hole energy density. This work establishes temporal geometry as a foundational mechanism behind gravitation and connects spacetime dynamics to internal time structure.