Emergent Relativistic Geometry from Time–Scalar Field Theory

We present a derivation of relativistic gravitational geometry emerging from Time–Scalar Field Theory (TSFT), formulated relative to a pre-geometric Lorentzian scaffold used to define propagation dynamics. The physically relevant spacetime geometry arises from scalar-field fluctuations through an emergent effective metric governing perturbation propagation. In this paper, we construct a non-circular derivation of relativistic geometry from scalar temporal dynamics. Beginning from a pre-geometric scalar field defined on an auxiliary background, we derive the propagation structure of temporal fluctuations and demonstrate that an effective spacetime metric emerges from the resulting wave dynamics. This emergent metric reduces to Minkowski spacetime in the uniform-field limit, recovering Special Relativity. We then analyze curvature properties of the emergent geometry and show that weak-field gravitational behavior consistent with General Relativity arises naturally from temporal field gradients. The Newtonian gravitational limit is recovered, and consistency with relativistic propagation effects is demonstrated.The resulting framework reproduces the weak-field predictions of General Relativity and admits a nonlinear gravitational closure consistent with the Einstein field equations. These results establish that relativistic spacetime geometry can emerge from scalar temporal dynamics, providing a unified interpretation of inertial motion, gravitational curvature, and relativistic behavior within Time–Scalar Field Theory.