Time Field Theory: Gravitational Waves — Monochromatic Solutions, Doppler Redshift, Geometric Optics, and Energy–Momentum Tensor

This work investigates the fundamental wave properties of gravitational radiation within the framework of Time Field Theory (TFT). Starting from the first-principles scalar wave equation of TFT, we derive the monochromatic plane-wave solutions, analyze the gravitational wave Doppler redshift, establish the geometric optics approximation in the short-wavelength limit, and construct the canonical energy–momentum tensor for the time-field. The results confirm that TFT gravitational waves propagate at the speed of light, obey standard relativistic frequency shifts, follow ray-like trajectories in the high-frequency regime, and carry positive-definite, conserved energy. These results establish the consistency and physical soundness of gravitational waves in Time Field Theory.