Time-Field Theory: Unified Derivation of Gravitational Redshift and Cosmological Redshift from Conformal Energy Conservation

Based on the core framework of Time-Field Theory (TFT) with flat spacetime and spacetime flux conservation, this paper rigorously derives the scaling relation between local photon frequency and the time-field strength, and establishes the conformal energy conservation theorem as the fundamental invariant of photon propagation. On this basis, we provide a unified physical explanation for both gravitational redshift and cosmological redshift. We prove that the conformal energy of photons is strictly conserved during propagation, independent of the spatial distribution and temporal evolution of the time field $\gamma(r,t)$. All observed redshift phenomena, whether gravitational or cosmological, are pure measurement effects arising from differences in local time benchmarks between the emitter and the observer. The physical mechanism is identical in both cases: the intrinsic conformal energy of the photon never changes; only observers at different positions or different cosmic epochs, using clocks that run at different rates, measure different frequencies for the same photon. This unified interpretation requires no spacetime curvature, no spatial expansion, and no additional entities beyond the scalar time field. ## Core Results 1. **Conformal Energy Conservation Theorem**: The conformal energy $E_{\text{conf}} = h\nu\gamma$ of electromagnetic radiation is an absolute invariant during propagation, independent of the spatial distribution and temporal evolution of the time field. 2. **Unified Redshift Mechanism**: Both gravitational redshift and cosmological redshift are pure measurement effects — the intrinsic conformal energy never changes; only the local time benchmarks of observers differ, leading to different measured frequencies. 3. **Conceptual Economy**: TFT explains both phenomena with a single scalar field and a single conservation law, replacing the two entirely different mechanisms (spacetime curvature and spatial expansion) required by standard theories. 4. **Cosmological Evolution**: The global time field $\gamma(t)$ increases monotonically and accelerates with cosmic time. Cosmological redshift is observational evidence of the speeding up of the global clock rate, not of spatial expansion. ## Theoretical Alignment This work is fully self-consistent with the metric system of the TFT foundational monograph (Version 2.1). All derivations strictly follow the core axioms of local intrinsic total velocity conservation and spacetime flux conservation, with no additional ad-hoc assumptions.