Frequency Compensation Analysis of GPS Satellite Atomic Clocks Based on Two-Dimensional Linear Matrix Wave Theory

Global Positioning System (GPS) satellite-borne atomic clocks require preset frequency offsets to synchronize timing with ground stations, and the relative frequency shift adopted in practical engineering is \(4.45 × 10⁻¹⁰\). This paper abandons the spacetime curvature geometric interpretation of general relativity and strictly derives the intrinsic frequency characteristics of satellite atomic clocks solely based on the three core axioms of the two-dimensional linear matrix wave theory. According to the radial distribution function of substrate wave frequency stipulated by the axioms: the closer to a massive celestial body, the higher the concentration of the substrate wave field, and the higher the intrinsic oscillation frequency of local atomic standing waves. The ground near the Earth’s surface is located in a high-concentration wave field, so ground atoms oscillate faster; by contrast, high-altitude satellites stay in a thinner wave field and their native atomic oscillation frequency is lower. When only the gravitational wave concentration gradient is taken into account, the relative frequency deficit of static satellites relative to ground clocks reaches \(5.285 × 10⁻¹⁰\). After superimposing the standing wave dilution effect induced by satellite orbital motion (\(8.349 × 10⁻¹¹\)), the total native relative frequency deficit of satellite atoms against ground references is \(4.450 × 10⁻¹⁰\). To unify the time standard defined by ground atomic oscillation counts (the SI second), the nominal output frequency of satellite oscillators must be artificially reduced for compensation. The required correction magnitude is quantitatively consistent with the measured calibration data of GPS engineering. The whole derivation process introduces no fitting parameters, and provides a complete microscopic wave dynamical mechanism to explain the frequency shifts caused by gravitational potential difference and relative orbital motion.