The article explores the differences in clock rates between Earth and the Moon based on their distinct orbits around a mutual center of mass. It introduces a locally freely-falling inertial frame to facilitate calculations of proper time elapsed on each celestial body. The gravitational potentials and relative velocities are considered, with an acknowledgment that the Sun contributes primarily through tidal effects. Key formulas are developed for comparing proper times on both surfaces, leading to insights about relativistic effects and clock discrepancies due to gravitational influences and positional variations.
To understand how time differs between Earth and Moon clocks, we consider that both bodies move in Keplerian orbits centered on their mutual center of mass, with their time rates influenced by gravitational fields and relative velocities.
Introducing a fictitious locally freely-falling inertial frame at the center of mass, we can calculate proper times for moving clocks on Earth and the Moon while simplifying the effects of tidal forces.
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