A Relativistic Framework to Estimate Clock Rates on the Moon

Abstract As humanity aspires to explore the solar system and investigate distant worlds such as the Moon, Mars, and beyond, there is a growing need to estimate and model the rate of clocks on these celestial bodies and compare them with the rate of standard clocks on Earth. According to Einstein’s theory of relativity, the rate of a standard clock is influenced by the gravitational potential at its location and its relative motion. A convenient choice of local reference frames allows for the comparison of local time variations of clocks due to gravitational and kinematic effects. We estimate the rate of clocks on the Moon using a locally freely falling reference frame coincident with the center of mass of the Earth–Moon system. A clock near the Moon’s selenoid ticks faster than one near the Earth’s geoid, accumulating an extra 56.02 μ s day −1 over the duration of a lunar orbit. This formalism is then used to compute the clock rates at Earth–Moon Lagrange points. Accurate estimation of the rate differences of coordinate times across celestial bodies and their intercomparisons using clocks on board orbiters at Lagrange points as time transfer links is crucial for establishing reliable communications infrastructure. This understanding also underpins precise navigation in cislunar space and on celestial bodies’ surfaces, thus playing a pivotal role in ensuring the interoperability of various position, navigation, and timing systems spanning from Earth to the Moon and to the farthest regions of the inner solar system.