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\u2019s 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\u2013Moon system. A clock near the Moon\u2019s selenoid ticks faster than one near the
Earth\u2019s geoid, accumulating an extra 56.02 \u03bcs day\u22121 over the duration of a lunar orbit. This formalism is then used
to compute the clock rates at Earth\u2013Moon 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\u2019 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.<\/p>\n\n\n\n