Lunar time in general relativity

We introduce the general-relativistic definition of lunar coordinate time (TCL) based on the International Astronomical Union 2000 resolutions that provide a framework for relativistic reference systems. From this foundation, we derive a transformation equation that describes the relative rate of TCL with respect to geocentric coordinate time (TCG) for various locations of a clock on the lunar surface. This equation serves as the cornerstone for constructing a relativistic TCL-TCG time conversion algorithm. Using this algorithm, we can compute both secular and periodic variations in the rate of an atomic clock placed on the Moon, relative to an identical clock on the Earth. The algorithm accounts for various effects, including time dilation caused by the Moon's orbital motion around Earth, gravitational potentials of both Earth and Moon, and direct and indirect time dilation effects due to tidal perturbations caused by the Sun and other major planets of the Solar System. Our approach provides exquisite details of the TCL-TCG transformation, achieving a precision of several nanoseconds within the spatial volume dominated by the Earth's gravitational field known as the Hill sphere. This sphere extends from the Earth to a distance of approximately $1.5\ifmmode\times\else\texttimes\fi{}{10}^{6}\text{ }\text{ }\mathrm{km}$, substantially encompassing the Moon's orbit. To validate our methodology for lunar coordinate time, we compare it with the mathematical formalism of local inertial frames applied to the Earth-Moon system, and confirm their equivalence.