Time transformation between the solar system barycenter and the surfaces of the Earth and Moon

The transformation of time between the surface of the Earth, the solar system barycenter, and the surface of the Moon involves relativistic corrections. For solar system Barycentric Dynamical Time (TDB), we also require that there be no rate difference between Terrestrial Time (TT) and TDB. The IAU has addressed these transformations with several resolutions. A series of robotic and crewed landings on the Moon are planned. The analogous transformation between TDB and time on the surface of the Moon (TL) needs a review and discussion. In this paper, we compute the rate terms involved in that transformation. We also present the TDB-compatible spatial scale and Lorentz contraction of Moon-centered positional coordinates. These transformations have been implemented in the JPL programs used to generate ephemerides of the Moon and planets. Finally, we provide expressions that can be used to synchronize TT and TL using either TDB or TT. The relevant transformations contain a small secular drift between the two time scales, along with additional small periodic terms that can be numerically evaluated using the solar system ephemerides.