High-precision Relativistic Timescales for Cislunar Navigation

Abstract We present a unified post-Newtonian framework for relativistic timing and coordinate transformations across the Barycentric, Geocentric, and Lunicentric Celestial Reference Systems ( LCRS ) and six timescales: Barycentric Coordinate Time ( TCB) , Geocentric Coordinate Time (TCG), Terrestrial Time (TT), Barycentric Dynamical Time ( TDB ), Lunicentric Coordinate Time ( TCL )​​​, and Lunar Time ( TL ). Following IAU Resolution II (2024), we implement an LCRS metric with time coordinate TCL , linked to TCB via the IAU B1.5 prescription with lunar quantities. We truncate all series to retain contributions above a fractional threshold of 5 × 10 −18 and timing terms exceeding 0.1 ps, expanding the lunar gravity to degree ℓ = 9 with Love number variations and including external tidal and inertial multipoles through the octupole. We derive closed-form mappings among TCB , TCG , TT , TDB , TCL, and TL, giving proper-to-coordinate time conversions and one-/two-way time-transfer corrections at subpicosecond accuracy. Secular rates and periodic perturbations from kinematic dilation, lunar monopole and multipoles, Earth tides, and gravitomagnetism are evaluated for clocks on the lunar surface in Very Low and Low Lunar Orbits, in elliptical lunar frozen orbits, at the Earth–Moon L1 point, and in Near-rectilinear Halo Orbits ( NRHOs ). Harmonics through ℓ = 9 and tides through ℓ = 8 suffice to meet 5 × 10 −18 stability in deep cislunar regimes (e.g., NRHO, L1), supporting subpicosend synchronization and centimeter-level navigation, whereas near-surface and Very Low Lunar Orbits require much higher degree (typically <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≳</mml:mo> <mml:mn>300</mml:mn> </mml:math> ). The framework implements the IAU LCRS/TCL prescription and supplies accuracy budgets for high-precision time/frequency transfer, relativistic geodesy, quantum links, and fundamental tests beyond low Earth orbit.