Lunar and Terrestrial Time Transformation Based on the Principle of General Relativity

TL;DR

The paper argues that a physically consistent lunar time standard requires a three-level, locality-aware relativistic framework rather than a two-level Earth-centric approach. It introduces a hierarchical decomposition into wide-area external and local internal problems, and adopts a Frenet frame as the non-rotating local reference to preserve conservation laws. The authors derive a hierarchical chain of coordinate-time transformations (TCB → TCC → TCG and TCB → TCC → TCL) with slowly varying coefficients L_CC and L_GL, enabling an independent lunar standard time TCL tied to Moon-based atomic clocks and Earth–Moon time comparisons. Operationally, they propose realizing SI seconds on both bodies, exchanging proper times, and iteratively refining the L_GL drift to maintain a synchronized lunar calendar, with the long-term drift of L_GL measurable via precise time comparisons. This framework aims to provide a conceptually clear, computationally tractable route to a universal lunar time standard and to quantify secular variations via Earth–Moon time metrology.

Abstract

Lunar time metrology necessitates a unified temporal framework beyond Earth, requiring an independent lunar system for timekeeping, dissemination, and calendrics. Recent American publications define Lunar Coordinate Time (LTC) within relativity and propose a Terrestrial Time (TT) to LTC conversion formula. However, this formula's derivation and assumptions are contested. The complex dynamics within the solar system can be simplified by decomposing relationships into hierarchical wide-area (external problem) and local-area (internal problem) levels. Grounded in the symmetry and conservation laws of physics, Einstein's general relativity emphasizes two key principles: (i) Equal weighting: Relationships among multi-level coordinate systems are independent and self-similar (analogous to fractals). (ii) *Locality*: The laws of physics retain invariant forms only in local coordinate systems. Specifically, a non-rotating system corresponds to the Frenet frame along a particle's geodesic. Preserving physical law invariance requires restricting rotating references strictly to the local domain; defining the orientation of an Earth-centered system using distant celestial bodies violates general relativity's locality principle. This work derives the relationship between coordinate time and proper time. Using the Earth-Moon system as an intermediary, it obtains a simplified transformation formula between LTC and TT. An independent and universal lunar standard time framework is proposed. Crucially, the derived coordinate time transformation coefficient exhibits long-term secular variation. This variation can be measured and predicted through precise Earth-Moon time comparisons.

Figures (3)

• Figure 1: In the solar system, the Earth and the Moon orbit around the barycenter they form, resulting in periodic changes in the gravitational potential exerted by the Sun.
• Figure 2: The idea of equal rights in general relativity principle and self-similarity in fractal geometry.
• Figure 3: Schematic representation of the Ferret frame of the curve.