Lunar time in general relativity: Kopeikin and Kaplan

What it proposes

1. A general-relativistic framework for Lunar Coordinate Time (TCL), rather than a rule of thumb for converting one clock reading into another.
2. Transformation equations for the rate of TCL against TCG at different locations on the lunar surface, covering both secular and periodic variations.
3. A model that carries time dilation from the Moon's orbital motion, the gravitational potentials of Earth and the Moon, and tidal perturbations from the Sun and the planets.
4. A framework anchored in the IAU 2000 resolutions, so that a lunar coordinate time is derived the way the terrestrial and barycentric ones already were.

This page restates what the paper carries. It does not grade the argument, and every model figure on it is cited to the work it comes from.

The subject is a framework for TCL, derived inside general relativity and grounded in the IAU resolutions of 2000 ^[1]. A framework is not a table of numbers: it is the set of equations from which the numbers for any particular clock follow.

What the equations deliver

The paper gives transformation equations for the rate of TCL against TCG, and it gives them for various locations on the lunar surface rather than for one nominal site ^[1]. From those equations both secular and periodic variations can be computed for the rate of an atomic clock placed on the Moon, relative to an identical clock on Earth ^[1].

Three groups of effect are carried in the model. Time dilation from the Moon’s orbital motion, the gravitational potentials of Earth and of the Moon, and tidal perturbations from the Sun and the planets ^[1]. Each of them moves the rate, and leaving any of them out changes the answer.

How far the framework reaches

The authors state a working range as well as a precision. The framework holds at the nanosecond level within Earth’s Hill sphere, the region around Earth in which Earth’s gravity dominates, given in the paper as about 1.5 million kilometres ^[1]. That covers the Moon, lunar orbit and the space in between.

They also state how the result was checked. It is validated by equivalence with the mathematics of a local inertial frame for the Earth-Moon system, which is a consistency test rather than a measurement ^[1].

Why the framework has to come first

Coordinate time is not read by any clock. It is a label attached to events by a chosen convention, and the convention has to be stated before any two clocks can be compared. The International Astronomical Union made that point directly in 2024, when it asked international organizations to agree the relations between a lunar reference time scale, a lunar coordinate time and UTC ^[3]Official.

Building on the 2000 resolutions matters for the same reason. Barycentric and geocentric coordinate times were defined there, and deriving a lunar one in the same framework keeps all three comparable instead of leaving a conversion to be invented later ^[1].

What it does not settle

A framework is not a scale. The IAU resolution of 2024 fixes the epoch of TCL, but it does not say which clocks realize anything or how often ^[2]Official. The rate figures that a realization needs were published separately, and are read in the lunar reference timescale paper ^[4].

Nor does a framework convert a timestamp on its own. Software that computes the transformation between TCL and the barycentric scales numerically, to a stated accuracy better than 0.15 nanoseconds before 2050, is read in the LTE440 lunar time ephemeris ^[5].

The proposal to read clock comparison itself as a measurement is in frequency differences between Earth and Moon clocks, and a later preprint titled simply Lunar Time ^[6] is covered in Lunar Time by Defraigne, Meynadier and Bourgoin. The rest of our readings are in the papers hub.

Key numbers

Every figure below is a model value taken from the source named in its own row. Nothing in this table is our own estimate.

Status

Peer-reviewed. Physical Review D 110, 084047, 5 July 2024^[1]. arXiv 2407.04862. The full text is at Lunar Time in General Relativity.

Why it matters

A coordinate time is a convention, and a lunar one is only usable if it is derived in the same relativistic framework as the scales it has to be compared with. This paper does that derivation for TCL, anchors it in resolutions that already exist, and states where its precision holds.