Frequency differences between clocks on Earth and the Moon
Key facts
- The preprint models the fractional frequency difference between Earth and lunar clocks through four time transformations. [1]
- The gravity potential difference between the two clocks affects the frequency difference at the 10⁻¹⁰ level, and the coordinate time ratio at the 10⁻¹¹ level. [1]
- On a single link the first-order Doppler term dominates at the 10⁻⁶ level and masks both of those terms. [1]
- The authors conclude that a Doppler-cancelling multi-link strategy is needed to extract the terms of interest. [1]
- It was first posted on 19 June 2025, and version 2 was revised on 12 June 2026. [1]
What it proposes
- A comprehensive model of the fractional frequency difference between a clock on Earth and a clock on the Moon, worked out for the pair of bodies rather than for one clock at a time.
- That model in four steps: proper time to coordinate time at each clock, the conversion between the Earth and Moon coordinate times, and the propagation of the signal between them.
- That such a comparison serves lunar geodesy, the study of the Moon's gravity field and shape, and not only timekeeping.
- A Doppler-cancelling multi-link strategy, because on a single link the first-order Doppler term hides the terms the comparison is after.
This page restates the model. Two rows of the table below are not this paper’s numbers and are marked with the work they come from: they are published model figures the comparison has to reproduce.
The starting point is an application, not a definition. Clock frequency comparisons already support geodesy on Earth, where they help determine the gravitational potential and realize a height system, and the paper extends that idea to the Moon [1].
Why a clock comparison is also a measurement
A clock keeps proper time — the time along its own path. Two clocks at rest in different gravitational potentials therefore run at different rates, which is gravitational time dilation. The size of the difference depends on the potential at each clock, so measuring the difference is a way of measuring the potential [2]Official.
On the Moon the potential at the surface is described by the selenoid, and its monopole term has a published value of 3.14 × 10⁻¹¹ [3]. A clock link accurate enough to see terms of that size would say something about the lunar gravity field, which is what makes the comparison geodetic as well as metrological [1].
The comparison in four steps
The model chains four transformations. Proper time to coordinate time for the Earth clock, and the same for the lunar clock, each tied to the local gravity potential; then the conversion between the Earth and Moon coordinate times; then the propagation of the time signal between the two clocks [1].
Inside those steps the authors evaluate the contributions of static, tidal and non-tidal potentials, of the self-rotation of each body, and of the different celestial bodies involved. The propagation step is broken down into Doppler, atmospheric and Shapiro delay effects [1].
Which terms are large and which are hidden
The sizes are what make the paper useful. The difference in gravity potential between the two clocks affects the frequency difference at the 10⁻¹⁰ level, and the coordinate time ratio contributes at the 10⁻¹¹ level [1].
The obstacle is larger than either. On a single link the first-order Doppler term dominates at the 10⁻⁶ level, and it masks both the gravity-potential and the coordinate-time terms [1]. The authors’ answer is a Doppler-cancelling multi-link strategy, which suppresses the propagation effect so the smaller terms can be extracted [1].
Where it stands next to the other work
This preprint and the reference timescale paper do not disagree. They answer different questions: one publishes the rate relations a scale needs, and this one asks what a measured rate difference tells us about the body underneath the clock [3]. The rate figures themselves are read in the lunar reference timescale paper, and the framework the coordinate times come from is read in lunar time in general relativity.
Fractional frequencies also turn up in the realization work, describing something else. The orbital clock proposal reports an offset of 6 × 10⁻¹⁵ for its simulated clock, which measures how closely that clock would track a chosen lunar reference rather than what lies under it [4]; it is read in two birds with one stone. The coordinate time conversion this model needs in step three is computed by the software read in the LTE440 lunar time ephemeris [5]. Every reading we publish is listed 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.
| Quantity | Value | Source |
|---|---|---|
| Effect of the gravity potential difference between the two clocks | 10⁻¹⁰ level | [1] |
| Effect of the coordinate time ratio | 10⁻¹¹ level | [1] |
| First-order Doppler term on a single link | 10⁻⁶ level | [1] |
| Time transformations the model chains together | 4 | [1] |
| Selenoid clock against geoid clock, Ashby and Patla | 56.0199(12) µs/day | [6] |
| Lunar monopole at the selenoid, Bourgoin and colleagues | 3.14 × 10⁻¹¹ | [3] |
Status
Preprint. arXiv preprint, 19 June 2025[1]. arXiv 2506.16377. The full text is at Frequency Differences between Clocks on the Earth and the Moon.
Why it matters
Comparing clocks is how two bodies' time scales are tied together in practice. This model sizes each term in that comparison, and it shows that the propagation of the signal is a larger effect than everything the comparison is trying to measure.
Sources
- Frequency Differences between Clocks on the Earth and the Moon
- What Time Is It on the Moon?
- Lunar reference timescale
- Two birds with one stone: simultaneous realization of both Lunar Coordinate Time and lunar geoid time by a single orbital clock
- Lunar Time Ephemeris LTE440: definitions, algorithm and performance
- A Relativistic Framework to Estimate Clock Rates on the Moon
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