⚛️ general relativity

The influence of lunar tidal potential on clock frequencies at different positions on Earth

This paper systematically investigates how lunar tidal potential induces fractional frequency shifts between Earth-based clocks at different positions, revealing that the Moon's longitude and latitude variations distinctly affect the phase and amplitude of these shifts, thereby offering critical insights for clock calibration and synchronization.

Original authors: Hongbin Zhang, Yanyue Gao, Baocheng Zhang

Published 2026-07-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Hongbin Zhang, Yanyue Gao, Baocheng Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine time as a river flowing through space. For decades, we thought this river flowed at a perfectly steady pace everywhere. But with our new, ultra-precise "time-rulers" (atomic clocks), we've discovered that the river actually ripples and swells slightly depending on where you are standing.

This paper is like a detailed map of those tiny ripples caused specifically by the Moon's gravity. Here is the story of what the authors found, explained simply:

The Setup: A Cosmic Dance

Think of the Earth and the Moon as dance partners. The Moon doesn't just pull on the oceans to create tides; it also pulls on time itself. Because the Moon is moving around the Earth, its gravitational "tug" changes depending on where you are on the planet and where the Moon is in the sky.

The authors used a special mathematical "lens" (called a geocentric Fermi frame) to look at how this tug affects two clocks sitting in different places on Earth. They wanted to see: If I have a clock in New York and one in London, how does the Moon's position change the way these two clocks tick relative to each other?

The Main Discovery: Location Matters

The paper reveals that the Moon's pull creates a tiny "frequency shift" (a change in how fast the clock ticks) that depends entirely on the geometry of the situation.

1. The "Same Street, Different Houses" Scenario (Same Longitude, Different Latitude)
Imagine two clocks standing on the same line of longitude (like one at the equator and one at the North Pole).

The Finding: The further apart they are in latitude (how far north or south they are), the bigger the difference in their ticking speed becomes.
The Moon's Role: If you move the Moon's position east or west (change its longitude), it acts like turning a volume knob—it changes the size of the effect, but not the pattern. However, if you move the Moon north or south (change its latitude), it acts like a conductor changing the rhythm (phase) and the volume of the effect.

2. The "Same Room, Different Corners" Scenario (Same Latitude, Different Longitude)
Now imagine two clocks sitting at the same latitude (like two cities on the equator) but far apart in longitude.

The Finding: The difference in their ticking speed grows as they get further apart in longitude (up to 90 degrees apart).
The Moon's Role: Here, the roles flip! If you move the Moon east or west (change its longitude), it changes both the rhythm and the volume of the effect. But if you move the Moon north or south (change its latitude), it only changes the volume.

The "Volume Knob" vs. The "Conductor" Analogy

To make this concrete, think of the Moon's gravity as a sound wave hitting the clocks.

Moon's Longitude (East/West movement):
- For clocks at the same latitude, moving the Moon East/West changes both the pitch (phase) and the loudness (amplitude) of the sound.
- For clocks at the same longitude, moving the Moon East/West only changes the loudness.
Moon's Latitude (North/South movement):
- For clocks at the same latitude, moving the Moon North/South only changes the loudness.
- For clocks at the same longitude, moving the Moon North/South changes both the pitch and the loudness.

Why Does This Matter?

The authors point out that these effects are incredibly small—about 17 zeros after the decimal point ( $10^{-17}$ ). For a long time, scientists ignored them because our clocks weren't good enough to hear them.

But now, our clocks are so precise that they can "hear" these whispers from the Moon. The paper concludes that if we want to synchronize clocks across the globe (for things like navigation or scientific experiments) with extreme precision, we can't just ignore the Moon's dance. We have to account for exactly where the Moon is and exactly where our clocks are standing, or our time measurements will be slightly out of sync.

In short: The Moon doesn't just pull the oceans; it pulls on time, and the way it pulls depends on a complex, shifting dance between the Moon's position and the location of your clock.

Problem
With clock timing technology advancing to the point where fractional frequency instabilities reach $10^{-18}$ or better, it has become critical to investigate subtle frequency shifts between clocks at different locations on Earth. While previous studies have addressed gravitational redshift, kinematic effects, and the influence of Earth's gravitational potential, the specific impact of the lunar tidal potential on the fractional frequency shift between ground-based clocks requires systematic analysis. Existing frameworks, such as the Geocentric Coordinate Reference System (GCRS), have been used to analyze these shifts, but there is a need to examine these effects within the geocentric Fermi frame to better account for relativistic effects and tidal potentials in a local free-fall context.

Methodology
The authors analyze the fractional frequency shift of clocks on Earth using the geocentric Fermi frame to the order of $c^{-2}$ . The study derives the proper time of a clock by incorporating the Earth's gravitational potential ( $\Phi_e$ ), the lunar tidal potential ( $\Phi_{tm}$ ), and the solar tidal potential ( $\Phi_{ts}$ ). While acknowledging that Earth's potential and solar tides contribute to frequency shifts, the paper focuses specifically on the lunar tidal potential due to its relative magnitude and the shorter orbital period of the Moon compared to the Earth's orbit around the Sun.

The study establishes a comparison method for two clocks, A and B, located at different positions on the Earth's surface. By expanding the proper time equations, the authors derive an expression for the fractional frequency shift ( $\Delta f / f_B$ ) that depends on the difference in gravitational and tidal potentials and the kinematic terms ( $v^2/2c^2$ ) between the two locations. The lunar tidal potential is modeled by expanding the Moon's gravitational potential, retaining terms up to the second order, and expressing the angle between the clock's position vector and the Moon's position vector in terms of latitude and longitude.

Key Contributions and Results
The paper provides a systematic calculation of the lunar tidal fractional frequency shift between two ground clocks under varying geometric configurations:

Clock Position Dependence:
- Same Longitude, Different Latitudes: When two clocks share the same longitude, the fractional frequency shift increases as the difference in latitude ( $\Delta \phi$ ) increases. The shift reaches a maximum of approximately $6 \times 10^{-17}$ when one clock is at the equator and the other at the North Pole. The magnitude of this shift is modulated by the cosine of twice the longitude ( $\cos(2\lambda)$ ).
- Same Latitude, Different Longitudes: When two clocks share the same latitude, the fractional frequency shift increases as the longitude difference ( $\Delta \lambda$ ) increases from $0^\circ$ to $90^\circ$ or decreases from $180^\circ$ to $90^\circ$ . The shift is zero when both clocks are at the North Pole ( $\phi = 90^\circ$ ).
Moon's Position Dependence:
- Same Longitude, Different Latitudes: Changes in the Moon's longitude ( $\lambda_m$ ) affect only the amplitude of the fractional frequency shift between the two clocks. Conversely, changes in the Moon's latitude ( $\phi_m$ ) affect both the phase and amplitude of the shift.
- Same Latitude, Different Longitudes: In this configuration, changes in the Moon's longitude affect both the phase and amplitude of the shift, while changes in the Moon's latitude affect only the amplitude.

The authors present these findings through analytical derivations and graphical representations (Figures 1–4), demonstrating how the lunar tidal signal varies with the relative geometry of the clocks and the Moon.

Significance
The paper claims that these results provide useful information for the calibration and synchronization of clocks on Earth. As clock precision improves, the ability to distinguish these subtle tidal effects becomes necessary for high-precision applications, including communication, navigation, and fundamental physics experiments such as testing the equivalence principle and detecting gravitational redshift. The study complements existing analyses by offering a detailed view of the position-dependent nature of lunar tidal frequency shifts within the geocentric Fermi frame.

The Setup: A Cosmic Dance

The Main Discovery: Location Matters

The "Volume Knob" vs. The "Conductor" Analogy

Why Does This Matter?

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