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Relativistic Time Scales and Transformations in the Solar System

This paper presents a unified 1PN documentation chain that integrates celestial reference system coordinate times, proper times, and transformation formulae across barycentric, Earth, Mars, and lunar systems to eliminate microsecond-level biases in deep space tracking and ensure multi-CRS consistency for solar system observables.

Original authors: Hong-Bo Jin, Jinsong Ping, Min Liu, Mingyuan Wang

Published 2026-07-02

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CC BY 4.0

Original authors: Hong-Bo Jin, Jinsong Ping, Min Liu, Mingyuan Wang

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 the Solar System as a massive, bustling train station. In the past, everyone agreed that "time" was just the clock on the wall at the main station (Earth). But now, we are building new stations on the Moon and Mars, and we are sending high-speed trains (spacecraft) between them.

The problem? Time doesn't tick at the same speed everywhere.

Just as a train moving very fast or sitting near a heavy mountain (gravity) experiences time differently than a person standing still, clocks on the Moon, Mars, and Earth actually run at slightly different speeds. If you don't account for this, your "GPS" for space will drift off by miles, and your communication signals will be out of sync.

This paper is essentially a universal instruction manual for translating time between these different "stations." Here is what the authors did, explained simply:

1. The Problem: Too Many Clocks, Too Many Rules

Right now, scientists use different rulebooks for different jobs.

One book tells you how to calculate time for Earth satellites.
Another book handles the Moon.
A third handles Mars.
And they all speak slightly different "languages" of math.

If you try to mix data from a Chinese Moon mission (Chang'e) and a Chinese Mars mission (Tianwen) without translating them perfectly, you get errors as big as a few microseconds. In space navigation, that sounds tiny, but it's actually huge—it's the difference between landing a rover safely or crashing it.

2. The Solution: One Unified "Time Translator"

The authors built a single, unified chain of math (a "documentation chain") that acts like a master translator. It connects:

Proper Time ( $\tau$ ): The time measured by the actual clock on the spacecraft or astronaut's wrist.
Coordinate Time: The "official" time of the solar system (like a master schedule).

They didn't just write a new theory; they took existing complex math (called the "Post-Newtonian" framework) and organized it into a clear, step-by-step recipe that software can follow.

3. The "Speed Bumps" of Space (Gravity and Speed)

The paper explains two main reasons why clocks get out of sync:

Gravity: The closer you are to a heavy object (like Earth or the Sun), the slower time moves. It's like walking through deep mud; time gets "stuck."
Speed: The faster you move, the slower time moves relative to someone standing still.

The authors calculated exactly how much these effects slow down clocks on the Moon and Mars compared to Earth.

Mars: A clock on the surface of Mars runs about 48 microseconds faster per day than a clock on Earth's surface.
The Moon: A clock on the Moon runs about 57 to 58 microseconds faster per day than a clock on Earth.

Analogy: Imagine Earth, the Moon, and Mars are runners on a track. Earth is running through thick mud (strong gravity), so it's slow. The Moon and Mars are running on smoother tracks, so they "finish the day" (tick off 24 hours) slightly faster. If you don't adjust your watch, you'll think the Moon runner is cheating because they finished early!

4. The "Shapiro Delay" (The Traffic Jam)

When we send a radio signal from Earth to a spacecraft, the signal has to travel through the "gravity field" of the Sun and planets. The paper explains that gravity acts like a traffic jam for light. The signal doesn't just travel in a straight line at a constant speed; it gets slightly delayed.

The authors created a formula to calculate this delay (called the Shapiro delay) so that when we measure how long a signal took to return, we know exactly how far the spacecraft is, rather than thinking it's further away just because of the "traffic."

5. Why This Matters for the Future

The paper is a "plumbing guide" for the future of space exploration.

For the Moon: The US and other nations are planning to build a "Lunar Time" standard. This paper provides the math to ensure that Lunar Time matches up correctly with Earth Time.
For Mars: As we send more missions to Mars, we need a "Mars Time" system. This paper sets the rules for how that time should be calculated so we don't get lost.

In short: This paper doesn't invent new physics; it takes the complex, scattered math of how time works in space and organizes it into a single, clear manual. This ensures that when we send robots to the Moon or Mars, their clocks are perfectly synchronized with ours, preventing navigation errors and keeping our deep-space communications on time.

Technical Summary: Relativistic Time Scales and Transformations in the Solar System

Problem Statement
Solar-system metrology currently relies on a fragmented documentation landscape where metric coefficients, coordinate scales, and radiometric models are distributed across separate manuals (e.g., IERS Conventions, Moyer models) and recent specialized papers. As interest in cislunar and Mars operations accelerates—driven by IAU resolutions on Lunar Celestial Reference Systems (LCRS) and US directives for Coordinated Lunar Time (LTC)—mission teams face a critical challenge. Merged tracking data from missions like Chang'e or Tianwen risks acquiring microsecond-level biases in range and Doppler measurements unless proper time ( $\tau$ ) is mapped consistently to barycentric (TCB) and body-centric coordinate times (e.g., TCG, TCL, MCG). Existing frameworks often lack a unified 1PN (first post-Newtonian) documentation chain that integrates harmonic Christoffel symbols, Fermi normal coordinates, and two-way observables into a single coherent structure for both Earth and other celestial bodies.

Methodology
The authors present a unified 1PN documentation chain that links the PPN harmonic metric to IAU coordinate times and body-centric reference systems. The methodology involves:

Metric Formalism: Utilizing the standard PPN metric in harmonic coordinates, extended to $O(c^{-4})$ . The paper tabulates harmonic Christoffel symbols (including gravitomagnetic potentials $V_i$ ) to support redshift and null-geodesic routines used in IERS-class software.
Coordinate Transformations: Establishing explicit transformation chains between the Barycentric Celestial Reference System (BCRS/TCB), Geocentric (GCRS/TCG), and provisional body-centric systems (LCRS/TCL for the Moon and MCRS/MCG for Mars). This includes integrating the clock hypothesis along world lines to derive coordinate-time rates.
Local Frames: Employing Fermi normal coordinates and tetrad transport to define local clock comparisons and handle non-geodesic motion (e.g., spacecraft thrust).
Observables: Deriving the light-time constraint, Shapiro delay, and a 1PN expansion of the two-way geometric range rate ( $\dot{\rho}$ ). This expansion explicitly includes Shapiro-rate terms and Sagnac-type contributions relevant to Mars-range sensitivities.
Application: Applying this chain in parallel to Mars (MCRS/MCG) and the Moon (LCRS/TCL) to calculate surface clock-rate differences relative to Earth's geoid.

Key Contributions
The paper's incremental contributions relative to prior works (e.g., Liu et al., Kopeikin & Kaplan, Ashby & Patla) include:

Tabulated Christoffel Symbols: A set of harmonic Christoffel symbols through $O(c^{-4})$ linked to redshift and null-geodesic routines, explicitly including gravitomagnetic terms ( $V_i$ ).
Two-Way Range Rate Expansion: An explicit 1PN expansion of the two-way geometric range rate $\dot{\rho}$ , incorporating Shapiro-rate and Sagnac-type terms at Mars-range sensitivity levels ( $10^{-12}$ – $10^{-13}$ ).
Unified Body-Centric Template: A parallel prescription for Mars (MCRS/MCG) and the Moon (LCRS/TCL) within the same metric-radiometry framework, allowing for cross-checked rate estimates against published nested coefficients.

Results

Mars (MCRS/MCG): The study calculates a Mars areoid–geoid instantaneous clock-rate difference of approximately 48.0 $\mu$ s day $^{-1}$ (derived from a monopole potential difference of $\sim 5.558 \times 10^{-10}$ ). The periodic modulation due to Mars' orbital motion and spin is found to be significantly smaller ( $\sim 2.1 \times 10^{-12}$ ).
Moon (LCRS/TCL): The lunar selenoid–geoid rate is estimated at 57.4 $\mu$ s day $^{-1}$ based on monopole surface potentials. This is compared with the nested TCG–TCL leading rate of 58.7 $\mu$ s day $^{-1}$ found in clock-comparison literature. The $\sim 2\%$ spread is attributed to selenocentre orbital velocity, external tidal potentials, and geoid–selenoid structural differences.
Relativistic Corrections: For Mars-range arcs, the 1PN range-rate law reveals gravitational and Shapiro-rate terms at the $10^{-12}$ – $10^{-13}$ level, which are comparable to current carrier-phase noise floors.
Consistency: The paper demonstrates that multi-CRS consistency relies on documented transformation chains rather than a single master clock, validating the use of separate body-centric systems steered to a common barycentric framework.

Significance and Claims
The paper claims to provide a necessary "unified documentation chain" rather than a replacement for existing IAU resolutions or specific nested transformations. Its primary significance lies in consolidating scattered metric, scale, and tracking formulae into a single place to prevent microsecond-level systematic biases in merged mission data. The authors assert that without explicit TDB tags, on-board proper-time maps, and documented transformation sequences (from $\tau$ to body-centric time to TCB/TT), merged Chang'e or Tianwen-class data will suffer from uncorrected drifts. The work serves as a technical foundation for operational implementation in lunar and Mars navigation, ensuring that relativistic effects are consistently modeled across different celestial bodies using established resolutions.

1. The Problem: Too Many Clocks, Too Many Rules

2. The Solution: One Unified "Time Translator"

3. The "Speed Bumps" of Space (Gravity and Speed)

4. The "Shapiro Delay" (The Traffic Jam)

5. Why This Matters for the Future

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