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arxiv: 2607.00550 · v1 · pith:LHGBASGSnew · submitted 2026-07-01 · 🌌 astro-ph.EP · astro-ph.IM

Relativistic Time Scales and Transformations in the Solar System

Hong-Bo Jin , Jinsong Ping , Min Liu , Mingyuan Wang This is my paper

Pith reviewed 2026-07-02 05:48 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IM
keywords relativistic time scalescoordinate time transformationssolar system ephemeridesMarsMoonclock rate differencesShapiro delay1PN approximation
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The pith

A single documented chain of 1PN transformations maps proper time to coordinate times across barycentric and body-centric systems for Mars and the Moon.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out a unified sequence of relativistic formulas that connects barycentric coordinate time to body-centric times for both the Moon and Mars. This sequence is built from tabulated Christoffel symbols, Fermi coordinates, and range-rate expansions so that proper time on a spacecraft or surface clock can be translated without leaving microsecond-scale mismatches. If the chain works as described, merged tracking data from different missions will avoid systematic range and Doppler biases at the level now targeted by lunar and Mars missions. The author shows the resulting metric clock-rate offsets on the areoid and selenoid match earlier nested calculations, while Shapiro terms in Mars ranging fall between 10 to the minus 12 and 10 to the minus 13.

Core claim

The paper claims that one consistent 1PN documentation chain—covering harmonic Christoffel symbols to order c to the minus 4, the barycentric-to-geocentric-to-terrestrial time sequence, Fermi normal coordinates, null-geodesic observables, and a 1PN two-way range-rate expansion—can be applied in parallel to Mars (MCRS/MCG) and lunar (LCRS/TCL) body-centric frames. When executed, the chain produces an areoid-geoid metric clock-rate difference of about 48 microseconds per day on Mars and selenoid-geoid rates of 57.4 to 58.7 microseconds per day on the Moon, while Shapiro-rate contributions in Mars ranging reach 10 to the minus 12 to 10 to the minus 13. Multi-CRS consistency is obtained by follo

What carries the argument

The unified 1PN documentation chain that assembles barycentric-geocentric-terrestrial time transformations, Fermi normal coordinates, and the two-way range-rate expansion for parallel use on Mars and lunar frames.

If this is right

  • Mars areoid clocks run slower than geoid clocks by approximately 48 microseconds per day.
  • Lunar selenoid clocks run slower than geoid clocks by 57.4 to 58.7 microseconds per day.
  • Shapiro-rate contributions in two-way Mars ranging appear at the 10 to the minus 12 to 10 to the minus 13 level.
  • Merged Chang'e- or Tianwen-class data sets remain free of microsecond-level range and Doppler biases only when the full transformation chain is applied uniformly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same chain could be written out for additional solar-system bodies once their body-centric frames are defined.
  • Navigation software that embeds the documented steps directly would reduce the chance of time-scale mismatches when combining data from separate missions.
  • Future tests could compare the predicted lunar rates against clocks on the lunar surface to check consistency at the 0.1 microsecond per day level.

Load-bearing premise

The IAU barycentric and body-centric hierarchy already supplies reference systems whose transformations can be merged without introducing extra higher-order terms at the microsecond level.

What would settle it

An independent high-precision measurement of the areoid-geoid clock-rate difference on Mars, obtained from a mission clock compared directly against a barycentric reference, would confirm or contradict the 48 microseconds per day value.

Figures

Figures reproduced from arXiv: 2607.00550 by Hong-Bo Jin, Jinsong Ping, Mingyuan Wang, Min Liu.

Figure 1
Figure 1. Figure 1: Coherent Two-Way DSN Geometry (t1: ground transmit; t2: spacecraft turnaround; t3: ground receive). Round-trip light time τRT = t3 − t1 is obtained from the light-time con￾straint (Eq. 39) with 1PN and Shapiro corrections (Eqs. 37, 34). with λR,T = wR,T /c2 evaluated at the respective events and kµ taken along the solved null ray. 7.5 Two-Way Doppler: 1PN Expansion of ρ˙ This subsection expands the two-way… view at source ↗
Figure 2
Figure 2. Figure 2: Mars areoid–geoid instantaneous clock￾rate difference over TM (schematic). Solid curve: monopole rate from (50); dashed curve: (51). Liu et al. (2026a). It is embedded in the BCRS through the lunar ephemeris ⃗rL(t) and velocity ⃗vL(t) from DE440-class integrations Park et al. (2021). Denote by ⃗r′ the position relative to the selenocentre. The 1PN line element is (12) with w = GML r ′ + wext(⃗r′ ), (52) Wi… view at source ↗
Figure 3
Figure 3. Figure 3: Solar-System Time-Scale Hierarchy. Solid arrows: IAU coordinate-time transformations; [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Lunar selenoid–geoid instantaneous clock-rate difference over TL (schematic). Solid curve: monopole rate from (58); dashed curve: (59). Markers at ∼57.4 and ∼58.7 µs day−1 follow Liu et al. (2026a); Kopeikin and Kaplan (2024); Ashby and Patla (2024). with AL = 1 2 (6.8 − 6.65) × 10−10 ≈ 7.5 × 10−12; a precision model would use the full sum (56) with ephemeris (Ak, Bk) Liu et al. (2026a). This met￾ric drift… view at source ↗
read the original abstract

Each solar-system observable is characterised by celestial reference system (CRS) coordinate time, proper time on its world line, and the transformation between them. Ephemerides and Deep Space Network (DSN) tracking use the International Astronomical Union (IAU) barycentric and body-centric hierarchy, now extended to cislunar and Mars work. The IERS Conventions, Moyer radiometric models, and recent lunar-time papers distribute metric, scale, and tracking formulae across separate manuals. Merged Chang'e- or Tianwen-class data can acquire microsecond-level range and Doppler biases unless proper time $\tau$ is mapped consistently to barycentric and body-centric coordinate times. We present a unified 1PN documentation chain: tabulated harmonic Christoffel symbols through $\mathcal{O}(c^{-4})$, the barycentric-geocentric-terrestrial coordinate-time sequence, Fermi normal coordinates, null-geodesic observables, and a 1PN two-way range-rate expansion, applied in parallel to Mars (MCRS/MCG) and lunar (LCRS/TCL) body-centric systems. The chain yields a Mars areoid-geoid metric clock-rate difference of $\sim$48~$\mu$s\,day$^{-1}$ and lunar selenoid-geoid rates of $\sim$57.4-58.7~$\mu$s\,day$^{-1}$ consistent with published nested coefficients. Mars-range Shapiro-rate terms reach $10^{-12}$-$10^{-13}$. Multi-CRS consistency relies on documented transformation chains rather than a single master clock.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents a unified 1PN documentation chain for relativistic time-scale transformations in the solar system, extending the IAU barycentric and body-centric hierarchy to Mars (MCRS/MCG) and lunar (LCRS/TCL) systems. It tabulates harmonic Christoffel symbols to O(c^{-4}), covers the barycentric-geocentric-terrestrial sequence, Fermi normal coordinates, null-geodesic observables, and a 1PN two-way range-rate expansion. The chain produces a Mars areoid-geoid metric clock-rate difference of ∼48 μs day^{-1}, lunar selenoid-geoid rates of ∼57.4-58.7 μs day^{-1} consistent with published nested coefficients, and Mars-range Shapiro-rate terms at 10^{-12}-10^{-13}. Multi-CRS consistency is asserted to rely on the documented chains rather than a single master clock, with application to merged Chang'e- or Tianwen-class data.

Significance. If the 1PN truncation and merging introduce no omitted cross terms at the microsecond level, the work supplies a practical unification of metric, scale, and tracking formulae previously distributed across IERS Conventions, Moyer models, and lunar-time papers. The explicit numerical outputs and direct consistency statements with prior nested coefficients constitute a concrete strength for precision DSN tracking and cislunar/Mars mission data analysis.

major comments (2)
  1. [Abstract] Abstract, paragraph on Merged Chang'e- or Tianwen-class data: the central claim that the documented 1PN chain (Christoffel symbols to O(c^{-4}), Fermi coordinates, null geodesics, two-way range-rate) produces the quoted μs day^{-1} rates with no unaccounted higher-order terms at the target precision is load-bearing, yet the manuscript supplies neither an explicit truncation-error bound nor a comparison against 1.5PN or gauge-dependent contributions that could accumulate over one day.
  2. [Abstract] Abstract: the reported consistency of the Mars areoid-geoid (∼48 μs day^{-1}) and lunar selenoid-geoid (∼57.4-58.7 μs day^{-1}) rates with published nested coefficients is asserted without showing the intermediate transformation steps or the quantitative match, leaving the verification of the merged-chain outputs unverifiable from the given material.
minor comments (1)
  1. [Abstract] The abstract alternates between ∼ and explicit ranges (57.4-58.7) for the lunar rates; a uniform notation would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation for major revision. We address each major comment below and will incorporate clarifications and additional material into the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph on Merged Chang'e- or Tianwen-class data: the central claim that the documented 1PN chain (Christoffel symbols to O(c^{-4}), Fermi coordinates, null geodesics, two-way range-rate) produces the quoted μs day^{-1} rates with no unaccounted higher-order terms at the target precision is load-bearing, yet the manuscript supplies neither an explicit truncation-error bound nor a comparison against 1.5PN or gauge-dependent contributions that could accumulate over one day.

    Authors: We agree that an explicit truncation-error estimate strengthens the central claim. In the revised manuscript we will add a dedicated paragraph (likely in Section 2 or a new appendix) providing order-of-magnitude bounds on omitted 1.5PN and gauge-dependent terms accumulated over one day, using the magnitudes of the metric potentials already tabulated. This will quantify that such contributions remain below the microsecond level for the quoted clock-rate differences. revision: yes

  2. Referee: [Abstract] Abstract: the reported consistency of the Mars areoid-geoid (∼48 μs day^{-1}) and lunar selenoid-geoid (∼57.4-58.7 μs day^{-1}) rates with published nested coefficients is asserted without showing the intermediate transformation steps or the quantitative match, leaving the verification of the merged-chain outputs unverifiable from the given material.

    Authors: The intermediate steps and quantitative matches are derived explicitly in the main text via the tabulated harmonic Christoffel symbols, the barycentric-to-body-centric sequences, and the Fermi-coordinate transformations (Sections 3–5). To improve verifiability from the abstract itself, we will insert a short parenthetical reference to the relevant sections and, if space permits, a one-sentence summary of the numerical agreement with the nested coefficients already published for the lunar case. revision: yes

Circularity Check

0 steps flagged

No circularity: standard IAU hierarchy chain documented with external consistency checks

full rationale

The paper documents a merged 1PN transformation chain (Christoffel symbols, Fermi coordinates, null geodesics, two-way range-rate) applied to IAU barycentric/body-centric systems extended to MCRS/MCG and LCRS/TCL. The output rates (Mars ~48 μs day^{-1}, lunar 57.4-58.7 μs day^{-1}, Shapiro 10^{-12}-10^{-13}) are stated as direct results of this chain and noted as consistent with published nested coefficients. No quoted equations show a parameter fitted to data then renamed as prediction, no self-definitional loop (X defined via Y where Y is the output), and no load-bearing premise that reduces solely to an unverified self-citation. The IAU hierarchy is treated as an external input, not derived within the paper. This is a documentation exercise whose central numbers are externally falsifiable against prior published coefficients.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters, new entities, or ad-hoc axioms are introduced; the paper rests on the standard 1PN approximation of general relativity and the established IAU reference-system hierarchy.

axioms (2)
  • domain assumption The 1PN (O(c^{-2})) approximation is adequate for microsecond-level solar-system time transformations.
    All expansions and tabulated symbols are limited to 1PN order throughout the documentation chain.
  • domain assumption The IAU barycentric and body-centric hierarchy extended to cislunar and Mars systems supplies the correct coordinate times for observables.
    The paper adopts and extends this hierarchy without deriving an alternative reference system.

pith-pipeline@v0.9.1-grok · 5809 in / 1561 out tokens · 25722 ms · 2026-07-02T05:48:27.719811+00:00 · methodology

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Reference graph

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