[Paper Review] Testing General Relativity with the ACES Mission

C. Le Poncin-Lafitte, S. Lambert|arXiv (Cornell University)|Oct 16, 2006

Advanced Frequency and Time Standards3 citations

TL;DR

This paper computes relativistic frequency shifts for atomic clocks aboard the International Space Station (ISS) to assess their detectability in the context of the Atomic Clock Ensemble in Space (ACES) mission. Using a detailed relativistic framework, it quantifies contributions from special and general relativity, including gravitational potential, Earth's multipole moments (J₂, J₄, J₆), and rotational effects, showing that shifts reach levels of 10⁻¹⁸, enabling high-precision tests of general relativity.

ABSTRACT

The new generation of atomic clocks will reach unprecedented uncertainties in frequency of $10^{-18}$. In order to prepare space missions such as ACES, we compute all relativistic frequency shifts detectable during this mission in the case of a clock aboard the International Space Station.

Motivation & Objective

• To evaluate the magnitude of relativistic frequency shifts expected for atomic clocks on the ISS during the ACES mission.
• To identify and quantify all significant relativistic effects—both special and general relativistic—that could influence clock comparisons at 10⁻¹⁸ fractional frequency accuracy.
• To provide numerical estimates of these shifts for a mid-latitude ground site (e.g., Paris) to support data processing and calibration in the ACES mission.
• To assess the feasibility of using the ACES mission as a testbed for general relativity at unprecedented precision.

Proposed method

• The paper applies a relativistic framework for frequency shift analysis, decomposing contributions into special-relativistic Doppler effects and gravitational effects.
• It expands the Earth's gravitational potential using spherical harmonics, including mass monopole (M), quadrupole (J₂), and higher-order multipole moments (J₄, J₆).
• The analysis includes velocity-dependent terms up to order 1/c⁴ and gravitational terms up to 1/c⁴, accounting for Earth's rotation and non-spherical mass distribution.
• Numerical evaluations are performed for an ISS-like orbit (350 km altitude, 51° inclination), with a ground station at 49°N latitude.
• The relativistic frequency shift is modeled as a sum of terms: (δν/ν) = (δν/ν)ₐ + (δν/ν)₉, where (δν/ν)ₐ is the special-relativistic Doppler shift and (δν/ν)₉ includes gravitational and mixed effects.
• Analytical expressions from prior literature (Blanchet et al. 2001; Linet & Teyssandier) are used to compute the gravitational contributions.

Experimental results

Research questions

• RQ1 What are the dominant relativistic frequency shifts affecting atomic clocks on the ISS during a pass over a mid-latitude ground station?
• RQ2 How do higher-order multipole moments of the Earth's gravitational field (J₂, J₄, J₆) contribute to frequency shifts at 10⁻¹⁸ accuracy?
• RQ3 What is the magnitude of special-relativistic Doppler shifts up to 1/c⁴ order for a low Earth orbiting clock?
• RQ4 Can general relativistic effects such as Earth's rotation (S-term) and mass quadrupole (J₂) be detected at the 10⁻¹⁸ level in the ACES mission?
• RQ5 How sensitive are these frequency shifts to orbital variations of the ISS, and what level of orbit control is required?

Key findings

• The first-order special-relativistic Doppler shift (1/c term) reaches a maximum magnitude of 1×10⁻⁵, which is well within measurable limits.
• Second-order special-relativistic terms (1/c²) contribute up to 2×10⁻¹⁰, while third-order terms (1/c³) reach up to 3×10⁻¹⁵.
• The dominant gravitational shift from Earth's monopole mass (M) contributes up to 3×10⁻¹¹ at 1/c².
• The J₂ quadrupole moment contributes up to 2×10⁻¹³ at 1/c², and higher-order moments (J₄, J₆) contribute at 3×10⁻¹⁶ and 1×10⁻¹⁶, respectively.
• Third-order gravitational terms (1/c³) from mass and J₂ contribute up to 2×10⁻¹⁴ and 1×10⁻¹⁸, respectively, with the latter being at the threshold of detectability.
• Fourth-order terms (1/c⁴), including Earth's rotation (S-term), are extremely small, with magnitudes ≤1×10⁻²², indicating negligible impact for current accuracy goals.