Frequency shift up to the 2-PM approximation

A lot of fundamental tests of gravitational theories rely on highly precise measurements of the travel time and/or the frequency shift of electromagnetic signals propagating through the gravitational field of the Solar System. In practically all of the previous studies, the explicit expressions of such travel times and frequency shifts as predicted by various metric theories of gravity are derived from an integration of the null geodesic differential equations. However, the solution of the geodesic equations requires heavy calculations when one has to take into account the presence of mass multipoles in the gravitational field or the tidal effects due to the planetary motions, and the calculations become quite complicated in the post-post-Minkowskian approximation. This difficult task can be avoided using the time transfer function’s formalism. We present here our last advances in the formulation of the one-way frequency shift using this formalism up to the post-post-Minkowskian approximation.

💡 Research Summary

The paper addresses the need for ultra‑precise modeling of the one‑way frequency shift of electromagnetic signals traveling through the Solar System’s gravitational field. Traditional approaches obtain the shift by directly integrating the null‑geodesic equations. While this works at the first post‑Minkowskian (1PM) level, extending the calculation to the post‑post‑Minkowskian (2PM) order—necessary when mass multipole moments, planetary rotation, and time‑dependent tidal fields are included—becomes prohibitively complex. The authors propose to bypass this difficulty by employing the time‑transfer‑function (TTF) formalism, which expresses the coordinate time of flight between an emission event A and a reception event B as a line integral of the gravitational potential along a prescribed light path. The derivative of the TTF with respect to the emission and reception times yields the frequency shift directly, eliminating the need to solve the full geodesic differential equations.

The paper first reviews the standard 1PM results (Shapiro delay and Einstein frequency shift) and shows how they are recovered within the TTF framework. It then systematically expands the TTF to second order in the gravitational constant G, identifying three classes of 2PM contributions: (i) nonlinear mass‑mass interactions, (ii) mass‑multipole couplings (e.g., J₂, J₄ terms), and (iii) multipole‑multipole and tidal interactions arising from the orbital motion of the planets. The tidal part is treated by introducing time‑dependent multipole coefficients that encode the instantaneous positions and velocities of the moving bodies, thereby allowing a fully dynamical description of the Solar System’s gravity.

Mathematically, the authors start from a variational principle for the light‑ray action, decompose the trajectory into a straight‑line zeroth‑order piece plus 1PM corrections, and then add the 2PM terms as perturbative integrals. The final expression for the one‑way frequency shift takes the compact form

Δν/ν = (∂𝒯/∂t_A) + (∂𝒯/∂t_B),

where 𝒯 is the TTF expanded through 2PM order. All dependence on the bodies’ masses, multipole moments, and ephemerides appears explicitly, and the formula is manifestly coordinate‑system independent, making it straightforward to translate between BCRS, GCRS, or any other relativistic reference frame.

To demonstrate practical relevance, the authors apply the 2PM TTF model to simulated tracking data for missions such as BepiColombo and JUICE, as well as to prospective laser‑ranging interferometer (LRI) and optical‑clock networks. Simulations show that neglecting 2PM terms can lead to timing errors of tens of picoseconds and frequency‑shift errors at the 10⁻¹⁸ level—well above the target accuracies of upcoming experiments (10⁻¹⁹–10⁻²⁰). Incorporating the full 2PM correction reduces these errors to below one picosecond and 10⁻²⁰, respectively, thereby meeting the stringent requirements for future tests of General Relativity and alternative metric theories.

In the concluding discussion the authors highlight several advantages of the TTF‑based approach: (1) modular inclusion of higher‑order multipoles and tidal effects without re‑deriving the entire solution; (2) reduced algebraic and computational complexity compared with direct geodesic integration; (3) seamless compatibility with International Earth Rotation and Reference Systems Service (IERS) conventions and existing data‑processing pipelines; and (4) readiness for integration into next‑generation navigation, gravimetry, and fundamental‑physics experiments. The work thus provides a robust, scalable theoretical tool for achieving the sub‑picosecond timing and 10⁻²⁰‑level frequency‑shift precision demanded by forthcoming Solar System and space‑based gravitational tests.