A Local Lorentz Invariance test with LAGEOS satellites

Strong theoretical arguments suggest that a breakdown of Lorentz Invariance could arise under some very particular conditions. From an experimental point of view, it is important to test the Local Lorentz Invariance with ever greater precision and in all contexts, regardless of the theoretical motivation for the possible violation. In this paper we discuss some aspects of the gravitational sector. Tests of Lorentz Invariance in the context of gravity are difficult and rare in the literature. Possible violations could arise from quantum physics applied to gravity or the presence of vector and tensor fields mediating the gravitational interaction together with the metric tensor of General Relativity. We present our results in the latter case. We analyzed the orbit of the LAGEOS and LAGEOS II satellites over a period of almost three decades. The effects of the possible preferred frame represented by the cosmic microwave background radiation on the mean argument of latitude of the satellites orbit were considered. These effects would manifest themselves mainly through the post-Newtonian parameter $α_1$, a parameter that has a null value in General Relativity. We constrain this parameterized post-Newtonian parameter down to the level of $α_1 \le 2\times10^{-5}$, improving a previous limit obtained through the Lunar Laser Ranging technique.

💡 Research Summary

The paper presents a new test of Local Lorentz Invariance (LLI) in the gravitational sector by exploiting the long‑term laser ranging data of the LAGEOS and LAGEOS II geodetic satellites. In the Parameterized Post‑Newtonian (PPN) framework, violations of LLI manifest as non‑zero values of the preferred‑frame parameters α₁, α₂ and α₃; General Relativity predicts all three to be exactly zero. The authors focus on α₁, which couples the motion of the Earth–Sun–satellite system with respect to a preferred frame identified with the Cosmic Microwave Background (CMB).

Starting from the Lagrangian term L_{α₁}=−(α₁/4c²)∑{a≠b}G_N m_a m_b /r{ab} (v_a·v_b), they derive the secular and periodic perturbations induced on the Keplerian elements of an artificial satellite. The most relevant observable is the mean argument of latitude ℓ₀ = ω + M (the sum of the argument of pericenter and the mean anomaly). By forming the time derivative ˙ℓ₀, the dominant Newtonian and relativistic perturbations cancel, leaving a clean annual sinusoidal term proportional to α₁·n·(w·v_⊕)/c², where w is the Sun’s absolute velocity with respect to the CMB (≈368 km s⁻¹, direction (β_PF, λ_PF) = (−11.13°, 171.55°)), v_⊕ is the Earth’s orbital velocity, and n is the satellite’s mean motion.

The authors processed nearly three decades of Satellite Laser Ranging (SLR) data for both satellites, applying state‑of‑the‑art precise orbit determination (POD). Their dynamical model includes the latest Earth gravity field (GRACE‑based), tidal effects, solar radiation pressure, atmospheric drag, and relativistic corrections. The POD yields orbit residuals, which are then examined for the expected annual sinusoid. A Fourier analysis extracts the amplitude and phase of the 1‑year component; the phase must match the predicted combination of the CMB direction and Earth’s orbital position.

Systematic errors are carefully quantified. Uncertainties in the low‑degree geopotential coefficients (J₂, J₄, etc.) can mimic an annual signal; however, using two satellites with different inclinations (≈110° and 52°) allows the authors to decorrelate these effects. Monte‑Carlo simulations propagate the uncertainties of the geopotential, solar radiation pressure coefficients, and measurement noise into the final α₁ estimate. Non‑gravitational perturbations (e.g., thermal thrust, Earth albedo) are also modeled and their residual impact assessed.

The analysis finds no statistically significant annual signal attributable to α₁. The resulting 95 % confidence interval is |α₁| ≤ 2 × 10⁻⁵. This improves upon the previous best bound from Lunar Laser Ranging (α₁ = (−7 ± 9) × 10⁻⁵) by roughly a factor of two and represents the first independent constraint on α₁ derived from Earth‑orbiting artificial satellites.

The paper situates this result within the broader landscape of Lorentz‑violation tests: pulsar timing provides limits on the “hat” versions of the parameters (e.g., \hat{α₁} ≈ (−0.4 ± 4) × 10⁻⁵), while solar‑system experiments bound α₂ to ≲ 2.4 × 10⁻⁷ and α₃ to ≲ 4 × 10⁻²⁰. The authors argue that satellite‑based tests complement these approaches because they probe a different dynamical regime (weak‑field, slow‑motion) and a distinct preferred‑frame orientation.

Looking forward, the authors note that upcoming missions such as GRACE‑Follow‑On, improvements in SLR precision (millimetre‑level), and the inclusion of additional laser‑tracked satellites (e.g., LARES) could push the α₁ limit toward 10⁻⁶. Combining multi‑satellite data with lunar laser ranging and interplanetary laser links would enable simultaneous constraints on all three preferred‑frame parameters across several characteristic frequencies (annual, semi‑annual, diurnal).

In summary, the study demonstrates that precise orbital analysis of passive geodetic satellites provides a powerful, independent probe of Local Lorentz Invariance in gravity, delivering the most stringent satellite‑based bound on the PPN parameter α₁ to date and opening a clear pathway for future, even tighter tests.