Temporal Equivalence Principle: Lunar Laser Ranging and the Nordtvedt Effect

The Temporal Equivalence Principle (TEP) is a scalar-tensor theory in which proper time is a dynamical field φ that couples universally to all matter via a conformal metric g̃μν = A²(φ) gμν. In deep gravitational potential wells, high ambient density suppresses the local gradient of this field — a mechanism called Temporal Shear screening — with the degree of suppression scaling with a body's gravitational compactness Φ/c². Because Earth and Moon have different compactness and interior shielding profiles, TEP motivates a compactness-dependent Strong Equivalence Principle response that would appear in Lunar Laser Ranging as a synodic Earth-Moon range modulation δr = 13η cos D, where D is the Moon-Sun elongation angle and η is the Nordtvedt parameter. The expected residual-channel amplitude is at the millimetre level for η ∼ 10⁻⁴. This work analyses 26,207 raw Lunar Laser Ranging O–C residuals from the public INPOP19a ephemeris archives (Paris Observatory, Geoazur), comprising N = 25,445 measurements from five international stations (1984–2019) after standard 6σ MAD outlier cleaning. The primary estimand is a precision-weighted synodic regression on the post-fit residual channel, with inverse station-variance weighting and a full systematic model that includes annual, monthly, and station-specific regressors. The primary precision-weighted residual-channel estimate is η = -3.91 × 10⁻⁴ ± 5.63 × 10⁻⁵ (6.94σ; 6.78σ cluster-robust). The amplitude stabilises in the band -3.2 to -4.1 × 10⁻⁴ across the estimator hierarchy, from cosD-only (5.25σ) through full-systematic OLS (6.17σ) to precision-weighted (6.94σ). Robustness checks — common-η mixed model, leave-one-station-out meta-analysis, wild cluster bootstrap, phase-locked differential, cross-ephemeris validation on DE430, parametric GR-null bootstrap, and a frequency null scan — all support residual-channel survival of the synodic component (Section 4). Within the standard Parametrized Post-Newtonian framework, the measured Nordtvedt parameter implies β = 0.999902 ± 1.07 × 10⁻⁵ from the joint (β, γ) contour, placing General Relativity (β = γ = 1) at Δχ² = 48.2, outside the 99% confidence contour. A full-sky directional scan on the residual channel (2,664 uniformly spaced directions, 5° grid) places the Planck CMB dipole axis at rank 226/2664 (top 8.5%); a scrambled-sky null with n = 1,000 Monte Carlo realizations yields a look-elsewhere-corrected p < 0.001. The result is therefore framed as a high-significance residual-channel candidate with a TEP interpretation, not as a completed replacement for direct-fit LLR bounds. Source-level numerical refits of the INPOP or DE430 integrators with η left free remain the critical open closure test. Ephemeris-absorption stress tests bound the residual-channel survival amplitude but do not replace a full dynamical integrator-level confirmation. Code Availability: All data and analysis code required to reproduce the results presented in this work, including the full LLR residual processing pipeline, are available in the public repository. Website: https://mlsmawfield.com/tep/llr/Repository: https://github.com/matthewsmawfield/TEP-LLR Keywords: temporal equivalence principle – lunar laser ranging – LLR – equivalence principle – Nordtvedt effect – post-Newtonian – scalar-tensor gravity – strong equivalence principle Open Science Statement: This work is a preprint and is open to community review, ideas, and collaboration. All materials required for full reproducibility—including data downloads, analysis scripts, code, and manuscripts—are open-source. Feedback and contributions to further test these results are welcome.