Testing for Lorentz Violation: Constraints on Standard-Model-Extension Parameters via Lunar Laser Ranging

We present constraints on violations of Lorentz invariance based on archival lunar laser-ranging (LLR) data. LLR measures the Earth-Moon separation by timing the round-trip travel of light between the two bodies and is currently accurate to the equivalent of a few centimeters (parts in ${10}^{11}$ of the total distance). By analyzing this LLR data under the standard-model extension (SME) framework, we derived six observational constraints on dimensionless SME parameters that describe potential Lorentz violation. We found no evidence for Lorentz violation at the ${10}^{-6}$ to ${10}^{-11}$ level in these parameters. This work constitutes the first LLR constraints on SME parameters.

Figure 1

The Sun-centered celestial-equatorial plane. The equator of the Earth defines the $XY$ plane and the Earth’s rotation axis is aligned with the $Z$ axis. The orbital plane of the Earth (the ecliptic) is inclined to the $XY$ plane by the obliquity angle, $\eta \approx 23.5°$. The Earth is shown at a descending node (a Vernal Equniox) which also defines the origin of our time coordinate, $t$. Reprinted figure with permission from 6. Copyright 2006 by the American Physical Society.

Figure 2

Lunar orbital parameters: here the Earth is shown translated to the center of the Sun-centered coordinate system. The lunar orbit is described by $r_0$, the mean distance between the Earth and Moon, $e$ (not labeled) the eccentricity of the lunar orbit, $\alpha$, the longitude of the ascending node, $\beta$, the angle between the normal to the lunar orbital plane and the normal to the Earth’s equatorial plane, and $\theta$, the angle, along the Lunar orbit, subtended by the ascending node line and the position of the Moon at $t=0$. Reprinted figure with permission from 6. Copyright 2006 by the American Physical Society.

Figure 3

The annual RMS residual between the LLR data and our best-fit model for the lunar range. The residual RMS in 1969 is over 300 cm. We omitted this point from the plot for clarity, but the two data points from that year were included in the analysis. Over this time span, the potential Lorentz-violating signals would all have undergone at least 34 cycles. As the number and capability of LLR ranging stations change with time so too does the LLR data rate and quality. For example, the sharp improvement in the model-data agreement around 1995 is due to the upgrade of the OCA station.