Ring Laser Gyroscope Tests of Lorentz Symmetry

TL;DR

This paper discusses how advanced interferometric gyroscopes can test Lorentz symmetry violations in gravity, linking experimental measurements to theoretical coefficients within the Standard-Model Extension framework.

Contribution

It provides a summary of methods connecting gyroscope measurements to Lorentz violation coefficients, enhancing the experimental search for fundamental symmetry violations.

Findings

Gyroscope measurements can constrain Lorentz violation coefficients.

Interferometric gyroscopes are effective for testing fundamental symmetries.

The work bridges experimental data with theoretical models in gravity.

Abstract

Interferometric gyroscope systems are being developed with the goal of measuring general-relativistic effects including frame-dragging effects. Such devices are also capable of performing searches for Lorentz violation. We summarize efforts that relate gyroscope measurements to coefficients for Lorentz violation in the gravity sector of the Standard-Model Extension.

Equations

$$g_{0j}$$

$$\Delta \tau \approx 2\oint g_{0j}d{x}^j,$$

$${\nu }_{\mathrm{LV}}^{(4)}$$

Full text

\bodymatter

1 Interferometric gyroscopes and Lorentz violation

Lorentz violation in the Standard-Model Extension (SME) can be sought[1] using interferometric gyroscopes based on light[2] or matter[3] waves. Here, we summarize some results from Ref. \refcitemoseley with a focus on light-based systems, which consist of beams traveling around a closed path in opposite directions. Effects that break the symmetry of the counter-propagating beams are encoded in their interference. The largest such effect routinely observed, the Sagnac effect, is due to rotation, which generates a beat frequency $\nu_{\rm s}=\frac{4A\vec{\Omega}\cdot{\hat{n}}}{P\lambda},$ where $A$ is the area enclosed, $\vec{\Omega}$ is the angular velocity, $\hat{n}$ is the vector normal to the loop, $P$ is its perimeter, and $\lambda$ is the wavelength of the light. A fixed system on Earth will experience several effects that alter the beat frequency, including the Sagnac effect of Earth’s rotation at angular frequency $\omega$ and the general-relativistic frame-dragging effect. Lorentz violation as described by the SME[4, 5] can also break the symmetry.

The contributions to a post-Newtonian expansion of the metric $g_{\mu\nu}$ in the SME[6] that are relevant for our analysis take the form

[TABLE]

where $U$ is the Newtonian potential, $\chi$ is the superpotential,[6] $U^{jk}$ is an additional post-Newtonian potential,[6] and $\overline{s}{}^{\mu\nu}$ is the $d=4$ coefficient for Lorentz violation that provides the relevant minimal effects. The $d=5$ effects are contained in ${\hat{Q}}^{j}=[q^{(5)0jk0l0m}+q^{(5)n0knljm}+q^{(5)njknl0m}]\partial_{k}\partial_{l}\partial_{m}$ where $q^{(5)\mu\rho\alpha\nu\beta\sigma\gamma}$ is the coefficient for Lorentz violation.[7]

2 Measuring Lorentz violation

The beat frequency measured by a light-based gyroscope[8] is related to the proper time difference in the lab $\Delta\tau$ between the paths taken by the counter-propagating beams via $\nu=\frac{\Delta\tau}{\lambda P}$, and at leading order in Lorentz violation we find in our post-Newtonian expansion:

[TABLE]

where the integral is taken around the closed interferometer loop. By analogy with Ampère’s law, we transform this line integral into an integration over area, which simplifies into a product. In order to derive a general result that is valid for any Earth-based laboratory and for any ring orientation within that laboratory, we use the angles $\theta$, $\phi$, $\alpha$, and $\beta$ shown in Fig. 1. Given $d=4$ coefficients for Lorentz violation, we find a beat frequency[1]

[TABLE]

where $M$ and $R$ are Earth’s mass and radius, respectively. The small terms suppressed by an Earth-revolution boost factor have been omitted. A sample special case of this result is found in Ref. \refcitens. For $d=5$ coefficients, we find a beat frequency with a similar form (omitted here for brevity), which can be written in terms of the 15 canonical coefficient combinations written as components of $K_{JKLM}$.[10] The angle $\phi$ varies at the sidereal frequency, which indicates that both dimension four and five coefficients for Lorentz violation will produce signals with this time dependence.

3 Experiments

A number of planned or ongoing experiments may be of interest in the context of this work. For example, the Gyroscopes IN GEneral Relativity (GINGER) experiment, which is designed to measure the de Sitter and Lense–Thirring effects of General Relativity,[11, 8] expects to obtain sensitivities to the angular velocity of the Earth via the Sagnac effect beyond the part in $10^{9}$ level reaching perhaps a few parts in $10^{12}$.[11] It is possible to generate crude estimates of the sensitivity to Lorentz violation that might be expected using these goals and the results outlined in Sec. 2. We find $\overline{s}{}^{TJ}\approx\frac{\epsilon\omega cR^{2}}{GM}\approx 10^{-6}$ and $K_{JKLM}\approx\frac{\epsilon\omega cR^{3}}{GM}\approx 10\,$m in SI units with $\epsilon=10^{-9}$ as the fractional sensitivity to $\omega$ and $c$ as the speed of light. For $\bar{s}^{TJ}$, these sensitivities are competitive with other laboratory experiments,[12] while for dimension five coefficients, these sensitivities are competitive with the best existing measurements from binary pulsars.[10]

Acknowledgments

The authors gratefully acknowledge financial support as follows: S.M. from Carleton Summer Science Fellows, N.S. from the St. Olaf CURI fund.