Atomic clocks and gravitational waves as probes of non-metricity

TL;DR

The paper tackles testing spacetime non-metricity within Weyl geometry, focusing on vectorial non-metricity as a minimal extension of GR. It derives a gauge-invariant observable based on the Weyl field strength $F_{\mu\nu}$ and a path-dependent length transport relation, $\frac{L_1}{L_2} \simeq 1 - \frac{\alpha}{2} \iint_S F_{\mu\nu} dS^{\mu\nu}$, linking length ratios to a closed-surface flux. Gravitational waves are analyzed by including the Weyl field as a dynamical degree of freedom with its energy-momentum tensor, showing that linear vacuum GW do not directly source $\omega_\mu$, but the Weyl sector backreacts on GW propagation to produce an anomalous strain. Using current GW data and precision atomic clocks, the study places strong bounds on non-metricity, e.g., current LIGO constraints give $\alpha^2\bar{\omega}_0 \lesssim 10^{-69}$ GeV, with future detectors improving by several orders of magnitude. Overall, the work demonstrates the complementary power of clocks and gravitational waves to test non-metricity and motivates extensions to broader non-metric theories such as $f(Q)$ gravity.

Abstract

Non-metricity provides a natural extension of Riemannian geometry, yet its experimental signatures remain largely unexplored. In this work we investigate how spacetime non-metricity can be probed through high-precision observations, focusing on atomic clocks and gravitational waves as complementary tools. Working within Weyl geometry as a minimal realization of vectorial non-metricity, we formulate observable effects in a gauge-invariant manner and show that they are associated with path-dependent length transport governed by the Weyl field strength. We derive constraints from atomic-clock experiments and demonstrate that, although gravitational waves do not directly source the Weyl field at linear order, its dynamical contribution induces a backreaction on gravitational-wave propagation, leading to an anomalous strain. As a result, the absence of deviations from General Relativity in current gravitational-wave observations already places meaningful and strong constraints on dynamical non-metric degrees of freedom.