Atomic clocks and gravitational waves as probes of non-metricity

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.

💡 Research Summary

The paper investigates the phenomenology of spacetime non‑metricity within the framework of Weyl geometry, focusing on two high‑precision observational probes: atomic clocks and gravitational‑wave (GW) detectors. In Weyl geometry the covariant derivative of the metric obeys ∇̃ₗg_{μν}=−α ωₗ g_{μν}, where ω_μ is the Weyl gauge field and α is a dimensionless coupling to matter. This relation leads to a path‑dependent rescaling of physical lengths: a spatial ruler transported along a curve γ experiences a fractional change dL/L=−(α/2) ω_λ dx^λ. Integrating around a closed loop C gives ΔL/L≈α²∬S F{μν} dS^{μν}, where F_{μν}=∂_μω_ν−∂νω_μ is the gauge‑invariant Weyl field strength and the integral is over any surface S bounded by C. Thus the observable effect of non‑metricity is entirely encoded in the flux of F{μν} through the loop, independent of the detailed shape of the paths.

Atomic clocks act as physical rulers because an atomic transition frequency scales inversely with the characteristic atomic length. Consequently a fractional length change translates directly into a fractional frequency shift Δf/f=−ΔL/L. Two identical clocks that follow different trajectories in a region with non‑zero Weyl flux will exhibit a relative frequency offset given by the same expression Δf/f≈α²∬S F dS. Using the largest terrestrial loop (the Earth’s radius) the authors estimate the maximal Weyl flux as ∼GM⊕/(R_⊕c²)≈10⁻⁹ for a massless Weyl boson. Current state‑of‑the‑art optical lattice and ion clocks reach fractional uncertainties of 10⁻¹⁸–10⁻¹⁹, implying a bound α≲10⁻⁹–10⁻¹⁰ for a massless field. If the Weyl boson carries a mass m_ω, the field is Yukawa‑suppressed beyond the scale m_ω⁻¹. For m_ω≫10⁻⁷ m⁻¹ (≈eV) the suppression renders terrestrial clock experiments essentially blind, so only very light (or massless) Weyl fields can be constrained by this method.

The second probe is the propagation of gravitational waves. While the linearized GW perturbation h_{μν} does not source the Weyl field at first order, the energy‑momentum tensor of the dynamical Weyl sector contributes at second order to the wave equation. This back‑reaction modifies the effective strain measured by interferometers, adding a term proportional to the same Weyl flux: Δh≈α²∬_S F dS. Current LIGO/Virgo detectors achieve strain sensitivities of 10⁻²¹–10⁻²², comparable to the amplitudes of observed events (e.g., GW150914, GW170817). Matching the observed strains with the Weyl‑induced correction yields an independent bound on α of order 10⁻⁹ for a massless Weyl boson, consistent with the atomic‑clock limit. For massive Weyl fields with m_ω ≳ eV, the Yukawa suppression again eliminates any observable effect in present GW data, but future detectors with improved low‑frequency sensitivity could probe lighter massive modes.

Overall, the work provides a gauge‑invariant formulation of Weyl‑induced non‑metricity effects, links them to measurable quantities in two complementary arenas, and derives quantitative constraints on the Weyl coupling α and the mass scale m_ω. It demonstrates that static precision tests alone cannot fully explore non‑metricity; dynamical probes such as gravitational waves are essential for a comprehensive search. The paper thus bridges theoretical extensions of gravity with concrete experimental strategies, highlighting the potential of next‑generation atomic‑clock networks and GW observatories to either detect or tightly bound vectorial non‑metricity in our universe.