Gravitational Holonomy in Sagnac Interferometry

We analyze the influence of gravitational waves on a Sagnac interferometer formed by the interference of two counter-propagating beams traversing a closed spatial loop. In addition to the well-known Sagnac phase shift, we identify an additional contribution originating from a relative rotation in the polarization vectors. We formulate this effect as a gravitational holonomy associated to the internal Lorentz group. The magnitude of both effects is computed due to gravitational waves generated by a localized source far from the detector, at leading order in the inverse distance. For freely falling observers, the phase shift is zero and the polarization rotation becomes the dominant effect.

💡 Research Summary

The paper investigates how gravitational waves (GWs) affect a Sagnac interferometer, focusing on two distinct physical effects: the conventional Sagnac phase shift and an additional rotation of the light’s polarization vectors. The authors formulate both effects within a unified framework as a gravitational holonomy of the internal Lorentz group, and they compute the magnitude of each contribution for a distant GW source, keeping terms up to order 1/r in the Bondi–Sachs expansion.

The analysis begins with the eikonal (geometrical optics) approximation for electromagnetic waves propagating on a curved spacetime. The vector potential is written as a rapidly varying phase S(x) multiplied by a slowly varying amplitude a(x) and a polarization vector fμ(x). In the short‑wavelength limit (ε → 0) the Maxwell equations reduce to the null condition kμkμ = 0 (null geodesics) and the parallel‑transport equation kν∇νfμ = 0 for the polarization. To isolate the physical two transverse degrees of freedom, the authors introduce an auxiliary null vector nμ satisfying k·n = –1 and define the transverse polarization pμ = (δμν + kμnν)fν. This construction is gauge‑invariant and yields a unit‑norm polarization that is orthogonal to the propagation direction.

Next, the paper describes a Sagnac interferometer in which two counter‑propagating beams travel the same closed spatial loop and are recombined. Using the eikonal fields, the electric and magnetic fields of each beam are expressed in terms of the transverse polarization p and the wavevector k̂. The total intensity measured at the detector contains three terms: the intensities of the individual beams and an interference term proportional to cosΔS·cosψ, where ΔS = S1 – S2 is the difference of the eikonal phases and ψ is the angle between the two polarization vectors (cosψ = p1·p2). Thus the interferometer is simultaneously sensitive to a phase shift (ΔS) and a polarization rotation (ψ).

The authors emphasize that the initial preparation of the beams can be chosen to enhance sensitivity to either effect. For example, setting an initial relative phase of π/2 while keeping the polarizations aligned makes the device primarily a phase detector, whereas preparing orthogonal polarizations with zero initial phase makes it a pure polarization‑rotation detector. This flexibility is crucial for isolating the new effect.

To evaluate the GW‑induced contributions, the authors work in Bondi–Sachs coordinates, expanding the metric to O(1/r) for a source far from the detector. They consider two classes of observers: a static observer (with a timelike Killing vector at infinity) and a freely falling observer. For the static observer the off‑diagonal g0i components generate a conventional Sagnac time delay, leading to a non‑zero ΔS proportional to the GW strain integrated along the loop. In contrast, for a freely falling observer the null geodesics experience no net time delay (kν∇νkμ = 0), so ΔS vanishes at leading order. However, the spin connection ωabμ, which governs the parallel transport of the polarization, does acquire a non‑trivial holonomy. This holonomy manifests as a net rotation ψ of the polarization after one circuit, independent of the light frequency and proportional to the GW “spin memory” – a permanent change in the asymptotic shear of spacetime.

The polarization rotation is frequency‑independent, opposite in sign for right‑ and left‑handed circular polarizations, and thus can be distinguished from the phase shift by separating the two helicities before recombination. The authors also discuss practical implementation: optical fibers can preserve polarization over long distances, allowing the construction of a fiber‑based Sagnac loop. By measuring the relative orientation of the output polarizations (e.g., with a polarimeter), one could detect the GW‑induced spin memory directly.

In summary, the paper demonstrates that a Sagnac interferometer, when appropriately configured, can serve as a detector of gravitational holonomy – a manifestation of the GW spin memory effect – in addition to the well‑known Sagnac time‑delay. For freely falling observers the polarization rotation dominates, providing a novel, gauge‑invariant observable of gravitational radiation that complements existing interferometric GW detectors. This work opens a new avenue for optical detection of memory effects and highlights the deep connection between spacetime geometry, gauge theory, and polarization optics.