Gravitational Holonomy in Sagnac Interferometry

TL;DR

This work analyzes how gravitational waves influence Sagnac interferometry by deriving two observable effects: the conventional Sagnac time delay and a novel polarization-rotation holonomy arising from gravitomagnetic aspects of the GW field. Using the eikonal approximation for Maxwell theory and Fermi normal coordinates anchored to Bondi–Sachs spacetimes, the authors compute leading $O(r^{-1})$ contributions to both effects for distant GW sources, showing that static observers experience a measurable phase shift while freely falling observers exhibit vanishing leading-order phase delay with polarization rotation dominating. The polarization rotation is shown to be gauge-invariant and frequency-independent, offering a distinctive signature of gravitational holonomy, whereas the phase shift scales with the light frequency, enabling separation from noise. The study also discusses GW memory effects, showing potential amplification for multiple-loop interferometry and highlighting the framework’s relevance for probing GW memory and gravitomagnetic phenomena with closed-loop interferometers and optical-fiber realizations.

Abstract

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.

Figures (4)

• Figure 1: Vast hierarchy of length scales appearing in this problem.
• Figure 2: Schematic figure of the Sagnac interferometer. A laser beam enters the beam splitter (BS) and is split into two beams, which travel in a closed spatial path in opposite directions. The path is guided by several mirrors. At the end of the trip, the recombine through the beam splitter and interfere on the detector's screen.
• Figure 3: Two light rays propagate along null worldlines $\gamma_1,\gamma_2$, starting at spacetime event $A$, and recombine at event $B$. The joint curve $\Gamma=\gamma_1\circ -\gamma_2$ is the closed spacetime curve starting from A, going forward in time along $\gamma_1$ and returning back to $A$ along $\gamma_2$.
• Figure 4: Orthonormal frame of the observer, adapted to outgoing light rays. The Sagnac loop is centered on the observer and has an area vector $\vec{\sigma}$.