Gravitational Holonomy in Sagnac Interferometry

Technical Summary: Gravitational Holonomy in Sagnac Interferometry

Problem Statement
The paper investigates the influence of gravitational waves (GWs) on a Sagnac interferometer, a device consisting of two counter-propagating light beams traversing a closed spatial loop. While the standard Sagnac effect is well-understood as a phase shift arising from relative time delays (often linked to rotation or gravitomagnetic fields), the authors identify a previously underemphasized contribution: an asymmetric rotation of the polarization vectors of the counter-propagating beams. The central problem is to characterize this polarization rotation as a gravitational holonomy associated with the internal Lorentz group and to quantify its magnitude relative to the standard phase shift, particularly in the context of GWs generated by distant localized sources.

Methodology
The authors employ a multi-scale analysis and a perturbative approach within the framework of General Relativity:

1. Eikonal Approximation: Given the hierarchy of scales where the electromagnetic wavelength ($\lambda_e$) is much smaller than the interferometer size ($|X|$), which is in turn much smaller than the GW wavelength ($\lambda_g$) and the distance to the source ($r$), the authors utilize the eikonal approximation. This allows the electromagnetic field to be decomposed into a rapidly varying phase and a slowly varying amplitude and polarization vector.
2. Interference Formalism: The interference pattern is analyzed by calculating the cross-correlation of the electric fields of the two beams. The authors demonstrate that the observable signal depends on two distinct angles: the difference in eikonal phases ($\Delta S$) and the angle ($\psi$) between the polarization vectors upon recombination. By adjusting initial conditions (e.g., orthogonal phases or orthogonal polarizations), the setup can be made sensitive specifically to the polarization rotation.
3. Gravitational Holonomy: The evolution of the polarization vector is treated as parallel transport along null geodesics. The authors formulate the net change in polarization after traversing a closed spacetime loop as a holonomy of the internal Lorentz group, represented by a path-ordered exponential of the spin connection.
4. Coordinate Systems: To handle the local laboratory frame, the authors construct Fermi Normal Coordinates (FNC) for observers. They analyze two specific observer types in the Bondi-Sachs formalism (describing asymptotically flat spacetimes with GWs):
   • A static (accelerated) observer.
   • A freely falling observer.
5. Perturbative Expansion: Calculations are performed to leading order in the inverse distance ($O(r^{-1})$) from the GW source. The metric is expanded in terms of the Bondi news tensor ($N_{AB}$) and its derivatives, utilizing the hierarchy $\lambda_e \ll |X| \ll \lambda_g \ll r$ to simplify the FNC metric and spin connections.

Key Contributions and Results

• Identification of Polarization Rotation: The paper identifies an additional effect beyond the standard time-delay-induced phase shift. Gravitational waves induce a relative rotation in the polarization vectors of counter-propagating beams. This effect is formulated as a gravitational holonomy.
• Frequency Independence: A crucial distinction is drawn between the two effects: the standard Sagnac phase shift is proportional to the light frequency ($\omega$), whereas the polarization rotation (holonomy) is frequency-independent.
• Behavior of Observers:
  • Static Observers: For observers held at fixed spatial coordinates (accelerated), both the time-delay phase shift and the polarization rotation are present.
  • Freely Falling Observers: For observers in free fall, the authors derive that the standard time-delay phase shift vanishes to leading order. Consequently, the polarization rotation becomes the dominant observable Sagnac effect for such observers.
• Gauge Invariance: The authors demonstrate that the relative polarization rotation is a gauge-invariant quantity, invariant under local Lorentz transformations of the tetrad, reinforcing its status as a physical observable.
• Explicit Computation: The magnitude of both effects is computed for GWs from a localized source in Bondi-Sachs spacetime. The results are expressed in terms of the Bondi news tensor and the geometry of the interferometer loop.

Significance and Claims
The authors position this work as a systematic study of gravitational memory effects within a Sagnac interferometer setup. Their primary claims are:

1. Theoretical Importance: The polarization rotation encodes a gauge-invariant holonomy of the internal Lorentz group. This provides a new observable channel for studying gravitational effects, distinct from the standard phase shift.
2. Memory Effects: The work connects the Sagnac effect to the broader class of gravitational memory effects (specifically spin memory and gyroscopic memory), offering a setup where these observables can be studied systematically.
3. Dominance in Free Fall: The finding that the polarization rotation dominates for freely falling observers (where the time delay vanishes) suggests that this effect could be the primary signature of GWs in specific experimental configurations involving inertial observers.
4. Proof of Principle: The paper serves as a proof of principle, computing these effects to leading order in asymptotically flat spacetimes. It does not propose a specific experimental realization but establishes the theoretical framework and magnitude of the effects, noting that while the polarization effect is subdominant in typical high-frequency GW scenarios compared to the phase shift, its unique frequency-independent nature makes it a distinct theoretical signature.

The paper concludes that the Sagnac interferometer, with appropriate initial conditions, can function as a detector for gravitational holonomy, providing a new perspective on the interaction between gravitational waves and electromagnetic propagation.