The Precession Caused by Gravitational Waves

TL;DR

This work shows that gravitational waves induce a permanent gyroscopic memory by causing freely falling gyroscopes to precess relative to distant stars. The authors formulate the problem in Bondi coordinates, construct a star-oriented tetrad, and derive the leading precession rate as governed by the dual covariant mass aspect, linking the effect to dual asymptotic symmetries. The orientation memory angle $\Phi$ decomposes into a spin-memory part plus an additional nonlinear term tied to gravitational electric-magnetic duality, and non-radiative spacetimes yield no memory. The results illuminate deep connections between gravitational memory, asymptotic symmetries, and dualities, and provide order-of-magnitude estimates suggesting observable effects may be feasible for extreme events like supermassive black hole mergers, with potential pulsar-based probes.

Abstract

We show that gravitational waves cause freely falling gyroscopes to precess relative to fixed distant stars, extending the stationary Lense-Thirring effect. The precession rate decays as the square of the inverse distance to the source, and is proportional to a suitable Noether current for dual asymptotic symmetries at null infinity. Integrating the rate over time yields a net rotation -- a `gyroscopic memory' -- whose angle reproduces the known spin memory effect but also contains an extra contribution due to the generator of gravitational electric-magnetic duality. The angle's order of magnitude for the first LIGO signal is estimated to be $Φ\sim 10^{-35}$ arcseconds near Earth, but the effect may be substantially larger for supermassive black hole mergers.

Figures (3)

• Figure 1: The world-line of a freely falling observer with a gyroscope, represented here (in red) in a Penrose diagram of near-Minkowski space. The gyroscope's spin is parallel-transported along its trajectory, but the passage of gravitational waves causes its orientation to change relative to fixed distant stars. Bondi coordinates $(u,r,\theta^a)$ are included; the source of radiation is located at the origin $r=0$.
• Figure 2: A cartoon of the typical local metric perturbation caused by gravitational waves. Even after the end of the disturbance, some metric components (typically some function of the shear $C_{ab}$ in Eq. \ref{['s6']}) differ from their initial value by an amount that depends on the waveform. This net offset has potentially observable consequences; one of them is the gyroscopic memory described here. See section \ref{['se3b']} for a more detailed discussion of this plot in the gyroscopic context.
• Figure 3: In Bondi coordinates, the most natural tetrad $\{{\boldsymbol{e}_{\hat{\mu}}}\}$ is a source-oriented one (with a radial vector ${\boldsymbol{e}_{\hat{r}}}$ aligned with outgoing null geodesics). Converting such a frame into a tetrad $\{{\boldsymbol{f}_{\hat{\mu}}}\}$ pointing towards fixed distant stars requires a local rotation $R(\theta)$, as in Eq. \ref{['s9']}.