Precession Caused by Gravitational Waves

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 Laser Interferometer Gravitational Wave Observatory signal is estimated to be $\mathrm{\Phi }\sim {10}^{-35}\mathrm{arc}\sec$ near Earth, but the effect may be substantially larger for supermassive black hole mergers.

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. (1)] 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 the last subsection for a more detailed discussion of this plot in the gyroscopic context.

Figure 3

In Bondi coordinates, the most natural tetrad $\{{\mathit{e}}_{\widehat{\mu }}\}$ is a source-oriented one (with a radial vector ${\mathit{e}}_{\widehat{r}}$ aligned with outgoing null geodesics). Converting such a frame into a tetrad $\{{\mathit{f}}_{\widehat{\mu }}\}$ pointing toward fixed distant stars requires a local rotation $R(\theta )$, as in Eq. (7).