Higher order analysis of the gravitational wave velocity memory effect between two free-falling gyroscopes in the plane wave spacetime

TL;DR

This paper analyzes the velocity memory effect caused by gravitational waves in plane wave spacetimes, revealing second-order contributions and quantifying the angular deviation between free-falling gyroscopes.

Contribution

It provides a higher-order analysis of the gravitational wave velocity memory effect, including polarization effects and contributions from initial conditions.

Findings

Second-order terms contribute to the velocity memory effect.

Angular deviation can reach 10^{-16} radians in black hole mergers.

Both polarization modes influence the memory effect.

Abstract

In the plane wave spacetime, when gravitational waves pass by, an angular deviation exists between two free-falling gyroscopes, which naturally corresponds to the velocity memory effect. In the shackwave spacetime background, the angular deviation between two free-falling gyroscopes is calculated, which is also found to correspond to the velocity memory effect. In the plane wave spacetime, with linear polarization taken into account, no contribution is made by the first-order terms of the initial separation distance \(L\) (or the initial separation velocity \(v_0\)), \(\bm{P}\) (or \(\bm{M}\)) of two free-falling gyroscopes to the velocity memory effect, while contributions are initiated from the second-order terms. Under certain circumstances, the second-order contribution of the initial separation distance \(L\) is of the same order of magnitude as the first-order contribution of the initial separation velocity \(v_0\). When both + polarization and \(\times\) polarization are taken into account, in the context of the merger of supermassive black holes and with the initial separation velocity approaching the speed of light, the order of magnitude of the angle is \(10^{-16}\) rads.