First-order velocity memory effect from compact binary coalescing sources

It has long been known that gravitational waves from compact binary coalescing sources are responsible for a first-order displacement memory effect experienced by a pair of freely falling test masses. This constant displacement is sourced from the nonvanishing final gravitational-wave strain present in the wave’s after zone, often referred to as the nonlinear memory effect, and is of the same order of magnitude as the strain from the outgoing quadrupole radiation. Hence, this prediction of general relativity is verifiable experimentally by measurement of the final relative separation between test masses that comprise gravitational-wave detectors. In a separate context, independent calculations have demonstrated that exact, sandwich, plane-wave spacetimes exhibit a velocity memory effect; a nonzero relative velocity, gained by a pair of test masses in free fall, after the passage of a gravitational wave. In this paper, we find that in addition to the known constant displacement memory effect test masses experience, a velocity memory effect at leading order arises due to the nonlinear nature of gravitational waves from compact binary sources. We discuss the magnitude of the first-order velocity memory effect in the context of observing gravitational-wave radiation from super massive binary black hole mergers in LISA.

Figure 1

This plot summarizes the results of numerically integrating the Brinkmann equations of motion versus approximating the equations of motion with perturbation theory. The red dotted curve (approximate) shows how much the $\mathcal{O}(h^2)$ energy density term contributes to the total velocity memory effect at a fixed amplitude of the sandwich wave. The solid blue curve (exact) is calculated from numerically integrating the metric, which includes all terms. We begin to see agreement between the two curves as the amplitude decreases below $A_s\sim 0.5{\sec }^{-2}$, implying that for weaker gravitational sandwich waves, the velocity memory is well approximated by just the energy density term. Higher-order terms are responsible for the disagreement between the two curves for larger amplitudes. For reference, the sandwhich wave studied in [46] had an amplitude of $A_s=1{\sec }^{-2}$.

Figure 2

This diagram summarizes the orientation convention used to integrate over the solid angle in the memory equation. We align the Earth along the CBC source’s $z^{\prime }$ axis. $\overrightarrow{L}$ is the direction of angular momentum of the CBC and is perpendicular to the plane of rotation.

Figure 3

A schematic diagram showing the coordinate transformations between the CBCs source frame and the following set of plane-wave spacetimes: Brinkmann, BJR, Local Lorentz, and transverse traceless. The explicit coordinate transformations are given in the four boxes to the upper left. Dashed lines correspond to approximate transformations, solid lines correspond to exact transformations and the dotted line corresponds to a transformation which can be characterized exactly, but in practice, cannot be implemented exactly. This diagram helps visualize the rich coordinate relations between plane-wave spacetimes.