Persistent gravitational wave observables: General framework

The gravitational wave memory effect is characterized by the permanent relative displacement of a pair of initially comoving test particles that is caused by the passage of a burst of gravitational waves. Recent research on this effect has clarified the physical origin and the interpretation of this gravitational phenomenon in terms of conserved charges at null infinity and “soft theorems.” In this paper, we describe a more general class of effects than the gravitational wave memory that are not necessarily associated with these charges and soft theorems, but that are, in principle, measurable. We shall refer to these effects as persistent gravitational wave observables. These observables vanish in nonradiative regions of a spacetime, and their effects “persist” after a region of spacetime which is radiating. We give three examples of such persistent observables, as well as general techniques to calculate them. These examples, for simplicity, restrict the class of nonradiative regions to those which are exactly flat. Our first example is a generalization of geodesic deviation that allows for arbitrary acceleration. The second example is a holonomy observable, which is defined in terms of a closed loop. It contains the usual “displacement” gravitational wave memory; three previously identified, though less well known memory effects (the proper time, velocity, and rotation memories); and additional new observables. Finally, the third example we give is an explicit procedure by which an observer could measure a persistent effect using a spinning test particle. We briefly discuss the ability of gravitational wave detectors (such as LIGO and Virgo) to measure these observables.

Figure 1

Two curves that have some initial separation ${\xi }^a$ and relative velocity ${\dot{\xi }}^a$, as well as accelerations ${\ddot{\gamma }}^{a^{\prime \prime }}$ and ${\ddot{\overline{\gamma }}}^{{\overline{a}}^{\prime \prime }}$. The curve deviation observable $\mathrm{\Delta }{\xi }_{\mathrm{CD}}^{a^{\prime }}$ is given by the difference between the measured final separation and the final separation that would be predicted if the gray region (containing gravitational waves) were also flat.

Figure 2

A loop about which observers can compute a holonomy that measures the effect of a burst of gravitational waves. The quantities $P^a$ and $J^{ab}$ are transported around this loop using Eq. (2.15) (with some set of parameters $\varkappa$) in the directions shown, thereby yielding the observables $\stackrel{\varkappa }{P}{}^a$ and $\stackrel{\varkappa }{J}{}^{ab}$.

Figure 3

An observer (left curve, $\gamma$) measuring properties of a spinning test particle (right curve, $\overline{\gamma }$). The test particle has some measured separation ${\xi }^a$, linear momentum $p^a$, and intrinsic spin $s^a$ before a burst of gravitational waves. After the burst, these quantities are all measured again and compared with their values before the burst, yielding $\mathrm{\Delta }{\xi }^{a^{\prime }}$, $\mathrm{\Delta }{p}^{a^{\prime }}$, and $\mathrm{\Delta }{s}^{a^{\prime }}$.

Figure 4

A nongeodesic triangle (generalizing Fig. 3 of [58]), where the two sides $\gamma$ and $\overline{\gamma }$ are arbitrary curves (with affine parameter lengths $\epsilon$ and $\overline{\epsilon }$), and where the third side is formed by joining the two end points by the unique geodesic extending between them.

Figure 5

A nongeodesic square, with two sides $\gamma$ and $\overline{\gamma }$ that are arbitrary curves (with equal affine parameter length $\epsilon$), and where the other two sides are the unique geodesics between the two initial and final points, respectively. A third unique geodesic forms the diagonal. We denote the tangents (normalized such that the total affine parameter lengths are 1) to these unique geodesics at $x$ by ${\xi }^a$ and ${\psi }^a$.

Figure 6

Two nearby worldlines $\gamma$ and $\overline{\gamma }$, joined by unique geodesics between the start and end points of $\gamma$ and $\overline{\gamma }$, respectively. The separation vector between the two worldlines is denoted by ${\xi }^a$.