The Memory Effect for Plane Gravitational Waves

We give an account of the gravitational memory effect in the presence of the exact plane wave solution of Einstein's vacuum equations. This allows an elementary but exact description of the soft gravitons and how their presence may be detected by observing the motion of freely falling particles. The theorem of Bondi and Pirani on caustics (for which we present a new proof) implies that the asymptotic relative velocity is constant but not zero, in contradiction with the permanent displacement claimed by Zel'dovich and Polnarev. A non-vanishing asymptotic relative velocity might be used to detect gravitational waves through the"velocity memory effect", considered by Braginsky, Thorne, Grishchuk, and Polnarev.


I. INTRODUCTION
By "gravitational memory effect" is meant that a system of freely falling particles (viewed as detectors) initially at relative rest are displaced after the passing of a burst of gravitational radiation [1,2]. The memory effect has risen to prominence recently because of the possibility of its observation using LISA [3,4] and/or aLIGO [5], and is also relevant to recent work by Hawking, Perry, and Strominger [6] on soft gravitons and black holes.
The original proposal of Zel'dovich and Polnarev [1] claimed that detectors originally at rest will be simply displaced and will have vanishingly small relative velocity, in contrast with later findings of Bondi and Pirani [7]. A "velocity memory effect" highlighted by nonvanishing asymptotic relative velocity was considered by Braginsky, Grishchuk, Thorne, and Polnarev [2,8,9] and more recently, by Lasenby [4].
The analysis of Braginsky and Grishchuk [2], who also coined the expression "memory effect", is concerned with the motion of test masses moving in a weak gravitational wave at the linear level. Non-linear generalizations were considered in [10][11][12][13]. In the neighborhood of a detector, far from the source, we may approximate the wave by an exact plane wave.
Carroll symmetry [14,15] provides us with a description of the trajectories, leading to a new proof of the Bondi-Pirani theorem [7]. Our results might have some relevance for detecting gravitational waves through the "velocity memory effect" [2,4,8,9,16].

II. PLANE GRAVITATIONAL WAVES
Plane gravitational waves are often described in Brinkmann Coordinates (B) [17,18] by the metric where the symmetric and traceless 2 × 2 matrix K(U ) = (K ij (U )) characterizes the profile of the wave. In this Letter we only consider linearly polarized waves with where A(U ) is an arbitrary function. Our particular interest lies in "sandwich waves" i.e.
such that K(U ) = 0 only for The metric (II.1) is an example of a "Bargmann" space which provides a general framework for describing non-relativistic dynamical systems : any natural non-relativistic system with an n dimensional configuration space may be "Eisenhart-lifted" to a system of null geodesics in an (n + 2)-dimensional Lorentzian spacetime endowed with a covariantly constant null Killing vector field ∂ V [19,20]. From the Bargmann point of view, (II.1)-(II.2) describe a non-relativistic particle subjected to an (attractive or repulsive) harmonic anisotropic oscillator potential, (II.2).
Returning to gravitational radiation and following the suggestion of [21], the wave produced by gravitational collapse might be modelled by the (approximate) sandwich wave, shown in fig.1.
In a (naive) model the "detector" could consist of freely falling particles initially at relative rest. This applies to the mirrors of the LIGO interferometers of those attached to the three satellites making up the proposed LISA detector [4].
The profile function in the geodesic equations 2 , acts as an effective potential which is attractive or repulsive depending on the sign of A.
Putting χ = det a 1 4 > 0 and ω 2 (u) = 1 8 Tr ((γ −1γ ) 2 ) where γ = χ −2 a, the vacuum Einstein equations reduce to the Sturm-Liouville equation see eqn (II.22) of [16]. Then (II.6) implies thatχ < 0, which in turn implies the vanishing of χ for some u 1 > u i , signalling a singularity of the metric (II.5) and illustrated by fig.2. Plane gravitational waves have a 5-parameter isometry group [7,14,18] identified as the Carroll group in 2 + 1 dimensions with broken rotations [15] and implemented in BJR coordinates as follows : u is fixed, and where H is the symmetric 2 × 2 matrix calculated using the transverse-space matrix, Then Noether's theorem provides us with 5 conserved quantities associated with these isometries [14,15], interpreted as conserved linear and boost-momentum, supplemented by m =u = 1. An extra constant of the motion is e = 1 2 g µνẋ µẋν . Geodesics are timelike, lightlike, or spacelike if e is negative, zero, or positive, respectively. The geodesics are thus given by [14,15] x(u) = H(u) p + k, with d a constant. The trajectories can be determined once the matrix-valued function H(u) in (II.9) had been calculated.
The relation between the two coordinates systems is given by [15,24] where U = u 3 , a(u) = P (u) T P (u) and Now we consider a sandwich wave. In the "beforezone" U < U i K = 0 and therefore eqn.
(II.13) can be solved by P (u) = 1 so that the Brinkmann and BJR coordinates coincide.
The crucial fact for us is that, by (II.10), the momentum of a particle at rest vanishes, p = 0 for u = U ≤ U i -and thus for all u = U including the "inside zone" U i ≤ U ≤ U f and the "afterzone" U f ≤ U , because p is conserved. By (II.11a) such geodesics are, for any H i.e. for any transverse metric a and therefore for any profile A, simply Thus the particles initially at rest are, when described in BJR coordinates, undisturbed by the passage of the wave 4 .
In Brinkmann coordinates both the waves and the geodesics are global with no singularity whatsoever. Solving the equations (II.13) numerically for P yields, cf. (II.12). Those complicated-looking trajectories in B coordinates in Fig.1 are recovered: the plots overlap perfectly up to the point where the BJR coordinate system becomes singular ; all the curling comes from P (u).
Our plots show that, consistently with the theorem of Bondi and Pirani [7], after the wave has passed the worldlines will first be focused in a finite and initial-position-independent time, and then diverge with constant but non-zero asymptotic velocity -contradicting the claim of Zel'dovich and Polnarev [1].
We can actually prove the Bondi-Pirani theorem independently using our framework.
First we note that A ≈ 0 for large values of U ; then it follows from the geodesic equations (II.4) that the acceleration vanishes, implying constant asymptotic velocity 5 .
The geodesics we are interested in are of the form (II.15); in the beforezone U ≤ U i we have P = 1 and the geodesics are at rest, X(U ) = x 0 . Let us now consider two such special geodesics, X a (U ) and X b (U ); we show that there exist x 0 a = x 0 b such that the two geodesics meet, X a (U 1 ) = X b (U 1 ) for some finite and universal value x 0 = 0, which happens when P (u 1 ) is a singular matrix, det(P (u 1 )) = 0. (II.16) But a = P T P so that det a = (det P ) 2 = χ 4 and therefore det P = 0 iff χ = 0. However, as recalled above, (II.6) implies that χ necessarily vanishes at some u 1 > u i -at the same value where the BJR coordinates become singular. This is not an accident: the singular BJR coordinate system can only be regularized by a singular transformation. Thus P (u 1 ) has a kernel and all geodesics for which x 0 a − x 0 b belongs to the kernel will meet at the same u 1 . If x 0 a − x 0 b does not belong to the kernel, then we get a caustic. For a linearly polarized wave as in (II.1)-(II.2), P = diag(P 11 , P 22 ) and the x 1 resp x 2 coordinates meet at critical (caustic) point when either P 11 or P 22 vanishes, which is where BJR coordinates become singular, cf. fig. 1.
In the flat afterzone u ≥ u f , in particular, we have the exact analytic solution [14,16]

III. ILLUSTRATION BY TISSOT DIAGRAM
The action of gravitational waves on a ring of freely falling particles can be illustrated by a series of time-frames showing how the ring is squashed and stretched as the wave passes over it [25]. An initially (i.e. before the pulse) circular disc of geodesics, X · X ≤ 1 for U < U i , projects to a time-independent circle, x · x ≤ 1, which extends to all u, i.e., during and after the sandwich, u ≥ u i . However their inverse images in Brinkmann coordinates are time dependent ellipses, as shown in Fig. 4.

FIG. 4:
Tissot diagram for the "collapse profile" (II.3). The initial circle is deformed to an ellipse which at u = u 1 degenerates to a line segment, and so on.

IV. SOFT GRAVITONS
We end with a description of the flat spacetime after the burst as being dressed by soft gravitons, a concept which plays a central role in the considerations of [6]. Before the burst a ij = δ ij , and the BJR coordinates coincide with standard inertial coordinates in Minkowski spacetime. After the burst a ij will be u-dependent but nevertheless will have vanishing curvature. Thus there exist coordinatesû,v,x such that the 2-metric isâ ij = δ ij . The transformation relating u, v, x to theû,v,x coordinates is given by (II.12) withP = 0, as follows from the vanishing of K. In the quantum field theory language adopted by Hawking et al. in [6], the coordinate transformation is a "gauge transformation which does not tend to the identity at infinity". In quantum field theory it is customary to distinguish states which so differ and regard them as containing "soft" [i.e. zero-energy] quanta. In the present case the quanta are gravitons given by pp-waves, whose metric is given by (II.5).

V. CONCLUSION
We found that test particles initially at rest move, after the wave has passed, with constant but non-zero relative velocity, contradicting Zel'dovich and Polnarev [1] but consistently with Bondi and Pirani [7], whose theorem we proved independently. The non-vanishing asymptotic relative velocity might allow to observe the velocity memory effect by improved Doppler tracking [2,8,9] 6 .
The intuitive explanation is that (II.1)-(II.2) is (as said above), the Bargmann metric of an anisotropic oscillator with time-dependent frequencies, one of which is attractive and the other repulsive. The components in the repulsive sector are pushed apart whereas in the attractive sector they are focused and then meet where the matrix P is singular.