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# Displacement within velocity effect in gravitational wave memory

P.-M Zhang, Peter Horvathy

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P.-M Zhang, Peter Horvathy. Displacement within velocity effect in gravitational wave memory. Annals of Physics, 2024, 470, pp.169784. 10.1016/j.aop.2024.169784. hal-04600713v2

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Distributed under a Creative Commons Attribution 4.0 International License
Displacement within velocity effect

in

gravitational wave memory*

P. M. Zhang¹†, P. A. Horvathy²,³‡,

¹ *School of Physics and Astronomy,*
*Sun Yat-sen University, Zhuhai, China*

² Institut Denis-Poisson CNRS/UMR 7013 - Université
de Tours - Université d'Orléans Parc de Grammont,
37200; Tours, FRANCE

³ Erwin Schrödinger Institute,
Vienna (Austria)

(Dated: May 14, 2025)

<footnote>* Extended version of the talks given by PAH at the Wigner Institute in Budapest (April 5 2024), at the Workshop *Carrollian Physics and Holography_CDFG_2024* at the Schrödinger Institute in Vienna (April 17 2024), and at the conference "*Conformal anomalies: theory and applications 2024*, https://indico.math.cnrs.fr/event/10718/ in Tours (May 7 2024).</footnote>

<footnote>† mailto:zhangpm5@mail.sysu.edu.cn</footnote>

<footnote>‡ mailto:horvathy@univ-tours.fr</footnote>
Abstract

Particles initially at rest hit by a passing sandwich gravitational wave exhibit, in general, the *velocity memory effect* (VM): they fly apart with constant velocity. For specific values of the wave parameters their motion can however become pure *displacement* (DM) as suggested by Zel'dovich and Polnarev. For such a “miraculous” value, the particle trajectory is composed of an integer number of (approximate) standing half-waves. Our statements are illustrated numerically by a Gaussian, and analytically by the Pöschl-Teller profiles.

Annals of Physics **470** (2024) 169784 https://doi.org/10.1016/j.aop.2024.169784

PACS numbers: 04.20.-q Classical general relativity;

04.30.-w Gravitational waves
# Contents

<table><tr><td>I. Introduction</td><td>3</td></tr><tr><td>II. Memory effect</td><td>4</td></tr><tr><td>III. Gravitational Wave with Gaussian profile</td><td>6</td></tr><tr><td>IV. Gravitational Wave with Pöschl-Teller profile</td><td>8</td></tr><tr><td>V. Longitudinal motion</td><td>12</td></tr><tr><td>VI. Extension to the massive case</td><td>13</td></tr><tr><td>VII. Carroll-symmetry-related approach</td><td>16</td></tr><tr><td>VIII. Conclusion</td><td>22</td></tr><tr><td>Acknowledgments</td><td>25</td></tr><tr><td>References</td><td>25</td></tr><tr><td>A. Geodesic motion of a massive particle</td><td>29</td></tr></table>

# I. INTRODUCTION

The *Memory Effect*, which may be a way to detect gravitational waves (GW), has two versions. The *displacement effect* (DM) proposed by Zel'dovich and Polnarev [1] for flyby suggests that particles initially at rest are hit by a burst of gravitational wave, *although the distance between free bodies will change, their relative velocity will become vanishingly small as the flyby concludes* [1–3].

Confirmation could be obtained by taking into account non-linear effects [4–7].

Earlier studies [8–14] argued instead in favor of a *velocity effect* (VM): the particles would be scattered apart with *constant velocity* by a burst of gravitational wave. Our previous investigations [15–19] confirmed VM however casted a doubt on the claim of Zel'dovich and Polnarev. In this paper we show that while VM is generic, *pure displacement may* indeed
arise — however only for exceptional values of the parameters : the trajectories must be composed of an integer number of half-waves (reminiscent of quantum conditions).

We illustrate our statement by two closely related examples, one numerical and the other analytical. Their profile is (i) either a Gaussian, (III.1), or (ii) the Pöschl-Teller potential [20], (IV.1). Their study sheds some light on the remarkable relation of the velocity and the displacement effects, VM and DM, respectively. Our result is curiously related to zero-energy time-independent solutions of the Schrödinger equation.

In this paper we consider $D = 1$ transverse dimension. Our investigations will be extended to more physical profiles and dimensions appropriate for flyby, gravitational collapse, etc in a follow-up paper [21].

## II. MEMORY EFFECT

We consider a plane gravitational wave with $D = 1$ transverse dimension whose metric is, in Brinkmann (B) coordinates $(X^\mu) = (X, U, V)$,

$$g_{\mu\nu}dX^{\mu}dX^{\nu} = (dX)^2 + 2dUdV - \frac{1}{2}\mathcal{A}(U)X^2(dU)^2, \qquad (II.1)$$

where $X$ is space-like and $U, V$ are light-cone coordinates [22–25]<fnref n="1" />. The wave is assumed to be a sandwich wave i.e., one whose profile $\mathcal{A}(U)$ is zero in both the Beforezone $U < U_b$ and in the Afterzone $U > U_a$ and is non-vanishing only in a short Wavezone $U_b < U < U_a$ (see [13, 14, 19] for the terminology, recalled in FIG.1).

The geodesic motion is described by,

$$\frac{d^2X}{dU^2} + \frac{1}{2}\mathcal{A}X = 0, \qquad (II.2a)$$

$$\frac{d^2V}{dU^2} - \frac{1}{4}\frac{d\mathcal{A}}{dU}(X)^2 - \frac{1}{2}\mathcal{A}\frac{d(X^2)}{dU} = 0. \qquad (II.2b)$$

The spacelike coordinate $X$ is decoupled from the lightlike coordinate $V$ and the projection of a trajectory into transverse space is independent of $V$. Conversely, our geodesic is a lift of $X(U)$ determined by eqn. (II.2a) alone, with $U$ viewed as Newtonian time [24, 25].

<footnote n="1">¹ Our toy example is not a vacuum gravitational wave because the coefficient of $dU^2$ is not traceless. It is a mere pp wawe - but this has no importance for us here.</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 1: The sandwich wave propagates downwind. The space-time is flat both in the yet undisturbed Beforezone $U < U_b$ and in the Afterzone $U > U_a$.</figcaption></figure>

Equation (II.2a) describes free motion both in the Before and in the Afterzones where $\mathcal{A} = 0$, but not in the Wavezone $\mathcal{A} \neq 0$, where we have a Sturm-Liouville problem [14, 19]. We first study the motion in the transverse space; that of $V$ is postponed to sec. V.

Geodesics admit a Jacobi invariant $^2$, whose rôle will be highlighted in sec. VI. Discarding tachyons,

$$ \mathfrak{m}^2 = -g_{\mu\nu} \dot{X}^{\mu} \dot{X}^{\nu} = \text{const.} \geq 0. \qquad (II.3) $$

For a massive relativistic particle initially at rest (as it is assumed in the study of the Memory Effect) the transverse initial conditions are,

$$ X(U_0) = X_0, \quad \dot{X}(U_0) = 0 \quad \text{for } U_0 \le U_b. \qquad (II.4) $$

This paper is devoted to answer the question: *When do we get pure displacement in the Afterzone ?* i.e.,

$$ \dot{X}(U) = 0 \quad \text{for } U > U_a. \qquad (II.5) $$

We first consider massless particles, $\mathfrak{m}^2 = 0$; the extension to the massive case $\mathfrak{m}^2 > 0$ will be discussed in sect. VI.

<footnote n="2">² The Jacobi invariant is indeed a Casimir-invariant which has re-emerged recently [26] in the E-D framework [23–25] for massive geodesics. It is reminiscent of Souriau’s internal energy [27].</footnote>
### III. GRAVITATIONAL WAVE WITH GAUSSIAN PROFILE

Our first example has a Gaussian profile. By rescaling the lightlike coordinate $U$ we can achieve that the wave has unit width,

$$ \mathcal{A} \equiv \mathcal{A}^G(U) = \frac{k}{\sqrt{\pi}}e^{-U^2}, \qquad (III.1) $$

and is normalized as

$$ \int \mathcal{A}(U) dU = k. \qquad (III.2) $$

The amplitude $k$ is thus the area below the profile.

Earlier work [8–19] indicated that after the passing of the gravitational wave the particles fly apart with non-zero velocity: we have VM. However now we show that *fine-tuning the amplitude* $k$ can lead to (approximate) DM, as it will be illustrated by a series of figures. Numerical investigations indicate, for example, that for $k = k_{crit} = 9.51455$ we get a “half-jump”, shown in FIG.2.

This “miracle” is explained, intuitively, by that at outside the (approximate) Wavezone $U_b < U < U_a$ both the *velocity and the force* vanish, — whereas the motion is governed by Newton’s laws.

<figure><img src="image_4.png" /><figcaption>FIG. 2: *Fine-tuning the amplitude to $k = k_{crit}$ provides us with the “half-wave displacement memory effect” with $m = 1$ standing half-wave. $X$: trajectory, $dX/dU$: velocity, $d^2X/dU^2$: force.*</figcaption></figure>

FIG.3 confirms VM for $k \neq k_{crit}$, but show no DM: the velocity does not vanish (even approximately) in the Afterzone. For $k < k_{crit}$ the force falls off before the oscillator reaches its return point; for $k > k_{crit}$ it pulls instead the particle back after reaching it.
<figure><img src="image_5.png" /><figcaption>FIG. 3: (a) For $k < k_{crit}$ the trajectory undershoots and (b) for $k > k_{crit}$ it overshoots before being straightened out.</figcaption></figure>

For $k_{crit} = k_1$ above we found precisely trajectory consisting of $m = 1$ one half-wave and one may wonder if DM with several half-waves can also be accommodated. The answer, obtained again by fine-tuning, says that DM can indeed arise with higher amplitudes when the Wavezone accommodates an integer number of half-waves, as illustrated in FIG. 4.

<figure><img src="image_6.png" /><figcaption>FIG. 4: Fine-tuning the amplitude yields DM with $m = 2$ and $m = 3$ half-waves as trajectories. NB: the plots have different scales.</figcaption></figure>

Emboldened by this success, further fine-tuning yields DM for other magic amplitudes,

$$k_1 \approx 9.5, m = 1, \quad k_2 \approx 30.7, m = 2, \quad k_3 \approx 63.1, m = 3, \quad k_4 \approx 106.7, m = 4, \dots \quad (\text{III.3})$$

The outgoing position depends on the parity of $m$:

$$X_{out} = (-1)^m X_{in}. \qquad (\text{III.4})$$

Higher wave number requires higher amplitude. The relation between $\sqrt{k}$ and $m$, depicted
in FIG.5, is approximately linear,

$$ \sqrt{k_m} \approx 0.78 + 2.38m. \qquad (III.5) $$

<figure><img src="image_7.png" /><figcaption>FIG. 5: The relation between m, the number of half-waves in the trajectory in the Wavezone and $\sqrt{k_{crit}}$ for DM is approximately linear.</figcaption></figure>

Intuitively, the coefficient of $dU^2$ in (II.1) is the potential of an oscillator and the particle initially at rest at $X_0 \neq 0$ is pulled towards the origin, picking up some speed. However passing $X = 0$ the force changes direction and starts to reduce the velocity. If the profile was $U$-independent, — it would be a genuine harmonic oscillator — then the particle would oscillate between $X_0$ and $-X_0$ forever. However for sandwich waves with bell-shaped potentials the pull progressively falls off with increasing $U$ and after a while the motion becomes free. If the residual velocity is non-zero, then we have VM as in FIG. 3. However if the velocity is zero which happens after an integer number of half-oscillations, then the particle stops, — and then it does not restart anymore by virtue of Newton’s laws which are valid in the Afterzone: we get DM.

Higher amplitude $k$ means stronger force which then requires more back-and-forth oscillations i.e. a larger $m$ before stopping.

These arguments are largely independent of the concrete profile as long as it is roughly bell-shaped, as it will be illustrated on the analytic example discussed below in sec.IV.

## IV. GRAVITATIONAL WAVE WITH PÖSCHL-TELLER PROFILE

For Gaussian profile the Sturm-Liouville equation (II.2a) has no analytic solution. However the shape of $A$ is strongly reminiscent of the [symmetric part of the] Pöschl-Teller (PT)
potential [20], considered, independently, also in [28] and in [29],

$$A^{PT}(U) = \frac{k}{2 \cosh^2 U}, \qquad (IV.1)$$

depicted in FIG.6. $A^{PT}$ is normalized as the Gaussian, (III.1), $\int A^{PT}(U)dU = k$, cf. (III.2).

<figure><img src="image_8.png" /><figcaption>FIG. 6: The Gaussian bell (dashed) can be approximated by the Pöschl-Teller potential (IV.1) (solid line), which admits analytic solutions. The parameters were chosen so that the area below both profiles be identical and equal to k.</figcaption></figure>

It has the advantage that the trajectories can be found analytically. Writing

$$k = k_m = 4m(m + 1), \qquad (IV.2)$$

where $m$ is a priori a real number, eqn. (II.2a) becomes that of a damped oscillator whose frequency $\omega^2 = \frac{m(m+1)}{\cosh^2 U}$ decreases with $U$,

$$\frac{d^2 X}{dU^2} + \frac{m(m+1)}{\cosh^2 U} X = 0. \qquad (IV.3)$$

The initial conditions for a particle at rest before the burst arrives are,

$$X(U = -\infty) = X_0, \quad \text{and} \quad \dot{X}(U = -\infty) = 0. \qquad (IV.4)$$

Putting $t = \tanh(U)$ into (IV.3) the Legendre equation is obtained,

$$(1 - t^2) \frac{d^2 X}{dt^2} - 2t \frac{dX}{dt} + m(m+1) X = 0. \qquad (IV.5)$$

Then DM means that $X(U)$ tend to a constant for $U \to \infty$ which amounts to requiring that the solution of (IV.5) should extend to $t = \pm 1$ which in turn implies that $m$ must be a positive integer, and the solution becomes proportional to a Legendre polynomial,

$$X(U) = X_m(U) = (-1)^m P_m(\tanh U) X_0, \quad m = 1, 2, \dots, \qquad (IV.6)$$
shown in FIGs. 7-8-9 should be compared with (III.3) and with FIGs.2 and 4 for the Gaussian). The trajectories (IV.6) are composed of m half-waves, as for the Gaussian. The

<figure><img src="image_9.png" /><figcaption>FIG. 7: For the Pöschl-Teller profile with $k_{crit} = k_1 = 8$ i.e. m = 1, the transverse trajectory is consistent with DM (to be compared with FIG.2).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 8: The trajectory (IV.6) for the Pöschl-Teller profile with (a) $k_2 = 24$ and (b) $k_3 = 48$ have standing waves with wave numbers m = 2 resp. m = 3, cf. FIG.4.</figcaption></figure>

Pöschl-Teller counterpart of the $k_{crit} \Leftrightarrow m$ relation (III.5),

$$k_m = 4m(m+1) \Rightarrow \sqrt{k_m} \approx 2m+1, \qquad (IV.7)$$

reminiscent of (III.3) is shown in FIG.10 to be compared with FIG.5.

For $k \neq k_{crit}$ we get again VM but no DM, as it could be illustrated by plots similar to those in FIG.3. It is instructive to plot also the velocities, FIG.11, which confirms that the
<figure><img src="image_11.png" /><figcaption>FIG. 9: Transverse trajectories for the Pöschl-Teller profile with $m = 1, \dots, 5$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIG. 10: The relation between the number of half-wave trajectories in the Wavezone, m, and $\sqrt{k_{crit}}$ for DM is approximately linear, as it is for the Gaussian in FIG. 5.</figcaption></figure>

number of half-waves is equal to the number of zeros of $dX_m/dU$ in the wave zone<fnref n="3" />.

We conclude this section by pointing out a curious relation with quantum mechanics. Let us indeed consider the one dimensional time-independent Schrödinger equation for the Pöschl-Teller potential,

$$ -\frac{\hbar^2}{2M} \frac{d^2\psi}{dx^2} - \frac{\hbar^2 m(m+1)}{2M \cosh^2 x} \psi = E\psi \quad (IV.8) $$

where the negative sign was chosen to get bound states. Redefining the energy as $\varepsilon = \frac{2M}{\hbar^2} E$ then yields

$$ -\frac{d^2\psi}{dx^2} - \frac{m(m+1)}{\cosh^2 x} \psi = \varepsilon\psi. \quad (IV.9) $$

Substituting here $\psi \to X$, $x \to U$ and putting $\varepsilon = 0$ we recognize our equation (IV.3): our DM trajectories correspond to zero-energy bound states of the quantum problem, shedding some light at the curious quantization of wave numbers, m. Such zero-energy solutions are

<footnote n="3">³ An analytic proof can be found by using the properties of the Legendre polynomials, and in particular $P'_n(x) = n \frac{P_{n-1}(x)-xP_n(x)}{1-x^2}$.</footnote>
<figure><img src="image_13.png" /><figcaption>FIG. 11: The transverse velocities for the Pöschl-Teller profile (IV.1)-(IV.2), for DM amplitudes $k = k_m$, shown for $m = 1, 2, 3$.</figcaption></figure>

discarded in quantum mechanics because they are not normalizable – but this condition is not required for our trajectories and the zero-energy solutions in (IV.6) which are thus admissible.

V. LONGITUDINAL MOTION

Returning to the general setting, now we complete our study of the $m = 0$ case by extending DM to the “vertical” component $V(U)$. Let us recall that eqn. (II.2b) is obtained by lifting the transversal trajectory $X(U)$ to Bargmann space,

$$ \hat{V}(U) = V_0 - \mathcal{I}(U), \quad \mathcal{I}(U) = \int_{-\infty}^{U} \mathcal{L}_{NR} du, \qquad (V.1) $$

where $\mathcal{L}_{NR}$ is the Lagrangian of a non-relativistic particle in 1+1 dimensions which moves in a possibly time-dependent oscillator potential; $\mathcal{I}(U)$ is the classical Hamiltonian action of the underlying NR model calculated along $X(U)$. We have chosen $M = 1$ for simplicity.

DM in transverse space requires the initial and final conditions

$$ X'(U = -\infty) = 0 = X'(U = \infty). \qquad (V.2) $$

The integral $\mathcal{I}$ in (V.1) makes the theory a priori non-local. However using the equations of
motion (II.2a) a calculation shows that along a trajectory,

$$ 
\begin{aligned}
\int \mathcal{L}_{NR} du &= \frac{1}{2} \int X'^2 du - \frac{1}{4} \int \mathcal{A} X^2 du = \frac{1}{2} \int X' dX - \frac{1}{4} \int X (-2X'') du \\
&= \frac{1}{2} XX' \Big|_{-\infty}^{\infty} - \frac{1}{2} \int XX'' du + \frac{1}{2} \int XX'' du = \frac{1}{2} XX' \Big|_{-\infty}^{\infty}
\end{aligned}
$$

so that $\mathcal{I}$ is in fact local. Moreover, for the DM boundary conditions (V.2) it vanishes,

$$ \mathcal{I} = \int_{-\infty}^{U} \mathcal{L}_{NR} du = 0 \quad \text{for } U > U_a. \qquad (V.3) $$

The (possibly) non-local term is thus eliminated, leaving us with,

$$ \hat{V}(U) = V_0 \quad \text{both for } U < U_b \quad \text{and} \quad U > U_a \qquad (V.4) $$

which extends, for $m = 0$, DM from the transverse coordinate to $V$ (with no $V$-displacement as all as a bonus).

Non-trivial motion arises only in the Wavezone, as shown in FIG.12.

Similar plots could be obtained (numerically) for the Gaussian.

We underline that (V.3) is valid only when the domain of integration contains the Wave zone. In the Pöschl-Teller case, for example,

$$ 
\left\{
\begin{array}{ll}
m = 1 & \hat{V}(U) = V_0 \qquad -\frac{1}{2} \frac{\sinh(U)}{\cosh^3(U)} X_0^2 \\
m = 2 & \hat{V}(U) = V_0 \qquad -\left(\frac{3}{2} \frac{\sinh(U)}{\cosh^3(U)} - \frac{9}{4} \frac{\sinh(U)}{\cosh^5(U)}\right) X_0^2.
\end{array}
\right.
\qquad (V.5) $$

However the non-trivial terms fall off to zero outside the (approximate) Wavezone, as illustrated in FIG.12.

Our plots indicate that while DM requires that the transverse trajectory be composed of $m$ half-waves, the “vertical” $V$ has $m$ full waves.

## VI. EXTENSION TO THE MASSIVE CASE

Requiring zero incoming total velocity, (II.4), is however *unphysical* for a massless particle: one can not stop a photon. Hence the importance of extending our investigations to particles with nonzero Jacobi invariant, $m \neq 0$, which do not follow null geodesics [23, 26, 30].
<figure><img src="image_14.png" /><figcaption>FIG. 12: For the Pöschl-Teller profile with $k_{crit} = k_m$ shown for **m** = 1, 2 both the **transverse**, **X**, and the **vertical**, **V**, trajectories behave, in the massless case **m** = 0, consistently with DM. **NB:** the scales for (a) and (b) are different.</figcaption></figure>

The V-equation (II.2b) can actually be solved for $\mathbf{m}^2 \ge 0$ $^4$. To this end we rewrite it by inserting $X(U)$ from (II.2a), as

$$ \frac{d^2 V}{dU^2} = -\frac{d}{dU} \underbrace{\left( \frac{1}{2} \left( \frac{dX}{dU} \right)^2 - \frac{1}{4} \mathcal{A} X^2 \right)}_{\mathcal{L}_{NR}}, \qquad (\text{VI.1}) $$

where in the bracket we recognize again the non-relativistic Lagrangian, $\mathcal{L}_{NR}$. Integrating twice then yields, consistently with previous results, [23, 26, 30],

$$ V = \left\{ V_0 - \int \mathcal{L}_{NR} dU \right\} + V_c U = \hat{V} + V_c U, \qquad (\text{VI.2}) $$

where $V_0$ and $V_c$ are integration constants. The meaning of $V_0$ is obvious, however what is the physical role of $V_c$? To this end, we first emphasise that our theory admits in fact two, different mass parameters. One of them is the relativistic mass, $\mathbf{m}$, defined by the Jacobi invariant (II.3). The other one we denote by $M$ is the conserved quantity associated with the “vertical” Killing vector $\partial_V$ which plays, in the E-D framework, the role of mass in the underlying NR theory [23–25].

We note that for an affine parameter $\lambda$ the Jacobi invariant is $g_{\mu\nu}dx^\mu dx^\nu = -\mathbf{m}^2 d\lambda^2$ and then switching to $U$ implies, after rearrangement,

$$ \frac{dV}{dU} = -\frac{1}{2} \left( \frac{dX}{dU} \right)^2 + \frac{1}{4} \mathcal{A} X^2 - \frac{1}{2} \frac{\mathbf{m}^2}{M^2} = -\mathcal{L}_{NR} - \frac{1}{2} \frac{\mathbf{m}^2}{M^2}, \qquad (\text{VI.3}) $$

<footnote>$^4$ We are grateful to J. Balog and G. Junker for calling our attention at this point.</footnote>
whose integration over a large-enough domain then yields (VI.2) with

$$V_c = -\frac{\mathbf{m}^2}{2M^2}. \qquad (VI.4)$$

At last, the DM boundary conditions for the transverse motion $X(U)$ in (V.2) imply that the integral term $\int \mathcal{L}_{NR}dU$ (which is independent from $\mathbf{m}$) drops out, cf. (V.3), and thus we end up, outside the wavezone, with the generalization of (V.4) to the massive case,

$$V \equiv V_{\mathbf{m}}(U) = V_0 - \frac{1}{2} \frac{\mathbf{m}^2}{M^2} U. \qquad (VI.5)$$

Do we get DM also in the massive case? At first sight, the answer seems to be negative : the linear-in-$U$ term tilts the vertical coordinate (VI.5) as shown in FIG.13.

<figure><img src="image_15.png" /><figcaption>FIG. 13: For non-zero relativistic mass $\mathbf{m}$ the lightlike “vertical” trajectory $V_{\mathbf{m}}(U)$ becomes tilted, as shown for $\mathbf{m} = M = 1$.</figcaption></figure>

However switching to the longitudinal coordinate $^5$,

$$Z = V + \frac{1}{2}U, \qquad (VI.6)$$

we get

$$Z = V_0 + \frac{1}{2} \left( 1 - \frac{\mathbf{m}^2}{M^2} \right) U,$$

<footnote n="5">⁵ Choosing instead $\tilde{Z}_{\mathbf{m}} = V + \frac{1}{2} \frac{\mathbf{m}^2}{M^2} U$ would work for DM however it would not preserve the form of the metric.</footnote>
and thus $U$ is eliminated when

$$ \mathbf{m} = M. \qquad (VI.7) $$

Then the trajectory is tipped back to horizontal, leaving us with

$$ Z(U) \equiv Z_{\mathbf{m}}(U) = V_0 = \text{const.} \qquad (VI.8) $$

In conclusion, we do obtain DM for all coordinates provided the relativistic and the non-relativistic masses are equal, (VI.7). The transverse trajectory $X(U)$ is independent of $V$ and the effect of $\mathbf{m}$ amounts to merely replacing $\hat{V}$ in FIG.12 by $Z_{\mathbf{m}}$ in FIG.14.

<figure><img src="image_16.png" /><figcaption>FIG. 14: When the two types of masses are equal, (VI.7), then swapping the light-cone coordinate $V_{\mathbf{m}}$ for the longitudinal one, $Z_{\mathbf{m}}$ in (VI.6), tips the trajectory back which becomes similar to $\hat{V}(U)$ for $\mathbf{m} = 0$ in FIG.12.</figcaption></figure>

The results of this section can also be presented from a slightly different point of view, see the Appendix.

## VII. CARROLL-SYMMETRY-RELATED APPROACH

In our paper [18] we argued that no permanent displacement is possible — whereas here we have just found two counter-examples. Now we clarify how does this come about.

First of all, we have shown in the previous sections that the relativistic mass $\mathbf{m}$ has no effect on the transverse dynamics and merely adds a linear-in-$U$ mass term $-(\mathbf{m}^2/2M^2)U$, (VI.4), to the $V$-dynamics [30]. Swapping the light-cone coordinate $V$ for the longitudinal $Z$ as in (VI.6) the problem then takes the same form as for $V$. Therefore we restrict henceforth our attention to lightlike geodesics, $\mathbf{m} = 0$, with NR mass parameter scaled to $M = 1$ for
simplicity. Then the Carroll-symmetry-based approach [17, 18, 31, 32] provides us with the following road map:

• We choose an $U_0$ and solve the Sturm-Liouville equation for the $1 \times 1$ “matrix” $Q^6$,

$$ \ddot{Q} + \frac{1}{2} A Q = 0 \quad \text{with initial conditions} \quad Q(u_0) = 1, \quad \dot{Q}(u_0) = 0. \qquad (\text{VII.1}) $$

• Using $Q$ we switch from Brinkmann (B) to Baldwin-Jeffery-Rosen (BJR) coordinates [33, 34],

$$ X = Qx, \quad U = u, \quad V = v - \frac{1}{4} \dot{a} x^2 \quad \text{where} \quad a(u) = Q^2(u). \qquad (\text{VII.2}) $$

$(X, U, V) \to (x, u, v)$ carries the metric (II.1) to the BJR form [9, 16, 19, 33, 34],

$$ a(u)dx^2 + 2dudv. \qquad (\text{VII.3}) $$

• The requirement of being at rest in the Beforzone i.e. before the sandwich wave arrives implies [17] that the transverse BJR trajectory is trivial,

$$ x(u) = x_0, \quad v(u) = v_0. \qquad (\text{VII.4}) $$

• Then the B-trajectory is,

$$ X(U) = Q(U)x_0, \quad V(U) = v_0 - \frac{1}{2}(Q\dot{Q})x_0^2. \qquad (\text{VII.5}) $$

This road map allowed us to conclude that no permanent displacement is possible : we have VM, but no DM [18]. Anticipating the details to come, we remind the reader of that while Brinkmann coordinates are defined for all $U$, the BJR coordinates are defined only in coordinate patches $I_k = (u_k, u_{k+1})$ [9, 17, 31]$^7$, distinguished by

$$ a(u_k) = 0 \quad \Leftrightarrow \quad Q(u_k) = 0, \quad k \ge 1 \text{ integer}. \qquad (\text{VII.6}) $$

The BJR coordinates become singular at the end points points. The B $\Leftrightarrow$ BJR correspondence (VII.2) works in each $I_k$ separately but should be matched at the contact points.

<footnote>$^6$ In $D$-dimensional transverse space $Q$ is a $D \times D$ matrix [9, 15]. The initial conditions in (VII.1) can be modified, as discussed e.g. in [19].</footnote>

<footnote>$^7$ The singularity of BJR coordinates was instrumental in the long-standing controversy about the physical existence of gravitational waves [35].</footnote>
Crossing such a contact point $u_k$ one has to restart the B $\Leftrightarrow$ BJR transcription – however the simple formulas (VII.4)-(VII.5), which are valid for zero initial velocity, have to be replaced by a considerably more complex procedure [9, 31].

In detail, comparing (VII.1) with (II.2a) shows that $Q(U)$ satisfies the same equation (II.2a) as $X(U)$ does for initial conditions $X(U_0) = 1$ and $\dot{X}(U_0) = 0$. The geodesics are conveniently found by switching to BJR and by using the conserved quantities generated by the symmetries of plane gravitational waves which, in addition to translations, involve also Carroll boosts [9, 31, 32]. These symmetries leave $u$ fixed and act on the BJR coordinates as,

$$x \rightarrow x + S(u)b + c, \quad \text{and} \quad v \rightarrow v - b x - \frac{1}{2} S(u) p^2 + f, \qquad (\text{VII.7})$$

where $c, b, f$ are real numbers and

$$S(u) = \int_{u_0}^{u} \frac{du}{a(u)} \qquad (\text{VII.8})$$

is the Souriau “matrix” [9, 31] (a scalar for our $D = 1$). (VII.7) preserves the BJR metric (VII.3) and generates by Noether’s theorem conserved linear and Carroll momenta,

$$p = a(u) \dot{x}(u), \quad M = 1, \quad k = x(u) - S(u)p, \qquad (\text{VII.9})$$

respectively. Conversely, these conserved quantities determine the BJR trajectories [31],

$$x(u) = k + S(u)p, \quad v(u) = v_0 - \frac{1}{2} S(u) p^2, \qquad (\text{VII.10})$$

where $v_0 = v(u_0)$.

Things are particularly simple when the incoming velocity is zero as it is required in the Beforezone. Then

$$0 = \dot{X}(U_0) = \dot{Q}(U_0)x_0 + Q(U_0)\dot{x}(U_0) = \dot{x}(U_0), \qquad (\text{VII.11})$$

which by (VII.9) implies the vanishing of the incoming conserved momentum,

$$p = 0. \qquad (\text{VII.12})$$

The Souriau-terms in (VII.10) are then switched off and the trajectory is just a fixed point, (VII.4) with $x_0 = k$, and the globally defined B-trajectory is recovered by pulling it back to Brinkmann by (VII.2)[17, 19]. However when the momentum does not vanish, $p \neq 0$, then (VII.4) and thus (VII.5) are not more valid and should be replaced by (VII.10) which requires to find $p$ and then to calculate the Souriau matrix.
We emphasise that all our investigations above and (VII.4) in particular are valid only where the BJR coordinates are valid, i.e., in the interval $I_k$ between two subsequent zeros of $Q(u)$. Then we are left with the task of gluing together the results obtained in neighboring domains $I_k$.

### Illustration: Pöschl-Teller in BJR

More insight is gained by illustrating the procedure by considering the Pöschl-Teller potential (IV.1)-(IV.2) with $M = 1$, for which we had found the analytic solutions $X(u) = -P_m(\tanh u)$ in (IV.6). The BJR profile, found by following our road map is plotted in FIG.15. It has $m$ zeros, and the procedure has to be restarted in each of the intervals $I_k$.

We start with a trajectory $X(U)$ viewed as $Q$ “matrix”. For an integer wave number $m$ the metric (VII.3) becomes, outside the wave zone, that of Minkowski, FIG.15 (whereas it diverges for $u \to \infty$ when $m$ is fractional FIG.16). The Souriau matrix (VII.8),

<figure><img src="image_17.png" /><figcaption>FIG. 15: The BJR profile for the Pöschl-Teller potential for wave numbers $m = 1, 2, 3$</figcaption></figure>

$$S(u) \equiv S_m(u) = \int_{u_0}^{u} \frac{du}{[P_m(\tanh u)]^2}, \qquad (VII.13)$$

is well-defined between two subsequent zeros of the denominator which are indeed those of the Legendre polynomial in sect. IV. For the $m = 1$, for example, the Brinkmann trajectory $Q(U) = X(U) = -\tanh U$ shown in FIG.7 yields,

$$S(u) \equiv S_{m=1}(u) = u - \coth u, \qquad (VII.14)$$

depicted in FIG.17. For $m = 2$ we have instead,

$$S_{m=2} = u + \frac{3 \sinh (2u)}{4 - 2 \cosh (2u)}. \qquad (VII.15)$$
<figure><img src="image_18.png" /><figcaption>FIG. 16: The BJR profile for the Pöschl-Teller potential for m is not an integer. The force falls off before the velocity is brought down to zero and then the particle flies off with constant velocity.</figcaption></figure>

Both in the Before and in the Afterzone, the Souriau matrix is approximately linear, as shown in FIGs. 17 and 18. Focusing our attention henceforth at $m = 1$, we note that the

<figure><img src="image_19.png" /><figcaption>FIG. 17: The Souriau matrix $S_{m=1}$ in (VII.14) is regular and approximately linear outside the wave zone both in $I_-$ and $I_+$ but diverges at their junction at $u_1 = 0$.</figcaption></figure>

Souriau matrix $S_{m=1}(u)$ is regular in both of the two domains

$$I_{-} = (-\infty, 0) \quad \text{and} \quad I_{+} = (0, \infty) \qquad (\text{VII.16})$$

but diverges at $u = 0$ — the point where the Brinkmann trajectory vanishes.

Eqn. (VII.10) is valid *separately* in both coordinate patches $I_\pm$,

$$x^{\pm}(u) = k^{\pm} + p^{\pm}(u - \coth u), \qquad v^{\pm}(u) = v_{0}^{\pm} - \frac{(p^{\pm})^2}{2}(u - \coth u), \qquad (\text{VII.17a})$$

$$\dot{x}^{\pm}(u) = p^{\pm}\left(1 + \frac{1}{\sinh^2 u}\right), \qquad \dot{v}^{\pm}(u) = -\frac{(p^{\pm})^2}{2}\left(1 + \frac{1}{\sinh^2 u}\right), \qquad (\text{VII.17b})$$
<figure><img src="image_20.png" /><figcaption>FIG. 18: The Souriau matrices for wave numbers m = 2, 3 are regular in m + 1 domains separated by the zeros of the B-trajectory, where they diverge.</figcaption></figure>

where $k^{\pm}$ and $p^{\pm}$ are arbitrarily chosen constants. Both BJR trajectories are regular in their respective domains (VII.16), but diverge at $u = 0$. However, pulling the BJR trajectory back to Brinkmann by (VII.2) removes the singularity :

$$X(U) = k^{\pm} \tanh U + p^{\pm} (U \tanh U - 1) \qquad (VII.18)$$

is regular. Then

1. The two BJR branches match at $u = 0$ when $p^{+} = p^{-}$.

2. From

$$\dot{X}(U) = \frac{k^{\pm}}{\cosh^2 U} + p^{\pm} \left( \tanh U + \frac{U}{\cosh^2 U} \right) \qquad (VII.19)$$

we deduce that no motion in the Beforezone, $\dot{X}(-\infty) = 0$, requires $p^{-} = 0$ and thus $p^{+} = p^{-} = p = 0$ as in (VII.12). The Souriau term is thus switched off.

3. By (VII.19) the velocities match when $k^{+} = k^{-} = k$.

In conclusion, for $k = -1$ we recover the solution (IV.6) with $m = 1$ i.e., $X_1(U) = -\tanh U$, shown in FIG.7. At last, (VII.5) is

$$V(U) = V_0 - \frac{1}{2} \frac{\sinh U}{\cosh^3 U} X_0^2 \rightarrow V_0 \qquad (VII.20)$$

consistently with (V.5).

We mention that the initial conditions in (VII.1) could actually be modified: the important condition for DM is (II.5). We illustrate this point by considering

$$\tilde{Q}(u) = u \tanh u - 1 \qquad (VII.21)$$
which is yet another solution of the Sturm-Liouville equation (VII.1). It is singular where $u \tanh u = 1$ and diverges at $\pm\infty$. Then our road map above yields the Souriau matrix

$$ \tilde{S}(u) = \frac{\tanh u}{1 - u \tanh u} \qquad (VII.22) $$

and from (VII.10) we deduce that,

$$ \dot{x}(u) = \frac{1}{[u \tanh u - 1]^2} p \quad \Rightarrow \quad \dot{x}(u = \pm\infty) = 0. \qquad (VII.23) $$

Thus the condition (II.5) is satisfied, and we do get DM, as shown in FIG.19.

<figure><img src="image_21.png" /><figcaption>FIG. 19: Trajectories for $\tilde{Q}$ in (VII.21) with wave number $m = 1$ in Brinkmann and in BJR coordinates.</figcaption></figure>

## VIII. CONCLUSION

Particles at rest before the arrival of a sandwich gravitational wave exhibit, generically, the velocity effect (VM) : the particles fly apart with diverging constant but non-zero velocity [8–11, 15–19]. Zel'dovich and Polnarev suggested instead [1] that flyby would generate (approximately) pure displacement (DM).

Our paper answers a question of the (unknown) referee of our previous paper [19] concerning the relation of VM and DM. In detail, we argue that for a judicious choice of the wave parameters, namely when the Wavezone contains an integer number of half-waves then we do get pure displacement. We illustrated our statement both by a numerical (sec.III) and analytical (sec.IV) examples.
Our results can be understood from yet another point of view [36]. In the linear approximation,

$$X(U) = X(0) + U\dot{X}_0 \quad \text{and} \quad \dot{X}(U) = \dot{X}_0, \qquad (\text{VIII.1})$$

where $\dot{X}_0$ is the initial velocity. After the wave had passed, the displacement and the velocity depend on three moments,

$$M_0 = \int \mathcal{A}(U)dU, \quad M_1 = \int U\mathcal{A}(U)dU, \quad M_2 = \int U^2\mathcal{A}(U)dU. \qquad (\text{VIII.2})$$

Then the motion is

$$\dot{X}_{sol}(U) = M_0X_0 + \dot{X}_0 + M_1\dot{X}_0, \qquad (\text{VIII.3a})$$

$$X_{sol}(U) = X_0 + U\dot{X}_{sol}(U) - M_1X_0 - M_2\dot{X}_0, \qquad (\text{VIII.3b})$$

where $X_0 = X(U_a)$ and $\dot{X}_0 = \dot{X}(U_a)$ are the initial position and velocity. In the Afterzone $U > U_a$ where $\mathcal{A}(U) \equiv 0$ all three moments vanish, $M_0 = M_1 = M_2 = 0$. Thus the motion is along straight lines with constant velocity [15], which vanishes if the incoming particle had zero velocity, $\dot{X}_0 = 0$. But then it stops for good,

$$X_{sol}(U) = X_0 = X(U_a) = \text{const.} \quad \text{for } U > U_a. \qquad (\text{VIII.4})$$

The difficulty is to find out for which values of the parameters does this happen. Our answer given to this question is : one should have an integer number of half-waves.

Our new results complete those previous ones [15–19] which are valid in a domain where the BJR coordinates are regular. The question is studied in detail and illustrated for the Pöschl-Teller profile in sec.VII.

Generalization to 4 dimensions with physical applications will be studied further in [21], confirming that reducing VM to DM is indeed possible also for more general plane gravitation waves which include those generated by flyby, gravitational collapse, etc as proposed in [13].

Both numerical and analytical evidence show, for example, a particular behavior for vacuum gravitational waves with a Brinkmann metric with 2 transverse direction

$$g_{\mu\nu}dX^{\mu}dX^{\nu} = \delta_{ij}dX^{i}dX^{j} + 2dUdV + \frac{1}{2}\mathcal{A}(U)((X^1)^2 - (X^2)^2)dU^2 \qquad (\text{VIII.5})$$

where $(X^i)$, $i = 1, 2$ are transverse and $U, V$ light-cone coordinates. The relative minus here is mandatory for a vacuum gravitational wave, therefore one of the components necessarily
<figure><img src="image_22.png" /><figcaption>FIG. 20: The geodesics for the Gaussian burst (III.1) in 2 transverse dimensions for various (blue/red/green) initial positions in the Beforezone. The potential is attractive in the $X^2$ but repulsive in the $X^1$ sector, implying diverging trajectories in the latter.</figcaption></figure>

diverges as shown by FIG.20 [FIG.3 of [18]] for the Gaussian profile (III.1) with $k = 1$. When looking for DM, the diverging coordinate should be discarded by putting it to identically zero, allowing for a “half DM” after fine-tuning.

We stress however that this “halfening” is unrelated to the question if our wave is a vacuum gravitational wave or not, but depends rather on the profile. Following [13], flyby should be described, for example, by a vacuum gravitational wave whose profile is the first derivative of the Gaussian. And it has full DM for both coordinates, as shown in FIG.21.

<figure><img src="image_23.png" /><figcaption>FIG. 21: For flyby with $m = 1$ half-wave we have full DM for all three components.</figcaption></figure>

This behavior corresponds indeed to a general pattern: even-order derivatives of the Gaussian exhibit half DM and odd-order derivatives exhibit full DM in both transverse directions [21].
Related results were discussed in [37–39] and more recently in [40, 41].

Acknowledgments

We are grateful to the anonymous referee of our previous paper [19] for directing our
interest at the VM vs DM problem. We are indebted to G. Gibbons and to J. Balog
for their insights and advices. The Appendix was contributed by J. Balog. Correspon-
dence is acknowledged to T. Damour and G. Junker. We profited from discussions with
P. C. Aichelburg, L. Diósi and M. Elbistan, Z. Silagadze and F. Ziegler. PAH thanks the
Schrödinger Institute (Vienna) for hospitality during the Workshop *Carrollian Physics and
Holography CDFG 2024* in April 2024. PMZ was partially supported by the National
Natural Science Foundation of China (Grant No. 12375084).

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Appendix A: Geodesic motion of a massive particle

The relativistic Lagrangian for geodesic motion in $D = 1$ transverse direction is, in Brinkmann coordinates,

$$ \mathcal{L}_{geo} = (\dot{X})^2 + 2\dot{U}\dot{V} - \frac{1}{2}\mathcal{A}(U) X^2 \dot{U}^2, \qquad (A.1) $$

where the dot denotes derivation w.r.t. an affine parameter $\lambda$. The Euler-Lagrange equations are,

$$ \begin{aligned} \ddot{X} &= -\frac{1}{2}\mathcal{A} X \dot{U}^2, \\ \ddot{U} &= 0, \\ \ddot{V} &= \frac{1}{4} \frac{d\mathcal{A}}{dU} X^2 \dot{U}^2 + \frac{1}{2} \mathcal{A}(\dot{X}^2) \dot{U}. \end{aligned} \qquad (A.2) $$

The $U$ equation is integrated at once, yielding the non-relativistic mass familiar in the Bargmann framework [24, 25], $\dot{U} = M = \text{const.}$

The Euler-Lagrange equations then imply that the Lagrangian is conserved along the geodesic: $\frac{d}{d\lambda}\mathcal{L}_{geo} = 0$ providing us with a constant of the motion,

$$ \mathcal{L}_{geo} = -\frac{\mathbf{m}^2}{M^2}, \qquad (A.3) $$

where $\mathbf{m}^2$ is the Jacobi invariant (II.3). Switching to longitudinal and relativistic time coordinates,

$$ z = V + \frac{1}{2}U \quad \text{and} \quad t = V - \frac{1}{2}U, \qquad (A.4) $$

respectively, three essentially different cases can be distinguished,

$$ \left\{ \begin{array}{ll} \mathbf{m}^2 > 0 & \text{timelike geodesic for massive particle} \\ \mathbf{m}^2 = 0 & \text{lightlike geodesic for massless particle} \\ \mathbf{m}^2 < 0 & \text{spacelike geodesic for tachyonic particle.} \end{array} \right. \qquad (A.5) $$

Dropping tachyons we consider henceforth $\mathbf{m}^2 \ge 0$ and scale $M$ to 1.

Replacing the affine parameter by $U$ and denoting $d/dU$ by prime, the transversal Sturm-Liouville equation, (II.2a)

$$ X'' = -\frac{1}{2}\mathcal{A}X, \qquad (A.6) $$

is obtained; integrating twice the $V$ equation (II.2b) yields,

$$ z(U) = V_0 + \frac{1}{2} \left( 1 - \left( \frac{\mathbf{m}}{M} \right)^2 \right) U - \frac{1}{2} X X'. \qquad (A.7) $$
For a particle initially at rest in the Before zone we must have $z = \text{const}$. Then using that $X' = 0$ in the Before zone, allows us to conclude that DM in the Afterzone requires that the two types of masses, $\mathbf{m}$ and $M$ be equal,

$$ \left(\frac{\mathbf{m}}{M}\right)^2 = 1 \qquad (A.8) $$

Then (A.7) reproduces (VI.8).

We underline that our proof does not apply in the lightlike case $\mathbf{m} = 0$ because the condition (A.8) can not be satisfied: photons can not be in rest. Requiring no motion for a massless particle would be unphysical anyway, as said before.

These considerations are completely general in the conclusion does not depend on the details of the $\mathcal{A}(U)$ profile and are valid in any transverse dimension including the physically relevant $D = 2$.