# Gravitational wave memory: Further examples

Q.-L. Zhao, P.-M. Zhang, M. Elbistan, P. Horvathy

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Q.-L. Zhao, P.-M. Zhang, M. Elbistan, P. Horvathy. Gravitational wave memory: Further examples. International Journal of Geometric Methods in Modern Physics, In press, 10.1142/S0219887825400195. [hal-05070593](https://hal.science/hal-05070593v1)

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Distributed under a Creative Commons Attribution 4.0 International License
Gravitational wave memory : further examples *

P.-M. Zhang¹†, Q.-L. Zhao¹‡, M. Elbistan²§, P. A. Horvathy³¶

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

² *Department of Energy Systems Engineering,*
*Istanbul, Bilgi University, Turkey*

³ *Institut Denis-Poisson CNRS/UMR 7013 - Université*
*de Tours - Université d'Orléans Parc de Grammont,*
*37200; Tours, France*

(Dated: May 14, 2025)

Abstract

Ehlers and Kundt [1] argued in favor of the velocity effect: particles initially at rest hit by a burst of gravitational waves should fly apart with constant velocity after the wave has passed. Zel'dovich and Polnarev [2] suggested instead that waves generated by flyby would be merely displaced. Their prediction is confirmed provided the wave parameters take some particular values.

key words: Plane Gravitational waves; Memory Effect; Displacement effect

PACS numbers: 04.20.-q Classical general relativity;

04.30.-w Gravitational waves

<footnote>* Talk given by P-M Zhang on July 15 at the 33rd/35th International Colloquium on Group Theo-
retical Methods in Physics (ICGTMP, Group33/35), held in Cotonou, Benin, July 15 - 19, (2024).
https://icgtmp.sciencesconf.org/</footnote>

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

<footnote>‡ mailto: zhaoqliang@mail2.sysu.edu.cn</footnote>

<footnote>§ mailto: elbistan@itu.edu.tr</footnote>

<footnote>¶ mailto:horvathy@univ-tours.fr</footnote>
# Contents

<table><tr><td>I. Introduction</td><td>2</td></tr><tr><td>II. The Memory effect</td><td>3</td></tr><tr><td>A. Gaussian or Pöschl-Teller profile</td><td>3</td></tr><tr><td>B. |U|<sup>-4</sup> profile</td><td>3</td></tr><tr><td>C. Flyby (and beyond)</td><td>5</td></tr><tr><td>III. Conclusion</td><td>8</td></tr><tr><td>Acknowledgments</td><td>8</td></tr><tr><td>References</td><td>8</td></tr></table>

# I. INTRODUCTION

The Velocity Memory (VM) effect was first considered by Ehlers and Kundt [1] who argued that particles initially at rest hit by a burst of gravitational waves would fly apart with constant velocity [1]. Their investigations were continued in [2–7]. Zel’dovich and Polnarev [2] suggested in particular that gravitational waves generated by flyby would exhibit the Displacement Effect (DM) : the particles would merely be displaced. Our earlier investigations [8–10] agreed with the general scheme [2, 2–7] but did not confirm their claim, as seen in FIG. # 12. of ref. [10]. However, recent study [11–13] indicates that for certain “miraculous” values of the wave parameters we do get DM. In this Note we illustrate the theory by several examples.

We start with the 4 dimensional, linearly polarized Brinkmann metric [14–16],

$$g_{\mu\nu}dX^{\mu}dX^{\nu} = \delta_{ij}dX^{i}dX^{j} + 2dUdV + \frac{1}{2}\mathcal{A}(U)\left((X^{1})^{2} - (X^{2})^{2}\right)(dU)^{2}, \quad (I.1)$$

where $X = (X^i)$ are transverse coordinates and $U, V$ are light-cone coordinates. Its geodesics are given by,

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

$$\frac{d^2 V}{dU^2} + \frac{1}{4} \frac{d\mathcal{A}}{dU} \left( (X^1)^2 - (X^2)^2 \right) + \mathcal{A} \left( X^1 \frac{dX^1}{dU} - X^2 \frac{dX^2}{dU} \right) = 0. \qquad (I.2b)$$
Having solved first eqns (I.2a), (I.2b) is solved by lifting to the gravitational wave spacetime [11, 13]. Henceforth we focus our attention at (I.2a). We recall that geodesic motion posses a conserved quantity referred to as the Jacobi invariant [17, 18],

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

Discarding tachions, we shall assume that $\mathfrak{m}^2 \le 0$.

## II. THE MEMORY EFFECT

### A. Gaussian or Pöschl-Teller profile

The Memory effect for Gaussian and Pöschl-Teller [19] profiles,

$$ \mathcal{A} \equiv \mathcal{A}^G(U) = \frac{k}{\sqrt{\pi}}e^{-U^2} \quad \text{and} \quad \mathcal{A}^{PT}(U) = \frac{k}{2\cosh^2 U}, \qquad (II.1) $$

was studied in [11, 13, 20]. DM can be achieved in the attractive, but not in the repulsive coordinate sector : we get “half DM”.

### B. $|U|^{-4}$ profile

Another example is obtained by considering the Brinkmann metric (I.1) with profile [21, 22],

$$ \mathcal{A}(U) = \frac{1}{U^4}. \qquad (II.2) $$

This is an approximate sandwich wave both for $U < 0$ and $U > 0$ but is singular at $U = 0$. The profile is even w.r.t. sign reversal,

$$ U \rightarrow -U. \qquad (II.3) $$

The sector of $X^1$ is repulsive, that of $X^2$ sector is attractive.

The geodesics of (II.2) can be found analytically. We recall first the linearly polarized vacuum gravitational wave of Brdicka [23], (I.1) with profile $\mathcal{A}(U) = \text{const}$. Its geodesics correspond to the Bargmann lift [10, 11, 15, 16] of the transverse trajectories

$$ \tilde{X}^1(\tilde{U}) = c_1 \cosh(\tilde{U}) + c_2 \sinh(\tilde{U}), \quad \tilde{X}^2(\tilde{U}) = c_3 \cos(\tilde{U}) + c_4 \sin(\tilde{U}). \qquad (II.4) $$
Then the conformal redefinition [24, 25]

$$U = -\frac{1}{\tilde{U}}, \quad \mathbf{X} = \frac{\tilde{\mathbf{X}}}{\tilde{U}}, \quad V = \tilde{V} + \frac{\tilde{\mathbf{X}}^2}{2\tilde{U}}, \qquad (II.5)$$

carries the Brdicka metric conformally into (I.1) with profile (II.2), whereas (II.4) is taken into,

$$X^1(U) = c_1 U \cosh \left( \frac{1}{U} \right) - c_2 U \sinh \left( \frac{1}{U} \right), \qquad (II.6)$$

$$X^2(U) = c_3 U \cos \left( \frac{1}{U} \right) - c_4 U \sin \left( \frac{1}{U} \right) . \qquad (II.7)$$

The terms with coefficients $c_1, c_3$ are odd and those with coefficients $c_2, c_4$ are even w.r.t. $U$-reversal, (II.3). The singularity at $U = 0$ can be removed when we posit

$$X^1(0) = X^2(0) = 0, \qquad (II.8)$$

allowing us to glue together the $U < 0$ and $U > 0$ branches. However we have a curious freedom for gluing, which can yield either even or odd trajectories. Let us first look at the attractive, $X^2$ sector. Starting with the even term for $U < 0$ and positing (II.8), we can continue by the same expression also for $U > 0$,

$$X_{\text{even}}^2(U) = \begin{cases} c_2 U \sin \left( \frac{1}{U} \right) & U < 0 \\ 0 & U = 0 \\ c_2 U \sin \left( \frac{1}{U} \right) & U > 0 \end{cases}, \qquad (II.9)$$

and get an even trajectory, shown in FIG.1. However, starting with the same branch for $U \le 0$ we can continue it for $U > 0$ also with minus the same expression, i.e.,

$$X_{\text{odd}}^2(U) = \begin{cases} c_2 U \sin \left( \frac{1}{U} \right) & U < 0 \\ 0 & U = 0 \\ -c_2 U \sin \left( \frac{1}{U} \right) & U > 0 \end{cases}, \qquad (II.10)$$

shown in FIG. 2(a). Both of these trajectories are continuous but non-differentiable at $U = 0$, consistently with the velocity figures FIGs.1(b) and 2(b).
<figure><img src="image_2.png" /><figcaption>Figure 1: 1(a): For $c_2 = 1$ as only nonzero parameter, the evenly-glued $X^2$ component (II.9) is symmetric w.r.t. U-reversal (II.3) and thus $X(-\infty) = 1 = X(+\infty)$. 1(b): At $U = \pm\infty$ the velocity goes to zero as it should.</figcaption></figure>

Both the evenly and the oddly - glued cases, (II.9) and (II.10), exhibit DM. However the different ways of glueing result in that their outgoing values alternate,

$$ \lim_{U \to \infty} X_{even}^2(U) = 1 \quad \text{and} \quad \lim_{U \to \infty} X_{odd}^2(U) = -1, \qquad (II.11) $$

consistently with the behavior w.r.t. parity [13].

Consistently with the general pattern in [13], no DM is possible in the repulsive, $X^1$ sector. Both the motion (II.7) and its velocity, shown in FIG. 3, diverge at $U = 0$ and can not be regularized unless $c_1 = c_2 = 0$ i.e., when $X^1(U) \equiv 0$: we get “half DM”.

### C. Flyby (and beyond)

Flyby [2] was thoroughly analysed for both the Gibbons - Hawking [3] and the Pöschl-Teller [19] derivative-profiles [11, 13],

$$ \mathcal{A}^G = \frac{d}{dU} \left( \frac{k}{\sqrt{\pi}} e^{-U^2} \right) \quad \text{and} \quad \mathcal{A}^{PT} = \frac{d}{dU} \left( \frac{k}{2 \cosh^2 U} \right), \qquad (II.12) $$

respectively. Odd parity, $\mathcal{A}^G(-U) = -\mathcal{A}^G(-U)$, implies that it is enough to consider one component only. Then DM is obtained in both coordinate sectors [13].
<figure><img src="image_3.png" /><figcaption>Figure 2: 2(a): For $c_2 = 1$ as only nonzero parameter, the oddly-glued component (II.10) is an-tisymmetric w.r.t. U-reversal (II.3) and is thus displaced from $X^2(-\infty) = 1$ to $X^2(+\infty) = -1$.
2(b): At $U = \pm\infty$ the velocity goes to zero, consistently with DM.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Both of the $c_2 = 1$ odd 3(a) and the $c_1 = 1$ even solution 3(b) diverge at $U = 0$ requiring $X^1(U) \equiv 0$ for all $U$.</figcaption></figure>

No analytic solutions are known for the profile $\mathcal{A}^G$ in (II.12) proposed by Gibbons and Hawking [3]. For the derivative of the Pöschl-Teller profile, $\mathcal{A}^{dPT}$ we do get analytic solutions, however they involve highly sophisticated manipulations with special functions [13]. Our clue is that the profiles in (II.12) can, although rather crudely, be approximated by the
double-square potential,

$$ \mathcal{A}(U) = \begin{cases} 0 & U < -a \\ 2h^2 & -a < U \le 0 \\ -2h^2 & 0 < U < a \\ 0, & U > a \end{cases} \qquad (II.13) $$

shown in FIG.4, which generalizes [26]. The initial conditions are choosen as,

<figure><img src="image_5.png" /><figcaption>Figure 4: Double-square approximation of the flyby profile.</figcaption></figure>

$$ X(U \le -a) = X_0, \quad V_0 = \frac{dX}{dU}\bigg|_{U \le -a} = 0. \qquad (II.14) $$

In the region $-a \le U \le 0$ the profile is that of an inverted oscillator, $\ddot{X} - h^2X = 0$. Thus,

$$ X_I(U) = X_0 \cosh(hU + ha). \qquad (II.15) $$

At the right boundary $U = 0$ we have,

$$ X_1 := X_I(U = 0) = X_0 \cosh(ha), \quad V_1 := \frac{dX_I}{dU}(U = 0) = hX_0 \sinh(ha) \qquad (II.16) $$

which serve as initial conditions in the right region $0 \le U \le a$, where we have a harmonic oscillator combining its sin, cos solutions. DM requires the velocity to vanish for $U = a$. This can be achieved exponentially, when

$$ ha = m\pi + \frac{\pi}{4}, \quad m = 0, 1, 2, 3, \dots \qquad (II.17) $$

Finally we find DM asymptotically,

$$ X(U) = \begin{cases} X_I(U) = X_0 \cosh(hU + ha), & -a < U < 0 \\ X_{II}(U) = X_0 [\cosh(ha) \cos(hU) + \sinh(ha) \sin(hU)], & 0 < U < a \end{cases} \qquad (II.18) $$
This behavior is reminiscent of the one we found for derivative-Pöschl-Teller.

Higher-derivative profiles can be also considered [3, 9]. The 2nd derivative of the Gaussian (or of Pöschl-Teller) describe, e.g. the Braginsky - Thorne system [3, 5], for which we get half DM. The 3rd derivative was proposed instead to describe gravitational collapse [3], for which we get again DM for both components [13].

### III. CONCLUSION

Zel'dovich and Polnarev suggested that flyby would generate pure displacement (DM) with vanishing relative velocity [2]. Our papers [11, 13] indicate that for judicious choices of the wave parameters does yield approximately pure displacement, refining the statement of Zel'dovich and Polnarev.

At last, Sect. 2-5.8 of [1] anticipates much of later work. Similar problems were considered also in ref. [27–30].

### Acknowledgments

Some of our results were obtained in collaboration with J. Balog. Discussions are acknowledged to P. C. Aichelburg, T. Damour, L. Diósi and G. Gibbons. PAH thanks the Erwin Schrödinger Institute (ESI, Vienna) for hospitality during the Workshop Carrollian Physics and Holography_CDFG_2024. PMZ was partially supported by the National Natural Science Foundation of China (Grant No. 12375084). M.E. is supported by Istanbul Bilgi University research fund BAP with grant no: 2024.01.00.

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