# Flyby-induced displacement: analytic solution

P.-M Zhang, Z.K Silagadze, P.A Horvathy

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P.-M Zhang, Z.K Silagadze, P.A Horvathy. Flyby-induced displacement: analytic solution. Phys.Lett.B, 2025, 868, pp.139687. 10.1016/j.physletb.2025.139687. hal-04966953

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# Flyby-induced displacement effect: an analytic solution

P.-M. Zhang¹*, Z. K. Silagadze²†, and P. A. Horvathy³‡

¹School of Physics and Astronomy,
Sun Yat-sen University,
Zhuhai 519082, (China)

² Budker Institute of Nuclear Physics and Novosibirsk State University,
630 090, Novosibirsk, (Russia)

³ Institut Denis Poisson
CNRS-UMR 7013 Tours University
Parc de Grandmont F-37200 Tours (France)

(Dated: May 15, 2025)

## Abstract

We describe analytically the scattering of particles by a sandwich gravitational wave generated during a flyby. The derivative-of-the-Gaussian profile proposed by Gibbons and Hawking is approximated by the hyperbolic Scarf potential, which allows for an exact analytic solution via the Nikiforov-Uvarov method. Our results confirm the prediction of Zel'dovich and Polnarev when the wave amplitude takes certain “magical” values.

Key words: gravitational waves; displacement memory effect; flyby; Nikiforov-Uvarov method.

PACS numbers: 04.20.-q Classical general relativity;

<footnote>* corresponding author. zhangpm5@mail.sysu.edu.cn</footnote>

<footnote>† silagadze@inp.nsk.su</footnote>

<footnote>‡ horvathy@univ-tours.fr</footnote>
# Contents

<table><tr><td>I. Introduction</td><td>2</td></tr><tr><td>II. Analytic solution for flyby</td><td>5</td></tr><tr><td>III. Transverse geodesics by the Nikiforov-Uvarov method</td><td>6</td></tr><tr><td>IV. Conclusion</td><td>9</td></tr><tr><td>Acknowledgments</td><td>10</td></tr><tr><td>References</td><td>10</td></tr><tr><td>A. Compendium on the Nikoforov-Uvarov method</td><td>14</td></tr></table>

# I. INTRODUCTION

One of the early proposals to observe gravitational waves, put forward by Braginsky and Thorne [1], was to study the motion of particles initially in rest after the passing of a burst of gravitational waves. They argued indeed that *Burst With Memory could be among the earliest kinds of gravitational waves detected* [1].

Initial studies [2] considered the *Velocity Memory* (VM) effect: after the wave has passed, the particles would fly apart with constant velocity along straight trajectories. Zel'dovich and Polnarev [3] advocated instead in favor of *flyby*: gravitational waves generated by a cluster of superdense stars (neutron stars or black holes) would yield the *Displacement Memory* (DM) effect: “...although the distance between a pair of two noninteracting bodies (such as satellites) will change, their relative velocity will become vanishingly small as the flyby concludes.”

The GW memory effect offers a way to test general relativity including the nonlinear nature of gravity, and provides additional information about the GW source. Preliminary studies have suggested that the detection of GW memory from individual mergers of massive binary black holes is expected with LISA [4]. However, more realistic estimates show that observing the GW memory effect at detectors such as LISA, although feasible, is a challenging task [5, 6]. Recent attempts to detect the effect in LIGO/Virgo's first transient
gravitational-wave catalog, as well as in a pulsar timing array experiment NANOGrav, have predictably yielded negative results [7, 8]. However the prospects for the next generation of gravitational wave detectors are brighter even for future ground-based detectors such as Cosmic Explorer and the Einstein Telescope (see [6, 8] and references therein).

Zel'dovich and Polnarev did not present any proof. Although their suggestion could be confirmed in [9–13], there were some doubts due to the apparent contradiction with the Bondi-Pirani theorem on the caustic property of plane gravitational waves [14, 15]. Then it came as a surprise that that the DM effect could be confirmed numerically [16–18] for plane waves — provided the amplitude takes some “magical” values. Below we present an analytic proof.

In Brinkmann coordinates, the metric is written as [19],

$$ds^2 = d\mathbf{X}^2 + 2dUdV + \mathcal{A}(U)(X_+^2 - X_-^2)dU^2. \quad (1)$$

Gibbons and Hawking [20] argued that in linear theory the Riemann tensor of a gravitational wave is proportional to the fourth time derivative of the quadrupole moment of the source [21]. This lead them to propose the derivative of the Gaussian,

$$\mathcal{A}^{dG} \propto \frac{d}{dU} \left( \exp[-U^2] \right) \quad (2)$$

as profile. The geodesic equations for (2) do not support any analytic solution, the profile can however be approximated by the derived Pöschl-Teller (dPT) potential [16, 22, 23],

$$\mathcal{A}^{dPT}(U) = \frac{d}{dU} \left( \frac{g}{2 \cosh^2 U} \right) = -g \frac{\sinh U}{\cosh^3 U}. \quad (3)$$

Then the geodesics can be studied semi-analytically [16, 18]. Our investigations led us to conjecture that DM arises when a quantization holds, namely when the wave zone contains an integer number of what we called half-waves.

In this note we present a third approximation of the flyby profile (2), namely by the hyperbolic Scarf potential [24]<fnref n="1" />,

$$\mathcal{A}^{\text{Scarf}}(U) = 2g \frac{d}{dU} \left( \frac{1}{\cosh U} \right) = -2g \frac{\sinh U}{\cosh^2 U}, \quad (4)$$

<footnote n="1">¹ The trigonometric Scarf potential was first introduced in [25] to mimic the potential of a real crystal. It turned out to be the tip of the iceberg of a whole class of potentials with interesting properties, ranging from classical and quantum integrability [26, 27] to supersymmetric quantum mechanics [28–32].</footnote>
to be compared with (3). For small values of $U$ the Scarf profile (4) behaves as the derivative of the Gaussian or of Pöschl-Teller, (2) or (3), but for large $U$ it falls off more slowly. However dilating $U$ appropriately almost perfect overlapping is obtained, as depicted in FIG.1. Their similarity follows from that all three of them are derivatives of similar bell-shaped functions. We note also all three profiles in (2)-(3)-(4) are derivatives of an even function and are therefore odd.

<figure><img src="image_2.png" /><figcaption>FIG. 1: The derived Pöschl-Teller profile (3) and its Scarf potential alter-ego (4) are both good approximations of the derived-Gaussian one proposed by Gibbons and Hawking [20] to describe flyby.</figcaption></figure>

The geodesics are, in Brinkmann coordinates, determined by the transverse equations [18],

$$ \frac{d^2 X^{\pm}}{dU^2} \mp \frac{1}{2} A X^{\pm} = 0, \qquad (5) $$

where the $\pm$ label refers to the upper resp. lower component. The sign of the second term comes from the relative minus between the coordinates in (1). The equation for the $V$ component follows from these [16, 18], and is therefore omitted. We recall that $U$ behaves, in the Bargmann framework [33, 34], as non-relativistic time. These Sturm-Liouville-type equations are completed with the DM boundary conditions,

$$ \frac{d\mathbf{X}}{dU}\bigg|_{(U = -\infty)} = 0 = \frac{d\mathbf{X}}{dU}\bigg|_{(U = +\infty)}. \qquad (6) $$

Our main result then says that for the profile (4) analytic solutions can be found which confirm the prediction of Zel'dovich and Polnarev [3], supplemented by the quantization condition mentioned above. Our clue is that the geodesic equation (5) can be viewed as a time-independent Schrödinger equation and the DM trajectories correspond to its real (non-normalizable) zero-energy ground states [16].
## II. ANALYTIC SOLUTION FOR FLYBY

We first recall how things go for the simple Pöschl-Teller case [16, 18]. Putting $t = \tanh U$ into eqn. (5) with profile $A^{PT} = (g/2) \cosh^{-2} U$ yields,

$$ (1 - t^2) \frac{d^2 X^{\pm}}{dt^2} - 2t \frac{dX^{\pm}}{dt} \mp \frac{g}{4} X^{\pm} = 0. \qquad (7) $$

Let us consider first one of the components, say $X^+$. Then DM is obtained in terms of Legendre polynomials [16] when the coefficient in the linear-in-$X^+$ term is quantized, as

$$ g = -4n(n+1) \quad \text{where } n > 0 \quad \text{is a natural integer.} \qquad (8) $$

However passing to the other component, the last, linear-in-$X^-$ term has opposite sign, therefore DM would be obtained for $-g$. But this is impossible because $g$ has already been fixed. In conclusion, one has what we called “half DM” [18]: we get a DM solution in one or in the other sector, but not for both in general — unless the “wrong” component is set to zero.

Solutions for the Scarf potential $A^{Scarf}$ (and similarly for derived Pöschl-Teller, $A^{dPT}$, [18]), are obtained by putting $t = \sinh U$. Then (5) is written

$$ (1 + t^2) \frac{d^2 X^{\pm}}{dt^2} + t \frac{dX^{\pm}}{dt} \mp g \frac{t}{1 + t^2} X^{\pm} = 0. \qquad (9) $$

The two equations here are uncoupled and can therefore be solved independently. Note however the coefficient $t$ in front of the linear-in-$X^\pm$ term whose sign change compensates the $\pm$ sign change between the components. Therefore the same $g$ works for both components, as it will be seen in sec. III.

Finding the geodesics for dPT is considerably more difficult, though, as for “underived” Pöschl-Teller; it can be obtained in terms of confluent Heun functions [16, 18] with numerically calculated approximate expressions, see eqn. (IV.16) of [18].

Turning to Scarf, we note that (9) is of the generalised hypergeometric type which can be solved analytically by the Nikiforov-Uvarov (NU) method [35]. Since a fairly detailed description of this simple yet elegant and powerful method can be found elsewhere [35–39], here we just list the main results. The algorithm, whose details are discussed in sec. III, yields the desired “magic” coefficient $g$ in (9),

$$ |g| = |g_n| = (2n + 1) \sqrt{n(n + 1)}, \qquad (10) $$
where n is a natural integer. Then the trajectory is,

$$X_n(t) = (1 + t^2)^{-\frac{n}{2}} e^{-\operatorname{sign}(g)} \sqrt{n(n+1)} \arctan t y_n(t), \quad (11)$$

where $y_n(t)$ is the polynomial

$$y_n(t) = \frac{B_n}{\rho(t)} \left[ (1 + t^2)^n \rho(t) \right]^{(n)} \quad (12)$$

with weight function

$$\rho(t) = (1 + t^2)^{-(n+\frac{1}{2})} e^{-2\operatorname{sign}(g)} \sqrt{n(n+1)} \arctan t. \quad (13)$$

The normalization constants $B_n$ is determined from the initial conditions.

The analytic solutions (11) reproduce almost identically those obtained numerically in FIG.s # 8 & 9 of [18] for the derived Pöschl-Teller case. For $n \ge 5$, though, the numerical procedure becomes tedious as it requires to calculate a high number of decimals. For high wave numbers the result becomes inaccurate because the procedure used in [18] is only an approximation. Plotting instead the analytic solution (11)-(12) yields accurate figures, underlining the advantage of the analytical approach.

### III. TRANSVERSE GEODESICS BY THE NIKIFOROV-UVAROV METHOD

The Nikoforov-Uvarov method (recalled in the Appendix) applies to second-order differential equations of the generalised hypergeometric type of the form,

$$u'' + \frac{\pi_1(z)}{\sigma(z)}u' + \frac{\sigma_1(z)}{\sigma^2(z)}u = 0. \quad (14)$$

In the Scarp case Eq. (9) $\sigma = 1 + t^2$, $\pi_1 = t$, $\sigma_1 = -gt$. Then,

$$\sigma_3 = \frac{t^2}{4} + gt + k(1 + t^2) = t^2 \left( k + \frac{1}{4} \right) + gt + k. \quad (15)$$

Therefore, the condition $\det(\sigma_3) = g^2 - 4k(k + \frac{1}{4}) = 0$ gives

$$k = \frac{\pm\sqrt{1 + 16g^2} - 1}{8} \quad \text{and} \quad |g| = 2\sqrt{k\left(k + \frac{1}{4}\right)}. \quad (16)$$

Thus we have

$$\sigma_3 = \frac{g^2}{4k} \left[ t^2 + \frac{4k}{g}t + \frac{4k^2}{g^2} \right] = \frac{g^2}{4k} \left( t + \frac{2k}{g} \right)^2, \quad \pi = \frac{t}{2} \pm \sqrt{\sigma_3} = \frac{t}{2} \pm \frac{|g|}{2\sqrt{k}} \left| t + \frac{2k}{g} \right|. \quad (17)$$
<figure><img src="image_3.png" /><figcaption>FIG. 2: The DM geodesics for the derived Pöschl-Teller (3) and for the Scarf profiles (4) are similar however not fully identical, as shown for wave numbers $n = 1$ and $n = 2$. (The trajectories were rescaled for better comparison).</figcaption></figure>

Then

$$ \tau = \pi_1 + 2\pi = 2t \pm \frac{|g|}{\sqrt{k}} \left| t + \frac{2k}{g} \right| \quad (18) $$

To have real solutions, we assume $k > 0$, that is we choose the upper sign $k = \frac{\sqrt{1+16g^2}-1}{8}$ in Eq.(16). On the other side, the quantization condition Eq. (A12) requires

$$ \lambda = -n(n+1) \mp \epsilon \frac{n|g|}{\sqrt{k}}, \quad (19) $$

since $\tau' = 2 \pm \epsilon \frac{|g|}{\sqrt{k}}$ with $\epsilon = \text{sign}(t + \frac{2k}{g})$ and we get, after some algebra,

$$ k = \lambda - \pi' = -n^2 - n - \frac{1}{2} \mp \epsilon \frac{|g|}{\sqrt{k}} \left( n + \frac{1}{2} \right), $$
and using Eq. (16) for $|g|/\sqrt{k}$, we get $\sqrt{k + \frac{1}{4}} = \mp\epsilon(n + \frac{1}{2})$ and we see that we must choose the lower signs in Eq. (17) and Eq. (18) to get $\mp\epsilon = 1$. Then the solution is $k_n = n(n + 1)$, and Eq. (16) determines the “magic” strengths labeled by the integer “quantum” number $n$, eqn. (10), for which we have polynomial solutions $y_n(t)$.

There remains to find the gauge function $\varphi(t)$ and the corresponding hypergeometric type polynomials $y_n$. We have,

$$ \pi(t) = -nt - \operatorname{sign}(g)\sqrt{n(n+1)}, \quad \tau(t) = -(2n-1)t - 2\operatorname{sign}(g)\sqrt{n(n+1)}. \quad (20) $$

The equation for the gauge function,

$$ \frac{d\varphi}{dt} = - \frac{nt + \operatorname{sign}(g)\sqrt{n(n+1)}}{1 + t^2}, \quad (21) $$

can be easily integrated to yield (up to an irrelevant additive constant which translates to a multiplicative constant in the solution $X(t)),

$$ \varphi(t) = -\frac{n}{2} \ln(1 + t^2) + \operatorname{sign}(g) \sqrt{n(n+1)} \arctan t. \quad (22) $$

Therefore, the corresponding geodesic trajectory is given by eqn. (11). The polynomial $y_n(t)$ is given by the Rodrigues formula Eq. (A13) which requires to calculate the weight function $\rho(t)$ in (13). Then we end up with eqn. (12).

Eqn. (13) implies that $\rho(t)$ does not change when $t$ and $g$ change signs simultaneously, while due to the derivatives in eqn. (12), the polynomials $y_n(t)$ acquire a $(-1)^n$ factor. Then eqn. (11) implies the parity behaviour

$$ X_n^{\pm}(-t) = (-1)^n X_n^{\mp}(t) \quad (23) $$

mentioned in sec.II. Numerical evidence suggests that $t = 0$ is not a root of $X_n^{\pm}$ neither for dPT nor for Scarf and, in fact, for a profile which is an order-odd derivative of an even function. On the contrary, it is a root when the profile is an order-even derivative.

The curious pairing of the components in (23) is reminiscent of and is indeed closely related to supersymmetry [42].

The $y_n(t)$ can be expressed in terms of Routh-Romanovski polynomials [24, 40] as $y_n(t) = R_n^{(\alpha, \beta)}(t)$ with $\alpha = -2\operatorname{sign}(g)\sqrt{n(n+1)}$ and $\beta = -n + \frac{1}{2}$. The first three of them are $y_1(t) \sim -t - 2\sqrt{2}\operatorname{sign}(g)$, $y_2(t) \sim 2t^2 + 8\sqrt{6}\operatorname{sign}(g)t + 23$, $y_3(t) \sim -6t^3 - 72\sqrt{3}\operatorname{sign}(g)t^2 - 423t - 172\sqrt{3}\operatorname{sign}(g)$.
IV. CONCLUSION

The profile (2) proposed by Hawking and Gibbons for flyby [20] can be approximated by both the derived-Pöschl-Teller (3) and by the Scarf potentials (4), as illustrated in FIG.1. The advantage of Scarf is that it supports analytic solutions, (11)-(12), labeled by an integer $n$, eqn (10). They are constructed by the Nikiforov-Uvarov method.

The displacement effect (DM) arises when the wave zone contains $n$ half-waves as observed made [18] for the Gaussian and Pöschl-Teller profiles. The number of half-waves is equal to that of the zeros of the polynomial $y_n(t)$.

Our Scarf solution is similar to but slightly different from those found semi-analytically for derived-Pöschl-Teller [18], as seen in FIG.2. It has also the advantage to be exact, while the one in [18] is only approximate, with error increasing for higher wave numbers.

The behavior of both the Scarf (4) and of the derived-Pöschl-Teller (3) is explained by their parity behaviour, (23), which explains in turn why do we get DM for both components. The same is true for any odd-order derivative of an even-profile function [16, 18, 46]. For even-order derivatives the analogous choices of the parameters would be contradictory, implying that one can gets DM only for one, but not for both components — unless the other is put to zero.

Physical applications of various incarnations of the Scarf potential are reviewed in ref. [40]. They include supersymmetry [24, 28, 32, 41, 42] and non-central potentials [30]. The recent paper [42], which is directly relevant for our purposes here, highlights the prominent role of SUSY QM for gravitational waves with Scarf profile. See also [43–46] for further related work.

We mention that our main tool here, namely the Nikiforov-Uvarov method, applies also to a number of other singular and periodic GW profiles, such as

$$ \mathcal{A} = 2g \left( 1 - \frac{1}{\tanh^2 U} \right), \quad \mathcal{A} = -2g \operatorname{cn}^2 \left( U, \frac{1}{\sqrt{2}} \right), \quad \mathcal{A} = 2g \tan U, \quad \mathcal{A} = \frac{2g}{\tan U}, \quad (24) $$

where $cn(z, k)$ is Jacobian elliptic cosine function. Interestingly, the last profile corresponds to the Coulomb potential in constant curvature space $S^3$, and hydrogenlike atoms in constant curvature spaces are most naturally considered by the Nikiforov-Uvarov method [47].

It has long been known (see, e.g., [24, 28, 30, 48, 49]) that the Pöschl-Teller and (hyperbolic) Scarf potentials are parts of the same supersymmetric partner potentials, following
from the shape invariant superpotential

$$W(U) = A \tanh(U) + B \sech(U). \qquad (25)$$

A recent paper [42] confirms our results using supersymmetric quantum mechanics with this superpotential.

Acknowledgments

The authors are grateful to Gary Gibbons and to M. Elbistan for discussions and to G. Natanson for correspondence. PMZ was partially supported by the National Natural Science Foundation of China (Grant No. 11975320).

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## Appendix A: Compendium on the Nikoforov-Uvarov method

As it was already mentioned in the text, the Nikoforov-Uvarov method applies to second-order differential equations of the generalised hypergeometric type of the form

$$u'' + \frac{\pi_1(z)}{\sigma(z)}u' + \frac{\sigma_1(z)}{\sigma^2(z)}u = 0, \qquad (A1)$$

where the prime denotes $d/dz$ where $z$ can be complex. $\pi_1(z)$ is a polynomial at most of the first degree, and $\sigma(z)$, $\sigma_1(z)$ are polynomials at most of the second degree [35].

The set of solutions of Eq. (A1) is invariant under “gauge” transformations $u(z) \to y(z)$, $u(z) = e^{\phi(z)}y(z)$, if the gauge function $\phi(z)$ satisfies the equation

$$\phi'(z) = \frac{\pi(z)}{\sigma(z)}, \qquad (A2)$$

where $\pi(z)$ is a polynomial at most of the first degree. In this case for $y(z)$ we obtain also a generalised hypergeometric-type equation:

$$y'' + \frac{\pi_2(z)}{\sigma(z)}y' + \frac{\sigma_2(z)}{\sigma^2(z)}y = 0, \qquad (A3)$$

where $\pi_2(z) = \pi_1(z) + 2\pi(z)$ is a polynomial of at most of the first degree, and

$$\sigma_2(z) = \sigma_1(z) + \pi^2(z) + \pi(z)[\pi_1(z) - \sigma'(z)] + \pi'(z)\sigma(z) \qquad (A4)$$

is a polynomial of at most of the second degree. Choosing $\pi(z)$ so that

$$\sigma_2(z) = \lambda\sigma(z) \qquad (A5)$$
where $\lambda$ is a constant, Eq. (A1) is simplified to an hypergeometric type equation,

$$ \sigma(z)y'' + \pi_2(z)y' + \lambda y = 0. \qquad (A6) $$

In the light of Eq. (A4), Eq. (A5) means

$$ \pi^2 + \pi[\pi_1 - \sigma'] + \sigma_1 - k\sigma = 0, \qquad (A7) $$

where $k = \lambda - \pi'$ is another constant. Eq. (A7) is solved by,

$$ \pi = \frac{\sigma' - \pi_1}{2} \pm \sqrt{\sigma_3(z)} \quad \text{where} \quad \sigma_3(z) = \left(\frac{\sigma' - \pi_1}{2}\right)^2 - \sigma_1 + k\sigma. \qquad (A8) $$

$\pi$ is a polynomial only when $\sigma_3(z)$ is the square of a first order polynomial. Hence it has a double root and its discriminant is zero, $\Delta(\sigma_3) = 0$, which determines the constant $k$, and hence the constant $\lambda$.

We want to find polynomial solutions of Eq. (A6). It can be shown that derivatives of $y(z)$, $v_n(z) = y^{(n)}(z)$, are also generalised hypergeometric-type functions,

$$ \sigma v_n'' + \tau_n(z)v_n' + \mu_n v_n = 0, \qquad (A9) $$

and we get the recurrence relations

$$ \tau_n(z) = \sigma'(z) + \tau_{n-1}(z), \quad \mu_n = \mu_{n-1} + \tau'_{n-1}, \qquad (A10) $$

with initial values $\tau(z) = \tau_0(z) = \pi_2(z)$, $\mu_0 = \lambda$. If $y(z) = y_n(z)$ is a polynomial of order $n$, then $v_n = \text{const}$ and Eq. (A9) is satisfied only if $\mu_n = 0$. But the repeated application of the recurrence relations Eq. (A10) produces

$$ \tau_n(z) = n\sigma'(z) + \tau(z), \quad \mu_n = \lambda + n\tau' + \frac{1}{2}n(n-1)\sigma''. \qquad (A11) $$

Getting a polynomial solution $y(z) = y_n(z)$ of Eq. (A6) thus requires the “quantization condition”,

$$ \lambda = \lambda_n = -n\tau' - \frac{1}{2}n(n-1)\sigma''. \qquad (A12) $$

It can be shown [35] that, when the quantization condition Eq. (A12) is satisfied, the corresponding polynomial solution of Eq. (A6) is given by the Rodrigues formula

$$ y_n(z) = \frac{B_n}{\rho(z)} [\sigma^n(z) \rho(z)]^{(n)}, \qquad (A13) $$

where $B_n$ is some (normalization) constant, and the weight function $\rho(z)$ satisfies Pearson's equation [50]

$$ (\sigma\rho)' = \rho\tau. \qquad (A14) $$