# Globally defined Carroll symmetry of gravitational waves

Mahmut Elbistan, Peng-Ming Zhang, Peter Horvathy

► To cite this version:

Mahmut Elbistan, Peng-Ming Zhang, Peter Horvathy. Globally defined Carroll symmetry of gravitational waves. 2025. hal-05347264

HAL Id: hal-05347264

https://hal.science/hal-05347264v1

Preprint submitted on 14 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Globally defined Carroll symmetry of gravitational waves

M. Elbistan¹*, P.-M. Zhang¹,²† P. A. Horvathy¹,³‡,

¹ Department of Energy Systems Engineering,
Istanbul Bilgi University, 34060, Eyupsultan, Istanbul, (Turkey)

² School of Physics and Astronomy,
Sun Yat-sen University, Zhuhai, China

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

(Dated: October 27, 2025)

Abstract

The locally defined Carroll symmetry of a gravitational wave is extended to a globally defined one by switching to Brinkmann coordinates. Translations and Carroll boosts associated with two independent globally defined solutions of a Sturm-Liouville equation allow us to describe the motions. The Displacement Memory Effect arises for particular choices of the parameters which yield trajectories with zero momentum. The relation to the Schwarzian derivative is highlighted. We illustrate our general statements by the Pöschl-Teller profile.

PACS numbers: 04.20.-q Classical general relativity

<footnote>* mailto:mahmut.elbistan@bilgi.edu.tr</footnote>

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

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

<table><tr><td>I. Introduction</td><td>2</td></tr><tr><td>II. From Brinkmann to BJR and back</td><td>3</td></tr><tr><td>III. Carroll symmetry in Brinkmann coordinates</td><td>5</td></tr><tr><td>IV. Schwarzian aspects</td><td>7</td></tr><tr><td>V. Illustration by the Pöschl - Teller profile</td><td>9</td></tr><tr><td>VI. Representation of the symmetry generators</td><td>11</td></tr><tr><td>VII. Conclusion</td><td>14</td></tr><tr><td>Acknowledgments</td><td>16</td></tr><tr><td>References</td><td>17</td></tr></table>

# I. INTRODUCTION

Braginsky and Thorne had suggested that gravitational waves might be observed by detecting the displacement of particles initially in rest, called later the *Memory Effect* [1]. While experimental verification is still in the making, an insight can be gained by taking advantage of the *Carroll symmetry* [2, 3]. Baldwin-Jeffery-Rosen (BJR) coordinates [4, 5] provide a simple description [6, 7], however suffer of being regular only in finite intervals, requiring to glue them together. In this paper we show that such complications can be avoided by switching to Brinkmann (B) coordinates [8] by solving a matrix Sturm-Liouville equation, (II.3) below. The price to pay is, though, to get more complicated-looking expressions.

We illustrate our general theory by the Pöschl-Teller profile [9–11], which is a very good *analytical* approximation of the widely studied Gaussian one. We follow mostly our review [12] which provides the reader also with further references.

Our notations are: $(\mathbf{X}, U, V)$ are Brinkmann coordinates and $(\boldsymbol{x}, u, v)$ are BJR coordinates on relativistic spacetime. $U$ is an affine parameter for geodesics. $\Theta^{\text{Brink}} = \xi \partial_{\mathbf{X}} + \eta \partial_{V}$ denote vector fields which generate the infinitesimal isometries on spacetime.
## II. FROM BRINKMANN TO BJR AND BACK

Plane gravitational waves are conveniently described by using globally defined Brinkmann coordinates [8, 13] in terms of which the metric is,

$$g_{\mu\nu}dX^{\mu}dX^{\nu} = d\mathbf{X}^{2} + 2dUdV + K_{ij}X^{i}X^{j}dU^{2}, \qquad (II.1)$$

where $\mathbf{X} = (X^1, \dots, X^D)$ parametrizes the transverse plane, endowed with the flat Euclidean metric $d\mathbf{X}^2 = \delta_{ij} dX^i dX^j$. U and V are light-cone coordinates and $(K_{ij})$ we call the profile is a symmetric $D \times D$ matrix whose entries depend only on U. Henceforth we focus our attention at $D = 1$ or $D = 2$ and assume, for simplicity, that is diagonal. In $D = 2$ we take

$$K_{ij}X^iX^j = \mathcal{A}(U)\left((X^+)^2 - (X^-)^2\right), \qquad (II.2)$$

which yields a vacuum solution of the Einstein equations [8, 13]. We consider “sandwich waves” [14], whose profile $\mathcal{A}(U)$ vanishes outside an interval $U_i < U < U_f$. The V-equation follows from the transverse ones [12] and will henceforth not be studied.

After the wave has passed, particles initially at rest exhibit the Velocity Memory Effect (VM): they move with constant velocity. Under certain “quantization” conditions the outgoing velocity can vanish, though, and we get the Displacement Memory Effect (DM) [15–18], which might well play a rôle in future observations [1, 17].

Plane gravitational waves have long been known to have a $2D + 1$ parameter symmetry group composed of $D + 1$ translations, completed by $D$ (rather mysterious) transformations [6, 19]. More recently [7], this group was identified with the Carroll group [2, 3], as young Lévy-Leblond called it jokingly. Its algebraic structure is readily determined by switching to Baldwin et al (BJR) coordinates [4–6] which are however only local. This note sheds further light at the Carroll–Memory relation using globally defined Brinkmann coordinates.

We start with considering real solutions of the Sturm-Liouville equations

$$P'' = \mathcal{A}(U)P, \qquad (P^T)P' = (P^T)'P \qquad (II.3)$$

for a $D \times D$ matrix $P(U) = (P_{ij}(U))$ [12]. The BJR coordinates $(x, u, v)$, are then obtained by

$$\mathbf{X} = P(u)\mathbf{x}, \quad U = u, \quad V = v - \frac{1}{4}\mathbf{x} \cdot (P^T P)'\mathbf{x}, \qquad (II.4)$$
in terms of which the metric is,

$$g_{\mu\nu}dx^{\mu}dx^{\nu} = (P(u)^{T}P(u))_{ij}(u)dx^{i}dx^{j} + 2du\,dv. \qquad (II.5)$$

Carroll symmetry consists, in BJR coordinates, of “vertical” (v) and transverse (x) translations with parameter h and c, completed by Carroll boosts with parameter b [6, 7],

$$\boldsymbol{x} \rightarrow \boldsymbol{x} + \boldsymbol{c} + S(u)\boldsymbol{b}, \quad u \rightarrow u, \quad v \rightarrow v - \boldsymbol{b} \cdot \boldsymbol{x} - \frac{1}{2}\boldsymbol{b} \cdot S(u)\boldsymbol{b} + h, \qquad (II.6)$$

where the symmetric 2 × 2 matrix

$$S(u) = \int_{u_0}^{u} (P^T P)^{-1}(t)dt \qquad (II.7)$$

is referred to as the Souriau matrix [6, 7, 12]. Infinitesimal boosts parametrized with $b_j$ are,

$$\theta^{BJR} = h\partial_v + c^j\partial_j + b_j(S^{ji}\partial_i - x^j\partial_v). \qquad (II.8)$$

Finding the trajectories is then straightforward. Noether's theorem provides us with 2D + 1 associated conserved quantities [6, 7],

$$\boldsymbol{p} = \boldsymbol{p}_0 = (P^T P)(u) \boldsymbol{x}'(u), \quad \boldsymbol{k} = \boldsymbol{k}_0 = \boldsymbol{x}(u) - S(u) \boldsymbol{p}, \qquad (II.9)$$

interpreted as conserved linear and boost-momentum, supplemented by $m = u' = 1$. Then the conserved quantities determine the geodesic,

$$\boldsymbol{x}(u) = \boldsymbol{k}_0 + S(u) \boldsymbol{p}_0. \qquad (II.10)$$

The subtlety comes from that the BJR coordinates are defined only in intervals $I_k = [u_{k-1}, u_k]$ between the (mandatory [6, 7]) zeros of the determinant,

$$\det P(u_k) = 0, \qquad (II.11)$$

which play a fundamentally important rôle in our investigation, as we shall see. The BJR coordinates are thus only local: both the metric (II.5) and the Souriau matrix S in (II.7) are singular at the junction points $u_k$, as will be illustrated in FIG.1 for the Pöschl-Teller profile. The singularity problem will be resolved in sect. III by switching to Brinkmann coordinates.

Now we set $P^T = P$ and we consider $D = 1$ (when P is a scalar function) for simplicity, although our results and conclusions could be extended to non-diagonal profiles and to any
dimension. Let us assume that $u < u_1$, the first zero of $\det(P)$. Requiring that our particles be in rest before the wave arrives,

$$ \boldsymbol{x}(-\infty) = x_0 \quad \text{and} \quad \boldsymbol{x}'(-\infty) = 0, \qquad (II.12) $$

imply that the momentum vanishes in $I_1$,

$$ \boldsymbol{p} = 0. \qquad (II.13) $$

The Souriau term is thus switched off from (II.10) and the transverse BJR “trajectory” is merely a fixed point.

$$ \boldsymbol{x}(u) = \boldsymbol{x}_0 = \text{const}, \qquad (II.14) $$

confirming, in BJR coordinates, the “no motion for Carroll” maxim [2]. In contrast, the Brinkmann trajectory is non-trivial : (II.4) yields,

$$ \boldsymbol{X}(U) = P(u) \boldsymbol{x}_0, \qquad (II.15) $$

with $U = u$. The initial conditions (II.12) then require :

$$ P(-\infty) = 1 \quad \text{and} \quad P'(-\infty) = 0. \qquad (II.16) $$

The Brinkmann trajectory, given by the $P$-matrix, may look quite complicated, depending on the Sturm-Liouville solution $P$ of (II.4) [12]. For appropriate “quantized” values of the wave parameter the wave zone contains an integer number of half-waves; then $P(u)$ tends to a constant matrix also for $u \to \infty$ : DM is obtained [16–18].

We emphasise that the investigations above are a priori valid only before the first zero of $\det P$ and should then be restarted until the next zero, and so on.

Another remarkable consequence of (II.15) is that when $\det P(u_0) = 0$ for some $u_0$, then all transverse trajectories which start from an $x_0 \in \text{Ker}(P(u_0))$ are focused at the origin, and we get a *caustic point* [12, 14]. The question will be further discussed in Sec.V and illustrated in FIG. 2.

### III. CARROLL SYMMETRY IN BRINKMANN COORDINATES

Simple as they are, BJR coordinates (II.4) suffer from being valid only in intervals between subsequent zeros of $P$, and the adjacent expressions should be glued together – which may
be laborious. A global approach can conveniently be given instead by using Brinkmann coordinates. For simplicity we restrict our attention at $D = 1$ transverse dimension and start with a fixed interval $I_k$. A straightforward computation then shows that pulling back the BJR expression (II.6) by using (II.4) [or its infinitesimal form (II.8)] yields the Carroll generators written in Brinkmann coordinates,

$$ \Theta^{\text{Brink}} = h \frac{\partial}{\partial V} + c \left( P \frac{\partial}{\partial X} - P' X \frac{\partial}{\partial V} \right) + b \left( (PS) \frac{\partial}{\partial X} - (PS)' X \frac{\partial}{\partial V} \right), \quad (\text{III.1}) $$

The coefficients of $\partial_X$ and of $\partial_V$ are indeed the symmetry generators $\xi$ and $\eta$, mentioned in the Introduction. In Sec. V they will be spelt out explicitly for Pöschl - Teller.

The P-matrix and its derivative $P'$ generate translations, whereas PS and $(PS)'$ generate Carroll boosts [23]. Their only nonzero commutator yields a vertical translation [20],

$$ \underbrace{\left[ P \frac{\partial}{\partial X} - P' X \frac{\partial}{\partial V} \right]}_{\text{translation}}, \quad \underbrace{\left( PS \frac{\partial}{\partial X} - (PS)' X \frac{\partial}{\partial V} \right)}_{\text{boost}} = -\partial_V. \quad (\text{III.2}) $$

The symmetry is thus the Heisenberg algebra.

P thus plays a double role: by (II.15) it determines the B trajectory, whereas (III.1) says that it also generates translations. The B trajectory is obtained indeed from $x_0 = \text{const}$ by a (generally U-dependent) translation.

The Brinkmann form (III.1) has an important advantage w.r.t. the BJR expression in (II.6): multiplication by P suppresses the singularity of Souriau matrix S [6, 12]:

$$ Q(U) = P(U)S(U) \quad (\text{III.3}) $$

is regular for all U. Thus (III.1) extends the Carroll symmetry (II.6) naturally from a fixed interval $I_k$ to the entire U-axis.

The remarkable fact is that both P and Q are solutions of the Sturm-Liouville equation (II.3) [12, 20, 21], as it can be verified by a direct calculation. The Wronskian is $W(P, Q) = P'Q - Q'P = -1$, therefore P and Q are independent. The expression (III.1) could also be verified independently, by checking the symmetry equation $L_Y g = 0$ [22, 23].

Contracting the Brinkmann metric (II.1) with (III.1) provides us with,

$$ \mathbf{p}_0 = P \mathbf{X}' - (P)' \mathbf{X} \quad \text{and} \quad \mathbf{k}_0 = -Q \mathbf{X}' + (Q)' \mathbf{X}, \quad (\text{III.4}) $$
where $\mathbf{p}_0$ and $\mathbf{k}_0$ are, a priori, new constants of the motion. However expressing them in BJR terms, (II.4), the previous expressions (II.9) are recovered; as anticipated by our notation.

The Sturm-Liouville solutions $P$ and $Q$ play also yet another rôle, though : either combining (II.10) with (II.4) or solving (III.4) directly yields the general Brinkmann trajectory,

$$ \mathbf{X}(U) = P(U)\mathbf{k}_0 + Q\mathbf{p}_0. \qquad (\text{III.5}) $$

which is our second main result. When $\mathbf{p}_0 = 0$ cf. (II.13), then the $Q$-term is switched off and we recover (II.15) with $\mathbf{x}_0 = \mathbf{k}_0$. For parameters which correspond to having an integer number of half-waves in the wavezone, we have also $P(+\infty) = \text{const}$ and we get DM.

## IV. SCHWARZIAN ASPECTS

For simplicity, we work in $D = 1$. Our results have an interesting relation with the Schwarzian derivative of a function $f$ of is some real variable $x$ [24–26],

$$ \mathcal{G}(f(x)) = \frac{f'''(x)}{f'(x)} - \frac{3}{2} \left( \frac{f''(x)}{f'(x)} \right)^2. \qquad (\text{IV.1}) $$

We note for later use that the Schwarzian depends only on $f'$, not on $f$ itself.

Consider now the Sturm-Liouville equation

$$ \varphi''(x) - \mathcal{A}(x)\varphi = 0, \qquad (\text{IV.2}) $$

cf. (II.3) for both $P$ or for $Q$ collectively. Let then $\varphi_1(x)$ and $\varphi_2(x)$ be two independent solutions and consider their quotient,

$$ f(x) = \frac{\varphi_1(x)}{\varphi_2(x)}. \qquad (\text{IV.3}) $$

It is then shown in [24] that the Sturm-Liouville profile can be recovered from the solutions in terms of the Schwarzian derivative,

$$ \mathcal{A} = -\frac{1}{2} \mathcal{G}(f). \qquad (\text{IV.4}) $$

Let us now apply these general formulas with the cast $x \rightsquigarrow U$, completed by,

$$ \varphi_1(U) = P(U) \quad \text{and} \quad \varphi_2(U) = P(U)S(U). \qquad (\text{IV.5}) $$
Thus

$$f = \frac{1}{S}. \qquad (IV.6)$$

The Wronskian $W(P, Q)$ is a constant, implying that $f'$ in (IV.1) is indeed proportional to $S^{-2}$. Thus by (IV.4) the profile is the Schwarzian derivative of the inverse of the Souriau matrix,

$$\mathcal{A} = -\frac{1}{2} \mathcal{G} \left( \frac{1}{S} \right). \qquad (IV.7)$$

The form (II.7) of $S$ then implies that $P$ determines also the second solution we called $Q$. Remarkably, we can also interchange the cast: choosing

$$\varphi_1(U) = Q = P(U)S(U) \quad \text{and} \quad \varphi_2(U) = P(U) \qquad (IV.8)$$

instead of (IV.5), it is $P$ which drops out, leaving us with the reciprocal relation for

$$\tilde{f} = \frac{Q}{P} = S, \qquad (IV.9)$$

instead of $f = P/Q = 1/S$ as in (IV.6) <fnref n="1" /> The new cast (IV.8) then yields, once again, a SL eqn. (IV.2) whose profile is now determined by the Souriau matrix <fnref n="2" />,

$$\mathcal{A}(U) = -2\mathcal{G}(S) = -2\frac{S'''(U)}{S'(U)} + 3\left(\frac{S''(U)}{S'(U)}\right)^2. \qquad (IV.10)$$

Somewhat surprisingly, inserting here $S$ from (II.7), the previous Sturm-Liouville equation (II.3) with the same profile $\mathcal{A}$ is recovered. These results indicate that the interchange

$$P, Q \to Q, P \Rightarrow S \to \frac{1}{S} \qquad (IV.11)$$

is a symmetry of the system, as it follows from the reciprocity relation

$$\mathcal{G}(f) = \mathcal{G} \left( \frac{1}{f} \right) \qquad (IV.12)$$

of the Schwarzian. The difference between the two casts comes from the initial conditions. Those for $P$ are dictated by the behaviour at $U = -\infty$ but those for $Q$ depend also on the integral (II.7).

<footnote n="1">¹ The singularities of $S$ has will be discussed in the next section for PT.</footnote>

<footnote n="2">² Eqn. (IV.10) can be viewed as an equation for the Souriau matrix $S$.</footnote>
V. ILLUSTRATION BY THE PÖSCHL - TELLER PROFILE

We illustrate our general theory by the Pöschl - Teller profile [9–11, 16] in $D = 1$,

$$ \mathcal{A} \equiv \mathcal{A}^{PT}(U) = -\frac{4m(m+1)}{\cosh^2 U}. \qquad (V.1) $$

The Sturm-Liouville equation (II.3) is solved by Legendre functions. DM trajectories arise when $m$ is an integer, and we get Legendre polynomials, $(-1)^m P_m(\tanh U)$ [16] which have $m$ zeros.

Turning to BJR by (II.4), the Souriau matrix $S$ in (II.7) is regular between two subsequent zeros of $P$, but diverges at the junction points. For $m = 1$, for example,

$$ S_{m=1}(u) = u - \coth u \qquad (V.2) $$

obtained for the choice $u_0^\pm \approx \pm 1.2$, is regular either in $I_- < 0$ or in $I_- > 0$, but diverges at $u = 0$, as depicted in FIG.1a. Similarly for $m = 2$,

$$ S_{m=2} = \frac{1}{4} \left( u + \frac{3 \sinh(2u)}{2(2 - \cosh(2u))} \right), \qquad (V.3) $$

is singular where the denominator vanishes, as shown in FIG.1b.

<figure><img src="image_2.png" /><figcaption>FIG. 1: For integer $m$ the Sturm-Liouville solutions **P** in (II.3) are DM trajectories, written in Brinkmann coordinates with wave numbers **m** = **1** and **m** = **2**, respectively. The Souriau matrix **S** diverges where **P** = **0**, but the pull-back to Brinkmann **Q** = **PS** is regular for all $U$.</figcaption></figure>

$Q_m = P_m S_m$, a second solution (III.3) of the Sturm-Liouville equation (II.3) shown in FIG.1, is a Legendre function. It is regular for all $U$, vanishes where the Souriau matrix does, and exhibits a VM behaviour where $P_m$ is DM.
The solutions $P_m$ and $Q_m$ are independent when the Wronskian does not vanish, $W(P,Q) \neq 0$. For $m = 1$ and $2$ we have, for example,

$$ P_1(U) = -\tanh U, \quad P_2(U) = \frac{1}{2}(3\tanh^2 U - 1), \qquad (V.4) \\ Q_1(U) = 1 - U \tanh U, \quad Q_2(U) = \frac{1}{2}(2U - 3\tanh U - 3U \sech^2 U). $$

The two Sturm-Liouville solutions are shown in FIG. 1 in red and in green.

The $P(U)$ found above provide us with DM geodesics which are consistent with (II.15). Their caustic behavior at the zeros of $P$ is manifest in FIG. 2.

<figure><img src="image_3.png" /><figcaption>FIG. 2: For a DM wave in $D=1$ transverse dimension Brinkmann trajectories focus to the caustic points determined by $P_m(U_k) = 0$, $k=1, \dots, m$. For $m=1$ there is one focal point at the origin, and for $m=2$, we have two of them, determined by $\tanh U_{\pm} = \pm 1/\sqrt{3}$.</figcaption></figure>

Coming to symmetries, the coefficients of the Carroll generators $P$ and $Q$ in (III.1) are shown in FIGs. 3 and 4. The globally defined Carroll symmetry generators (III.1) are

<figure><img src="image_4.png" /><figcaption>FIG. 3: For wave number $m=1$, the (a) translation generator $P_1$ [which is also a trajectory] is odd. (b) the generator of a Carroll boost, $Q_1 = P_1 S_1$, is even. The only zero of $P_1$ is at $U=0$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>FIG. 4: Infinitesimal Carroll generators (V.6) with wave number $\mathbf{m} = 2$. (a) For a translation, the transverse component $\mathbf{P}_2$ is even, and has two zeros, at $U_±$. (b) For a boost, $\mathbf{Q}_2 = \mathbf{P}_2\mathbf{S}_2$ is odd.</figcaption></figure>

$$ \Theta_{m=1}^{\text{Brink}} = h \frac{\partial}{\partial V} + c \left( \tanh U \frac{\partial}{\partial X} - (\sech^2 U) X \frac{\partial}{\partial V} \right) + b \left( (U \tanh U - 1) \frac{\partial}{\partial X} - (\tanh U + U \sech^2 U) X \frac{\partial}{\partial V} \right). \quad (\text{V.5}) $$

$$ \Theta_{m=2}^{\text{Brink}} = h \frac{\partial}{\partial V} + c \left( (3 \tanh^2 U - 1) \frac{\partial}{\partial X} - (6 \tanh U \sech^2(U)) X \frac{\partial}{\partial V} \right) + b \left( \frac{1}{4}(2U - 3 \tanh U - 3U \sech^2 U) \frac{\partial}{\partial X} - \frac{1}{2}(1 - 3 \sech^2 U + 3U \sech^2 U \tanh U) X \frac{\partial}{\partial V} \right). \quad (\text{V.6}) $$

## VI. REPRESENTATION OF THE SYMMETRY GENERATORS

The symmetry-generating vectors along the trajectories, shown in FIGs. 3 and 4, are conveniently viewed in a co-moving tangent space – which is a plane carried along the trajectory. In FIGs. 5–8 the trajectory is “hidden” in yellow “blobs” and the coordinates $\xi$ and $\eta$ represent the coefficients of $\partial_X$ and of $\partial_V$ of the symmetry generators in (III.1), $\Theta^{\text{Brink}} = \xi \partial_X + \eta \partial_V$, spelt out in (V.5)-(V.6).

FIG. 5 shows that the translation vector starts at $\mathbf{U} = -\infty$ from $\xi = -1, \eta = 0$. It leaves the caustic point for $\mathbf{U} = 0$ at $\mathbf{X} = 0, V = 0$ invariant. Arriving into the Afterzone, the translation vectors change sign and end, for $\mathbf{U} = +\infty$, at $\xi = +1, \eta = 0$. Thus they act again as usual translations – but with reversed sign.

In FIG. 6, the longest arrow (in brown) is both the initial and the final boost vector for $\mathbf{U} = \mp\infty$. The shortest red arrow ($\xi = 1, \eta = 0$) is reached for $\mathbf{U} = 0$ at the caustic point $\mathbf{X} = 0, V = 0$, where the boost shifts all trajectories by the same amount.

According to FIG. 7, the translation generator (with $c = 1/2$) is, for $\mathbf{U} = -\infty$, ($\xi =$
<figure><img src="image_6.png" /><figcaption>FIG. 5: Infinitesimal translations $(\xi, \eta)$ along a DM Brinkmann trajectory $\mathbf{X}(U)$ with $\mathbf{m} = \mathbf{1}$, shown in the co-moving tangent space. The trajectory is “hidden” in the yellow blob.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 6: Infinitesimal boosts along the trajectory “hidden behind the yellow blob” with $\mathbf{m} = \mathbf{1}$, shown in the co-moving tangent space.</figcaption></figure>

1, $\eta = 0$). FIGs. 3 and 4 confirm that, consistently with (V.5), the two focal points of the trajectory, at $U_+$, are left invariant. For $U = +\infty$ the “8-shaped” curve returns to $(\xi = 1, \eta = 0)$, where it started from.

Boosts along the $m = 2$ trajectory are shown in FIG. 8. The longest vector on the left (in brown) is, consistently with (V.5) and (V.6) with $b = 1/2$ and $c = 2$, the boost for $U = -\infty$, and the longest one on the right (also in brown) is for $U = \infty$. The two shortest blue arrows ($\xi = \pm 1, \eta = 0$) show the boost acting as a translations by $\mp b$ at the caustic points $U_\mp$. Combining FIGs. 3 with FIG. 5 provides us with the outfolding to 3dimensions, shown in FIG. 9.

The translations implemented along the trajectories vanish where $P(U) = 0$. They return to their initial value for $m$ odd and their transverse component changes sign for $m$ even.
<figure><img src="image_8.png" /><figcaption>FIG. 7: Translations act along the $m = 2$ DM trajectory by (V.6) The trajectory $X(U)$ with $m = 2$ is “hidden” in the yellow blob.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. 8: Infinitesimal Carroll boosts in Brinkmann coordinates along a chosen DM trajectory marked by a yellow blob, with wave number $m = 2$.</figcaption></figure>

Carroll boosts implemented along the trajectories behave at pure translations at $P(U) = 0$. They return to their initial value for $m$ odd and their transverse component changes sign for $m$ even.

Conversely, we duly recover the Pöschl - Teller profile (V.1) from the Sturm-Liouville solutions by using the Schwarzian formulas in sect.IV. We show this here for the simplest, $m = 1$ solution (V.4), with the cast

$$ \varphi_1(U) = P(U) = -\tanh U, \qquad \varphi_2(U) = Q(U) = 1 - U \tanh U. \qquad (\text{VI.1}) $$

The Wronskian equals to $(-1)$ everywhere. Then inserting $f'(U) = -\cosh^2 U + 1$ into the Schwarzian (IV.4) reproduces the $m = 1$ profile $A(U) = -\frac{8}{\cosh^2(U)}$ in (V.1).
<figure><img src="image_10.png" /><figcaption>Translation (m = 2)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIG. 9: The translations implemented along the trajectories with m = 1 and m = 2.</figcaption></figure>

# VII. CONCLUSION

Our results shed further light on the intimate relation of Carroll symmetry and the Memory Effect : the matrix *P* with initial conditions (II.16) yields our geodesics. The general expression (III.5) which involves both DM and VM is one of our main results.
<figure><img src="image_12.png" /><figcaption>FIG. 10: Carroll boosts implemented along the trajectories with $m = 1$ and $m = 2$.</figcaption></figure>

When the parameter $m$ is an integer, then the wavezone accommodates $m$ half-waves and we get DM. For $m = 2\ell + 1$ $P(U)$ is odd and we do get net displacement. For $m = 2\ell$, though, $P(U)$ is even and the particle returns, after non-trivial motion in the wave zone, to its initial position: we get DM with no final displacement, consistently with (II.15) and confirmed by FIGs 2, 3a and 4a.
DM trajectories are distinguished by having zero momentum and can be found by putting
$p_0 = 0$ into (III.5). Turning on the momentum, $p_0 \neq 0$, distorts the trajectory to VM.

The merit of BJR coordinates is their simplicity [6, 7]. For our initial conditions (II.16),
the “motion” reduces to a fixed point, given by the conserved boost momentum $\mathbf{x}_0 = \mathbf{k}$ in
(II.10). The entire dynamics is carried by $P(u)$. The price to pay is that the BJR description
works only between zeros of the $P$-matrix, and must then be fitted together. These same
points, distinguished by the vanishing of $\det(P)$.

The advantage of the Brinkmann form (III.1) is that it replaces the necessarily singular
Souriau matrix $S$ [6] by the globally defined Sturm-Liouville solution $Q$ – another one of our
principal results,

Examplified by the Pöschl - Teller profile, it allowed us to extend the locally given Car-
rollian Killing vectors to the whole spacetime. The clue is that the multiplying by $P$ cures
the singularity of the Souriau matrix $S$ and thus that of the BJR coordinate system. The
Brinkmann trajectories are focused at the zeros of $\det(P)$, as shown in FIG.2.

The Carroll symmetry generators, first identified by Souriau [6], look more complicated
in Brinkmann coordinates as their BJR counterparts do. They are however defined globally.
They are also directly related to geodesics and allow us to unify the Displacement and
Velocity Memory effects by making use of the two independent solutions $P$ and $Q$ of the
Sturm-Liouville problem, see (III.5). The intimate relationship of geodesics with Carroll
symmetry is highlighted by their common use of the Sturm-Liouville equation [12, 23].

Figs. 5–10 illustrate the curious behavior of the Carroll generators along the trajectories.

Further insight is provided by the Schwarzian framework [24–26]. This approach confirms
that $Q = PS$ is indeed a global, independent, 2nd solution of the Sturm-Liouville problem
which does not involve the locally defined Souriau matrix $S$.

The supersymmetric extension [27] sheds further light on the Memory Effect.

Our results go beyond $D = 1$ dimensional profiles and the Pöschl - Teller example. Similar
results hold for the Scarf profile in $D = 2$, which yields an analytic approximation of flyby
[15, 18]. Details will be presented in a forthcoming comprehensive review [28].

Acknowledgments

The authors are indebted to J. Balog, J. Ben Achour and Z. Silagadze for correspondance.
PMZ was partially supported by the National Natural Science Foundation of China (Grant No. 11975320). ME is supported by Istanbul Bilgi University research fund (BAP) with grant no: 2024.01.009 and by The Scientific and Technological Research Council of Turkey (TÜBITAK) under grant number 125F021. PMZ and PAH benefitted of TÜBITAK 2221 grant and hospitality at Bilgi University during their visit in Istanbul.

[1] V B Braginsky and K. S. Thorne, “Gravitational-wave bursts with memory experiments and experimental prospects”, Nature **327**, 123 (1987)

[2] J. M. Lévy-Leblond, “Une nouvelle limite non-relativiste du group de Poincaré,” Ann. Inst. H Poincaré **3** (1965) 1;

[3] V. D. Sen Gupta, “On an Analogue of the Galileo Group,” Il Nuovo Cimento **54** (1966) 512.

[4] O. R. Baldwin and G. B. Jeffery, “The Relativity Theory of Plane Waves,” Proc. R. Soc. London **A111**, 95 (1926);

[5] N. Rosen, “Plane polarized waves in the general theory of relativity,” Phys. Z. Sowjetunion, **12**, 366 (1937).

[6] J-M. Souriau, “Ondes et radiations gravitationnelles,” Colloques Internationaux du CNRS No 220, pp. 243-256. Paris (1973).

[7] C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, “Carroll symmetry of plane gravitational waves,” Class. Quant. Grav. **34** (2017) no.17, 175003 doi:10.1088/1361-6382/aa7f62 [arXiv:1702.08284 [gr-qc]].

[8] M. W. Brinkmann, “Einstein spaces which are mapped conformally on each other,” Math. Ann. **94** (1925) 119–145.

[9] G. Pöschl and E. Teller, “Bemerkungen zur Quantenmechanik des anharmonischen Oszillators,” Z. Phys. **83** (1933), 143-151 doi:10.1007/BF01331132

[10] F. Correa, V. Jakubsky and M. S. Plyushchay, “Aharonov-Bohm effect on AdS(2) and nonlinear supersymmetry of reflectionless Poschl-Teller system,” Annals Phys. **324** (2009), 1078-1094 doi:10.1016/j.aop.2009.01.009 [arXiv:0809.2854 [hep-th]].

[11] I. Chakraborty and S. Kar, “Geodesic congruences in exact plane wave spacetimes and the memory effect,” Phys. Rev. D **101** (2020) no.6, 064022 doi:10.1103/PhysRevD.101.064022 [arXiv:1901.11236 [gr-qc]].
[12] M. Elbistan, P. M. Zhang and P. A. Horvathy, “Memory effect & Carroll symmetry, 50 years later,” Annals Phys. **459** (2023), 169535 doi:10.1016/j.aop.2023.169535 [arXiv:2306.14271 [gr-qc]].

[13] D. Kramer, H. Stephani, M. McCallum, E. Herlt, “Exact solutions of Einstein’s field equations,” Cambridge Univ. Press (1980).

[14] H. Bondi and F. A. E. Pirani, “Gravitational Waves in General Relativity. 13: Caustic Property of Plane Waves,” Proc. Roy. Soc. Lond. A **421** (1989) 395.

[15] Ya. B. Zel'dovich and A. G. Polnarev, “Radiation of gravitational waves by a cluster of superdense stars,” Astron. Zh. **51**, 30 (1974) [Sov. Astron. **18** 17 (1974)].

[16] P. M. Zhang and P. A. Horvathy, “Displacement within velocity effect in gravitational wave memory,” Annals of Physics **470** (2024) 169784 [arXiv:2405.12928 [gr-qc]].

[17] P. M. Zhang, Q. L. Zhao, J. Balog and P. A. Horvathy, “Displacement memory for flyby,” Annals Phys. **473** (2025), 169890 doi:10.1016/j.aop.2024.169890 [arXiv:2407.10787 [gr-qc]].

[18] P. M. Zhang, Z. K. Silagadze and P. A. Horvathy, “Flyby-induced displacement: analytic solution,” Phys. Lett. B **868** (2025), 139687 doi:10.1016/j.physletb.2025.139687 [arXiv:2502.01326 [gr-qc]].

[19] H. Bondi, F. A. E. Pirani and I. Robinson, “Gravitational waves in general relativity. 3. Exact plane waves,” Proc. Roy. Soc. Lond. A **251** (1959) 519.

[20] M. Blau, “Plane Waves and Penrose Limits.” Version of December 13, 2024. http://www.blau.itp.unibe.ch/Lecturenotes.html

[21] J. Ben Achour and J. P. Uzan, “Displacement versus velocity memory effects from a gravitational plane wave,” JCAP **08** (2024), 004 doi:10.1088/1475-7516/2024/08/004 [arXiv:2406.07106 [gr-qc]].

[22] C. G. Torre, “Gravitational waves: just plane symmetry,” Gen. Rel. Grav. **38** (2006) 653 [gr-qc/9907089].

[23] P. M. Zhang, M. Elbistan, G. W. Gibbons and P. A. Horvathy, “Sturm-Liouville and Carroll: at the heart of the memory effect,” Gen. Rel. Grav. **50** (2018) no.9, 107 doi:10.1007/s10714-018-2430-0 [arXiv:1803.09640 [gr-qc]].

[24] V. Ovsienko and S. Tabachnikov, “What is the Schwarzian Derivative?” Notices of the AMS, **56**, 34 (2009).

[25] P. Zhang, Q. Zhao and P. A. Horvathy, “Gravitational waves and conformal time transfor-
mations,” Annals Phys. **440** (2022), 168833 doi:10.1016/j.aop.2022.168833 [arXiv:2112.09589 [gr-qc]].

[26] N. Vlahakis, “The Schwarzian Approach in Sturm–Liouville Problems,” Symmetry **16** (2024) no.6, 648 doi:10.3390/sym16060648 [arXiv:2405.12549 [math-ph]].

[27] E. Catak, M. Elbistan and M. Mullahasanoglu, “Displacement memory effect from supersymmetry,” Eur. Phys. J. Plus **140** (2025) no.6, 540 doi:10.1140/epjp/s13360-025-06516-5 [arXiv:2504.05043 [gr-qc]].

[28] P. M. Zhang, M. Elbistan, and P. A. Horvathy, “Carroll Symmetry and the Gravitational Memory Effect,” (in preparation).