# Scaling and conformal symmetries for plane gravitational waves

P.-M. Zhang, M. Cariglia, M. Elbistan, Peter Horvathy

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P.-M. Zhang, M. Cariglia, M. Elbistan, Peter Horvathy. Scaling and conformal symmetries for plane gravitational waves. Journal of Mathematical Physics, 2020, 61 (2), pp.022502. 10.1063/1.5136078. hal-02144335v2

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Distributed under a Creative Commons Attribution 4.0 International License
Scaling and conformal symmetries
for plane gravitational waves

P.-M. Zhang<sup>1,2*</sup>, M. Cariglia<sup>3†</sup>, M. Elbistan<sup>2,4‡</sup>, P. A. Horvathy<sup>2,4§</sup>

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

<sup>2</sup>Institute of Modern Physics,
Chinese Academy of Sciences
Lanzhou, China

<sup>3</sup>DEFIS, Universidade Federal de Ouro Preto, MG-Brasil,

<sup>4</sup> Institut Denis-Poisson CNRS/UMR 7013 - Université
de Tours - Université d'Orléans Parc de Grammont,
37200, Tours, (France)

(Dated: May 16, 2025)

<footnote>* e-mail: zhangpm5@mail.sysu.edu.cn</footnote>

<footnote>† e-mail: marco.cariglia@gmail.com</footnote>

<footnote>‡ mailto: mahmut.Elbistan@lmpt.univ-tours.fr</footnote>

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

The isometries of an exact plane gravitational wave are symmetries for both massive and mass-
less particles. Their conformal extensions are in fact chrono-projective transformations (intro-
duced earlier by Duval et al), and are symmetries for massless particles. Homotheties are universal
chrono-projective symmetries for any profile. Chrono-projective transformations also generate new
conserved quantities for the underlying non-relativistic systems in the Bargmann framework. Homo-
theties play a similar role for the lightlike "vertical" coordinate as isometries play for the transverse
coordinates.

J. Math. Phys. 61, 022502 (2020); https://doi.org/10.1063/1.5136078

arXiv:1905.08661v5 [gr-qc]

PACS numbers: 04.20.-q Classical general relativity;

02.20.Sv Lie algebras of Lie groups;

04.30.-w Gravitational waves
# 1. INTRODUCTION

A 4-dimensional plane wave spacetime is given, in Brinkmann coordinates [1], by

$$g_{\mu\nu}dX^{\mu}dX^{\nu} = \delta_{ij}dX^{i}dX^{j} + 2dUdV + K_{ij}(U)X^{i}X^{j}dU^{2}, \quad (1.1)$$

where $X \in \mathbb{R}^2$ and $U, V$ are the transversal resp. light-cone coordinates. (1.1) admits a covariantly constant, null Killing vector $\xi = \partial_V$ and it has a symmetric profile $K_{ij}(U)$. If, in addition, $K_{ij}$ is traceless then the elements of the Ricci tensor vanish identically, $R_{\mu\nu} = 0$, and (1.1) satisfy vacuum Einstein equations, i.e., it is an exact plane gravitational wave [2].

Recent insight into the “Memory Effect” for gravitational waves [3–11] was brought about by a better understanding of their symmetries. For exact plane waves (1.1) the implicitly known isometry group [2, 12–15] maps geodesics to geodesics and yields conserved quantities ; conversely, the latter determine the transverse motion (motion in the X-plane) of test particles in the gravitational wave background [12, 16].

The isometries of plane gravitational waves (1.1) have been identified as Lévy-Leblond’s “Carroll” group with broken rotations [16–22]. However homotheties, $h : M \to M$,

$$U \to U, \quad X \to \chi X, \quad V \to \chi^2 V, \quad \chi = \text{const.} \quad (1.2)$$

play also an important role, namely for the integrability of the geodesic equations [23–25]. The homothety is not an isometry though ; it is a conformal transformation, i.e., infinitesimally

$$L_Y g_{\mu\nu} = 2\omega g_{\mu\nu} \quad (1.3)$$

for some function $\omega$. For the homothety (1.2) $\omega = 1$. Finding all conformal symmetries of (1.1) is a difficult task which requires a series of constraints to be satisfied whose solution depends on the chosen profile and is found only case-by-case [26–31].

On the other hand, plane gravitational waves endowed with a covariantly constant null vector $\xi$ (1.1) can also be viewed as a “Bargmann manifold” for a non-relativistic system in one less dimensions. The underlying non-relativistic motions can be “Eisenhart-Duval (E-D) lifted” as null geodesics [32–35]. The Bargmann point of view provides a powerful framework to investigate the symmetries of the associated non-relativistic system.

Each conformal vector field, (1.3), of the metric generates a conserved quantity $Q$ for null geodesics. If $Y$ preserves, in addition, the vertical vector $\xi$,

$$L_Y \xi = 0, \quad (1.4)$$
then $Q$ projects to a conserved quantity for the underlying non-relativistic dynamics. Conformal vector fields which satisfy also (1.4) generate the “extended Schrödinger group”; such isometries span the “Bargmann group” [32, 33].

Up in the Bargmann space a conserved quantity is associated with any conformal vector field, even if the latter does not preserve $\xi$, though. For instance, the generator $Y_h = (Y_h^\mu)$ of the homothety (1.2) preserves only the direction of the vertical vector

$$L_{Y_h} \xi = \psi \xi, \quad \psi = -2\chi = \text{const.} \qquad (1.5)$$

and generates the charge (5.1) whose conservation determines the vertical coordinates, see sec. 5 below.

The first relation here is in fact a paradigm of the “chrono-projective condition”

$$L_Y \xi = \psi \xi \quad \psi = \text{const.} \qquad (1.6)$$

which plays a fundamental rôle in our investigations. The constant $\psi$ is called the chrono-projective factor.

This condition has been considered at various instances.

Firstly, it was put forward by Duval et al. [33, 37]. Remarkably, these authors introduced it originally as a geometric property related to the Newton-Cartan structure of $d+1$-dimensional non-relativistic spacetime. They called it “chrono-projective property”. Later these same authors realized that chrono-projectivity can actually be derived by light-like reduction from a $d+1, 1$ dimensional relativistic spacetime — their Bargmann space [37, 38], — namely as in (1.6). See also eqn. # (4.4) of [33] or # (5.17)-(5.21) of [38]. In [40] it was rebaptized as the conformal Newton-Cartan group; in [41] it was rediscovered under the name of “enlarged Schrödinger group”. In this paper we return to the original terminology proposed in [36].

The condition (1.6) is almost identical to a property noticed by Hall et al [30], who pointed out that for pp waves it follows from studying the Weyl tensor – however with $\psi$ a function, rather than a constant. The original definition made in [37] would allow $\psi$ to be a function. However, the additional condition # (4.8) these authors impose on the connection $\Gamma$ implies that $\psi$ is necessarily a constant. It is (1.6) (and not the formula of [30]) that reproduces the chrono-projectivity after reduction.

In sec. 2B we show that all special conformal Killing vectors of a non-flat pp-wave (1.1)
are *chro no-projective*. Thus all conformal Killing vector of an Einstein vacuum solution satisfies (1.6).

In this paper we take, on the one hand, advantage of the chronoprojective condition (1.6) to simplify the procedure of finding all conformal vectors of gravitational waves and derive, on the other hand, the associated conserved charges.

Our paper is organized as follows: In Sec.2, after recalling exact plane waves and conformal transformations, we briefly outline the Bargmann [alias Eisenhart-Duval] approach. Chrono-projective transformations are then introduced. Conformal transformations and their subgroups in flat space are spelled out in sec. 3. In sec. 4, conserved quantities of null geodesics related to conformal transformations are discussed. (Symmetries of time-like geodesics were considered recently using non-local conservation laws [43]). In sec. 5, new types of conserved quantities associated with chrono-projective transformations in the Bargmann framework, generalizing those in [39] are considered. The chrono-projective transformations of exact plane gravitational waves are studied in sec.6, using Baldwin-Jeffery-Rosen (BJR) [42] coordinates. In sec.7 we illustrate our general theory on various examples.

## 2. EXACT PLANE GRAVITATIONAL WAVES

### A. Gravitational waves and conformal transformations

For generic profile $K_{ij}(U)$, the isometries of an exact gravitational plane wave (1.1) i.e. diffeomorphisms of spacetime, $f: M \to M$ s.t.

$$f^*g_{\mu\nu} = g_{\mu\nu} \quad \text{infinitesimally} \quad L_Y g_{\mu\nu} = 0 \qquad (2.1)$$

span a 5-parameter group [2, 12–15], which is in fact the subgroup of the Carroll group in 2 + 1 dimensions with broken rotations [12, 16–19]. However the homothety (1.2) [14, 23] generated by the vector field

$$Y_{hom} = X^i \partial_i + 2V \partial_V \qquad (2.2)$$

is not an isometry but a conformal transformation of the pp-wave metric (1.1),

$$f^*g_{\mu\nu} = \Omega^2 g_{\mu\nu} \quad \text{infinitesimally} \quad L_Y g_{\mu\nu} = 2\omega g_{\mu\nu}. \qquad (2.3)$$

For the homothety (1.2) $\Omega^2 = \chi^2 = \text{const.}$ Its role may be understood by looking at the
geodesic motion. For the profile

$$K_{ij}(U)X^i X^j = \frac{1}{2} \mathcal{A}_+(U) ((X^1)^2 - (X^2)^2) + \mathcal{A}_\times(U) X^1 X^2, \quad (2.4)$$

where $\mathcal{A}_+$ and $\mathcal{A}_\times$ are the + and $\times$ polarization-state amplitudes [1, 2, 13, 15], the geodesics are described by

$$\frac{d^2 \mathbf{X}}{dU^2} - \frac{1}{2} \begin{pmatrix} \mathcal{A}_+ & \mathcal{A}_\times \\ \mathcal{A}_\times & -\mathcal{A}_+ \end{pmatrix} \mathbf{X} = 0, \quad (2.5a)$$

$$\frac{d^2 V}{dU^2} + \frac{1}{4} \frac{d\mathcal{A}_+}{dU} \left( (X^1)^2 - (X^2)^2 \right) + \mathcal{A}_+ \left( X^1 \frac{dX^1}{dU} - X^2 \frac{dX^2}{dU} \right) \\ + \frac{1}{2} \frac{d\mathcal{A}_\times}{dU} X^1 X^2 + \mathcal{A}_\times \left( X^2 \frac{dX^1}{dU} + X^1 \frac{dX^2}{dU} \right) = 0. \quad (2.5b)$$

Then the homothety (1.2) multiplies the $\mathbf{X}$ - equation by $\chi$ and the $V$-equation by $\chi^2$; trajectories are therefore taken to trajectories, as illustrated on fig.2.

Alternatively, the geodesic Lagrangian

$$\mathcal{L}_{\text{geo}} = \frac{1}{2} \delta_{ij} \dot{X}^i \dot{X}^j + \dot{U} \dot{V} + \frac{1}{2} K_{ij}(U) X^i X^j \dot{U}^2, \quad (2.6)$$

where the dot means derivation w.r.t. an affine parameter $\sigma$<fnref n="1" /> scales under as,

$$\mathcal{L}_{\text{geo}} \rightarrow \chi^2 \mathcal{L}_{\text{geo}}, \quad (2.7)$$

implying again that the trajectories go into trajectories: The geodesic motion in such a background is thus *scale invariant*. We note that all of these 4D trajectories project to the same curve $\mathbf{X}(U)$ in the transverse plane. Let us record for further use that

$$\mathcal{L}_{\text{geo}} = 0 \quad (2.8)$$

for *null geodesics*, which are thus homothety-invariant by (2.7).

We note that the transverse equations (2.5a) are decoupled from the “vertical” one, (2.5b), and can be solved separately. Once $\mathbf{X}(U)$ has been determined, the result should be inserted into (2.5b) which then can be integrated. Analytic solutions are difficult to find, and therefore the best is to use numerical integration [10]. As it will be further discussed in sec.5

<footnote n="1">¹ We mostly choose $\{\cdot\} = d/dU$.</footnote>
the “new” conserved charge $Q_{hom}$ associated with the homothety provides an alternative way to derive the vertical motion.

For further use, we record some basic facts about the conformal transformations (2.3) : for a conformally flat spacetime, the number of conformal transformations is 15, but for a non-conformally-flat spacetime, their maximum number is 7 [30]. 5 of them are isometries and there is 1 homothety. There may or may not be a 7th transformation which may or may not be an isometry depending on special conditions, see sec. 7.

## B. All conformal Killing vectors of vacuum pp-waves are chrono-projective

The aim of this subsection is to prove the theorem stated in the title.

In their seminal paper Maartens and Maharaj [28] have shown that for a pp-wave the conformal factor of the most general conformal Killing vector $Y$ is (cf. their Eqns. # (29-32))

$$ \omega(U, X, V) = \mu V + a'_{i}(U)X^{i} + b(U), \qquad (2.9) $$

where the constant $\mu$ and the functions $a_i(U)$ and $b(U)$ can be determined by a case-by-case calculation using the additional constraints. Now,

$$ L_Y \partial_V = - \left[ 2\mu V + a'_{i}(U)X^{i} + 2b(U) - a'(U) \right] \partial_V - \left[ \mu X^{i} + a_{i}(U) \right] \partial_{i}, \qquad (2.10) $$

which does not seem to be parallel to $\xi = \partial_V$. Accordingly, the conformal factor (2.9) depends on all coordinates. However, in order to conclude whether $Y$ is chrono-projective or not, one needs to deal with profile-dependent integrability conditions. Taking into account the additional integrability constraints, Maharaj and Maartens found, after tedious calculations, that for a special conformal Killing vector $W$ in a non-flat pp-wave. (A conformal Killing vector is called special when its conformal factor satisfies $\omega_{;\mu\nu} = 0$.)

$$ W = \rho(U^2 \partial_U + \frac{1}{2} \delta_{ij} X^i X^j \partial_V + U X^i \partial_i) + Z \qquad (2.11a) $$

$$ Z = \phi(2V \partial_V + X^i \partial_i) + X, \qquad (2.11b) $$

$$ X = (\alpha U + \beta) \partial_U + (\lambda - \alpha V + c'_{i}(U) X^{i}) \partial_V + (c_i + \gamma \epsilon_{ij} X^j) \partial_i \qquad (2.11c) $$

where $\rho, \phi, \alpha, \beta, \gamma$ are constants and $c_i(U)$ is a function, cf. their eqn. # (56). The first term in $Z$ is a homothety; $X$ is a Killing vector. An additional integrability condition, their eqn. # (55), should also be satisfied.
The special conformal Killing vector **W** is chrono-projective, with conformal and chrono-projective factors

$$ \omega = \omega(U) = \rho U + \phi \quad \text{and} \quad \psi = \alpha - 2\phi, \qquad (2.12) $$

respectively. To complete the proof it is enough to remember that every conformal Killing vector of an Einstein vacuum pp-wave is special conformal [28].

In an alternative approach inspired by [30], one starts with a conformal vector field $Y^\mu$, $L_Y g_{\mu\nu} = 2\omega g_{\mu\nu}$, which satisfies $L_Y C_{\nu\rho\sigma}^\mu = 0$, where $C_{\nu\rho\sigma}^\mu$ is the Weyl tensor. $k^\mu$ is called a principal null direction when $C_{\mu\nu\rho\sigma}k^\sigma = 0$. Then $L_Y [C_{\mu\nu\rho\sigma}k^\sigma] = C_{\mu\nu\rho\sigma}L_Y k^\sigma = 0$. Assuming that the spacetime is non-flat (which excludes, e.g. the Minkowski space) the Weyl tensor is non trivial, allowing us to conclude that the Lie derivative of the null direction should be again a null direction. Now a pp wave is known to be of Petrov type N and have just one principal null direction, namely our “vertical vector” $\xi$. Therefore $L_Y \xi$ is proportional to $\xi$ itself,

$$ L_Y \xi = \alpha(X^\mu) \xi \qquad (2.13) $$

where $\alpha(X^\mu)$ is a function. One can only know it if the conformal vector and spacetime are given. This is as far as that we can go with no further assumptions. The chrono-projective property (1.6), i.e., $\alpha(X) = \psi = \text{const.}$ may or may not be satisfied at this level, as it is manifest from (2.10).

We just mention that calculating the components of the Weyl tensor could lead to an alternative proof of our statement.

### C. The “Bargmann” point of view

Further insight can be gained using the “Bargmann” framework [32, 33]. We first recall that the space-time of a 4-dimensional gravitational wave with metric (1.1) we denote by $(M, g_{\mu\nu})$ can be viewed as the “Bargmann space” for a non-relativistic system in 2 + 1 dimensional non-relativistic spacetime, obtained by factoring out the integral curves of the covariantly constant “vertical” vector $\xi = \partial_V$. The 4 dimensional Bargmann manifold will be referred to as “upstairs” and the underlying non-relativistic 2 + 1-d system will be “downstairs”.

The relativistic (metric) structure of Bargmann space projects to a non-relativistic Newton-Cartan structure [32]. The factor space has coordinates $(U, \mathbf{X})$ with $U$ playing
the rôle of non-relativistic time ; the classical motions “downstairs" are the projections of
the null geodesics “upstairs". See [32, 33] for precise definitions and details.

For example, the null geodesics of 4D flat Minkowski spacetime written in light-cone
coordinates project to free non relativistic motions in (2+1) dimensions.

More generally, let us consider<fnref n="2" />,

$$
ds^2 = d\mathbf{X}^2 + 2dUdV - 2\Phi(U, \mathbf{X}) dU^2 \qquad (2.15)
$$

The geodesics are described by the action $S = \int \mathcal{L}_{geo} d\sigma$ with $\mathcal{L}_{geo}$ in (2.6). The equations of motion are

$$
\ddot{\mathbf{X}} = -(\dot{U})^2 \frac{\partial \Phi}{\partial \mathbf{X}}, \quad \ddot{U} = 0, \quad \frac{d}{d\sigma} (\dot{V} - 2\Phi \dot{U}) = -\frac{\partial \Phi}{\partial U} \dot{U}^2. \quad (2.16)
$$

The related 4D geodesic Hamiltonian is

$$
\mathcal{H} = \frac{1}{2} \mathbf{P}^2 + P_U P_V + \Phi(\mathbf{X}, U) P_V^2, \qquad (2.17)
$$

where $\mathbf{P} = \dot{\mathbf{X}}$, $P_U = \dot{V} - 2\Phi\dot{U}$ and $P_V = \dot{U}$ is a constant. The Hamiltonian (2.17) and the Lagrangian (2.6) are in fact identical. As they do not depend explicitly upon $\sigma$, we also have the constraint

$$
\dot{\mathbf{X}}^2 + 2\dot{U}\dot{V} - 2\Phi(\mathbf{X})(\dot{U})^2 = -\epsilon, \qquad (2.18)
$$

where $\epsilon = 1$ for timelike geodesics and $\epsilon = 0$ for null geodesics.

Focusing our attention at null-geodesics by requiring $\mathcal{H} \equiv 0$ and $P_V = M$ yields the
non-relativistic $2+1$-d Hamiltonian “downstairs”,

$$
H_{NR} = \frac{\mathbf{P}^2}{2M} + M\Phi(\mathbf{X}, U) = -P_U. \qquad (2.19)
$$

$-P_U$ is the Hamiltonian for a non-relativistic particle of mass $M$ downstairs, and $U = M\sigma$
plays the role of Newtonian time. $\Phi(\mathbf{X}, U)$ is identified as a [possibly “time”-dependent]
scalar potential. The projected motion is governed by the single equation in (2.16).

$$
ds^2 = G_{ij}(U, \mathbf{X})dX^i dX^j + 2dUdV - 2\Phi(U, \mathbf{X})dU^2. \qquad (2.14)
$$

<footnote n="2">² In 4D, the metric of any solution of the vacuum Einstein equations $R_{\mu\nu} = 0$ which is conformal to some vacuum Einstein solution can be brought to the form [1, 33],</footnote>

<footnote>where $G_{ij}(U, \mathbf{X})$ is a possibly $U$ [but not $V$] dependent metric on transverse space. This is however *not*
true in $D \ge 5$ dimensions, allowing for more freedom [1, 33].</footnote>
As seen from (2.18), the condition $\mathcal{H} = 0$ implies

$$ \dot{V} = -\dot{U}\left(\frac{1}{2}\frac{\dot{\mathbf{X}}^2}{\dot{U}^2} - \Phi(X, U)\right) = -\left(\frac{1}{2M}\left(\frac{d\mathbf{X}}{dU}\right)^2 - M\Phi(\mathbf{X}, U)\right) = -L_{NR}, \quad (2.20) $$

where $L_{NR}$ is the non-relativistic Lagrangian. Therefore the vertical coordinate is essentially (minus) the classical action along the path $\mathbf{X}(\sigma)$,

$$ V = V_0 - S, \quad S = \int L_{NR} d\sigma, \quad (2.21) $$

as noticed already by Eisenhart [35].

When the potential $\Phi$ happens not to depend on $U$ explicitly, $\partial\Phi/\partial U = 0$, eqn. (2.16) implies that $(\dot{V} - 2\Phi\dot{U})$ is also conserved; eliminating $\dot{V}$ using (2.20) yields the constant of the motion $E = \frac{1}{2}(\dot{\mathbf{X}}/\dot{U})^2 + \Phi$, identified as the conserved energy for unit mass of the projected motion. The special choice (1.1),

$$ \Phi(U, \mathbf{X}) = -\frac{1}{2}K_{ij}(U)X^i X^j \quad (2.22) $$

where $K_{ij}$ is a traceless symmetric matrix, represents, in Bargmann terms, a time-dependent anisotropic (attractive or repulsive) harmonic oscillator in the transverse plane [5, 10, 16].

For a general Bargmann space, those isometries (resp. conformal transformations) which preserve in addition the vertical vector $\xi = \partial_V$, i.e., which satisfy (2.1) resp. (2.3), with the additional condition

$$ f_*\xi = \xi \quad \text{infinitesimally} \quad L_Y\xi = 0 \quad (2.23) $$

span the [generalized] Bargmann (alias extended Galilei) resp. the [generalized] extended Schrödinger group/algebra. One can prove that the conformal factors $\Omega$ resp. $\omega$ depend only on $U$ [32, 33]. The homothety (1.2) belongs to the [extended] chrono-projective group [33, 37, 38] 3 defined, in general, by weakening the constraint (2.23),

$$ f^*g_{\mu\nu} = \Omega^2(U)g_{\mu\nu} \quad \text{infinitesimally} \quad L_Yg_{\mu\nu} = 2\omega(U)g_{\mu\nu} \quad (2.24a) $$

$$ f_*\xi = \Psi\xi \quad \text{infinitesimally} \quad L_Y\xi = \psi\xi \quad (2.24b) $$

where $\Psi$ resp. $\psi$ are constants. It is a further 1-parameter (non-central) extension of the (centrally extended) Schrödinger group.

<footnote n="3">3 The original definition [36] is in the Newton-Cartan structure of non-relativistic spacetime.</footnote>
### 3. THE MINKOWSKI CASE

Here we list the generators of the conformal transformations for flat Minkowski spacetime $dX^2 + 2dUdV$. Plane gravitational waves with non-trivial profile $K_{ij}$ will be studied in sec. 6. The flat metric can, alternatively, be thought as the Bargmann space for a 2 + 1 dimensional NR particle (2.19). The various subalgebras/subgroups can be identified by listing the generators of $O(4, 2)$ [44],

$$Y_U = \partial_U, \quad Y_T^i = -\partial^i, \quad Y_V = -\partial_V, \qquad \text{translations,} \qquad (3.1a)$$

$$Y_{12} = X^1 \partial^2 - X^2 \partial^1, \qquad X^1 - X^2 \text{ rotation,} \qquad (3.1b)$$

$$Y_B^i = U \partial^i - X^i \partial_V, \qquad \text{galilean boosts,} \qquad (3.1c)$$

$$Y_{AB}^i = X^i \partial_U - V \partial^i, \qquad \text{“antiboosts”,} \qquad (3.1d)$$

$$Y_{UV} = U \partial_U - V \partial_V, \qquad \text{U-V boost,} \qquad (3.1e)$$

$$Y_D = 2U \partial_U + X^i \partial^i, \qquad \text{Sch dilatation,} \qquad (3.1f)$$

$$Y_K = U^2 \partial_U + U X^i \partial^i - \frac{\mathbf{X}^2}{2} \partial_V, \qquad \text{Sch expansion,} \qquad (3.1g)$$

$$Y_{C1} = \frac{\mathbf{X}^2}{2} \partial_U - V X^i \partial^i - V^2 \partial_V, \qquad C_1, \qquad (3.1h)$$

$$Y_{C2}^i = X^i U \partial_U + X^i V \partial_V - \left(\frac{\mathbf{X}^2}{2} + UV\right) \partial^i + X^i (X^j \partial^j) \qquad C_2^i. \qquad (3.1i)$$

* The 4D **Poincaré group** $P_4$ is the 10-parameter group of isometries generated by

$$\{Y_U, Y_T^i, Y_V, Y_{12}, Y_B^i, Y_{AB}^i, Y_{UV}\}, \qquad (3.2)$$

some of which do not preserve the vertical vector $\partial_V$, $[Y_{AB}^i, \partial_V] = \partial^i$, $[Y_{UV}, \partial_V] = \partial_V$.

* The isometries which do preserve the vertical vector $\xi = \partial_V$ (2.23) provide us with the 7-parameter **Bargmann group** [32, 33] whose Lie algebra defined by $(\mathcal{L}_{Yg})_{\mu\nu} = 0$, $[Y, \partial_V] = 0$ is spanned by

$$\{Y_U, Y_T^i, Y_V, Y_{12}, Y_B^i\}. \qquad (3.3)$$

* The **extended Schrödinger group** includes conformal transformations which preserve the vertical vector $\partial_V$. The additional non-isometric transformations are non-
relativistic dilations $Y_D$ (3.1f) and expansions $Y_K$ (3.1g) [45] which act as

$$ (\mathcal{L}_{Y_D} g)_{\mu\nu} = 2\Lambda g_{\mu\nu}, \quad [Y_D, \partial_V] = 0, \qquad (3.4a) $$

$$ (\mathcal{L}_{Y_K} g)_{\mu\nu} = 2\Lambda U g_{\mu\nu}, \quad [Y_K, \partial_V] = 0. \qquad (3.4b) $$

The extended Schrödinger group has thus 9 parameters, namely,

$$ \{Y_U, Y_T^i, Y_V, Y_{12}, Y_B^i, Y_D, Y_K\}. \qquad (3.5) $$

A detailed discussion can be found in [32, 33].

* The **chrono-projective group** [33, 36–38, 40] is a subgroup of the [relativistic] conformal group, (2.3), defined by the condition $[Y, \partial_V] = \psi \partial_V$, cf. (2.24b). For 4D Minkowski space it has 10-parameters and is spanned by [33, 37, 38, 40]

$$ \{Y_U, Y_T^i, Y_V, Y_{12}, Y_B^i, Y_D, Y_K, Y_{UV}\}. \qquad (3.6) $$

The $U-V$ boost $Y_{UV}$ in (3.1e) is, in particular, a **chrono-projective isometry**: it satisfies (2.24) with $\omega = 0$ and $\psi = 1$, respectively. The homothety (2.2) can be expressed as

$$ Y_{hom} = Y_D - 2Y_{UV}, \qquad (3.7) $$

therefore belongs to the **chrono-projective algebra**.

These “Bargmannian” expressions $Y_K$, $Y_{UV}$ and $Y_{hom}$ appear in the literature independently, under the name of “special conformal Killing vectors of the pp wave spacetime” [28, 31].

* The (2 + 1)D **Carroll group** [17–19, 33] is the restriction of the Bargmann group to the 3D submanifold $\mathcal{C}$ defined by the constraint $U = 0$,

$$ f^*g = g, \quad f_*\xi = \xi, \quad f(\mathcal{C}) \subset \mathcal{C}. \qquad (3.8) $$

It is a 6 parameter subgroup embedded into $P_4$. $U$-translations $Y_U$ are no more allowed. Its generators are,

$$ \{Y_T^i, Y_V, Y_B^i, Y_{12}\}, \quad U = 0. \qquad (3.9) $$

The Bargmann framework, whose primary aim is to provide a “relativistic” description of non-relativistic physics, has additional bonuses. One of them is to consider, instead
of projecting from 4 to 3 dimensions, the pull-back of a given Bargmann metric to the 3-dimensional submanifold $U = 0$ $^4$. $\mathcal{C}$ has coordinates $(\mathbf{X}, V)$ and carries a *Carroll structure*. $\xi = \partial_V$; the coordinate $V$ is interpreted as “*Carrollian time*” [18, 33].

* The **Schrödinger-Carroll group** is the conformal extension of the Carroll group within the conformal group $O(4, 2)$, obtained by relaxing the isometry condition in (3.8) but still requiring that $\partial_V$ be preserved,

$$ (\mathcal{L}_Y g)_{\mu\nu} = \Omega^2 g_{\mu\nu}, \quad [Y, \partial_V] = 0, \quad U = 0. \qquad (3.10) $$

It has 8 generators, namely those of the Carroll isometries, augmented by non-relativistic dilations and expansions,

$$ \{Y_T^i, Y_V, Y_B^i, Y_{12}, Y_D, Y_K\}, \quad U = 0. \qquad (3.11) $$

and is represented by vectorfield

$$ (\omega_j^i X^j + \gamma^i + \lambda X^i) \frac{\partial}{\partial X^i} + T(\mathbf{X}) \partial_V, \quad T(\mathbf{X}) = \nu - \beta \cdot \mathbf{X} + \frac{1}{2} \kappa \mathbf{X}^2, \qquad (3.12) $$

where $T(\mathbf{X})$ is called, borrowing the Bondi-Metzner-Sachs (BMS) - inspired terminology [46, 47], *supertranslations*.

* The **Chrono-Carroll group** is a 1-parameter extension of the Schrödinger-Carroll group with the weakened condition (2.24b), $[Y, \partial_V] = \psi \partial_V$. This adds $Y_{UV}$ to the Schrödinger-Carroll algebra, yielding 9 generators

$$ \{Y_T^i, Y_V, Y_B^i, Y_{12}, Y_D, Y_K, Y_{UV}\}, \quad U = 0. \qquad (3.13) $$

Infinitesimally, (3.12) is generalized to $Y_V = \psi V + \nu - \beta \cdot \mathbf{X} + \frac{1}{2} \kappa \mathbf{X}^2$.

# 4. GEODESICS AND THEIR SYMMETRIES

In this section we revisit some aspects of geodesics and the conserved quantities associated with Killing and reps. conformal Killing vectors.

<footnote n="4">⁴ The embedding $U = \text{const.}$ would yield an equivalent construction. See also [30].</footnote>
A. Affinely parametrised geodesics

A fully covariant action for a particle (and the only one for a massless particle) is

$$S = \int g_{\mu\nu} \frac{dX^{\mu}}{d\sigma} \frac{dX^{\nu}}{d\sigma} d\sigma. \qquad (4.1)$$

Variation w.r.t. $X^\mu$ gives the geodesic equations in the form

$$\frac{d^2 X^\mu}{d\sigma^2} + \Gamma^\mu_{\alpha\beta} \frac{dX^\alpha}{d\sigma} \frac{dX^\beta}{d\sigma} = 0. \qquad (4.2)$$

Here the $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols of the metric $g_{\mu\nu}$. Because there is no explicit dependence on $\sigma$, we have the constraint (2.18). Choosing $\epsilon$ to be $-m^2 \le 0$,

• When $m \neq 0$, one sees that

$$|m| d\sigma = d\tau, \qquad (4.3)$$

where $m$ is the relativistic mass and $\tau$ is proper time along the curve $X^\mu = X^\mu(\sigma)$.

• However when our geodesic is massless, $m^2 = 0$, then $\sigma$ is called an *affine parameter*, and is defined only up to an affine transformation.

The constraint (2.18) may be written as $g^{\mu\nu}P_\mu P_\nu = -m^2$, where $P_\mu = g_{\mu\nu} \frac{dX^\nu}{d\sigma}$ is the 4-momentum. If $m^2 \neq 0$ one has $P_\mu = |m|g_{\mu\nu} \frac{dX^\nu}{d\tau}$. In flat space one may set $P_0 = -E$, and obtains the well known formulae $E^2 - \mathbf{P}^2 = m^2$, $E = \sqrt{m^2 + \mathbf{P}^2}$.

For the general plane gravitational wave (1.1) the constraint is

$$-K_{ij}(U)X^i X^j P_V^2 + 2P_U P_V + m^2 + P_i P_i = 0. \qquad (4.4)$$

In general the only conserved quantity is $P_V$. If in addition $K_{ij}$ is independent of $U$, we have an additional conserved quantity,

$$P_U = \frac{1}{2P_V} (K_{ij} X^i X^j P_V^2 - m^2 - P_i P_i). \qquad (4.5)$$

We mention for completeness that null geodesics lying in the null hypersurfaces $U = \text{const.}$, referred to as the *null geodesic generators of the null hypersurfaces $U = \text{const.}$*; they may be related to lifts of isotropic geodesics in Newton-Cartan spacetimes [36, 46] for which $V$ is an affine parameter.
## B. Killing resp. conformal Killing vectors

We first recall what happens for *Killing vectors*. If we define the tangent vector of a curve with general parameter $\lambda$ by $T^\mu = \frac{dX^\mu}{d\lambda}$, then a geodesic satisfies

$$ T^\alpha T_{;\alpha}^\mu = h(\lambda) T^\mu \qquad (4.6) $$

for some function $h(\lambda)$, where the “semicolon ; $\alpha$” denotes covariant derivative.

### Killing vectors.

Suppose first that $Y^\mu$ is a *Killing vector field*; then it satisfies Killing’s equations

$$ Y_{\mu;\alpha} + Y_{\alpha;\mu} = 0. \qquad (4.7) $$

It follows that

$$ \mathcal{E} = Y_{\mu} T^{\mu} = g_{\mu\nu} T^{\mu} Y^{\nu} \quad \text{satisfies} \quad \mathcal{E}_{;\alpha} T^{\alpha} = \frac{d\mathcal{E}}{d\lambda} = h(\lambda) \mathcal{E}. \qquad (4.8) $$

Then we get a conserved quantity for the geodesic motion,

$$ Q_Y = g_{\mu\nu} \frac{dX^\mu}{d\sigma} Y^\nu = g_{\mu\nu} \frac{dX^\mu}{d\lambda} \frac{d\lambda}{d\sigma} Y^\nu, \quad \frac{dQ_Y}{d\sigma} = 0. \qquad (4.9) $$

Translations along the “vertical” vector $\xi = \partial_V$ are isometries for any metric of the form (1.1). The associated conserved quantity $P_V = M$ in (4.4) is identified, in the Bargmann framework, with the mass downstairs (as mentioned in section 2 C).

### Conformal Killing vectors.

Now we suppose that we have instead a *conformal Killing vector* $Y^\mu$, i.e., one for which

$$ Y_{\mu;\nu} + Y_{\nu;\mu} = 2\omega g_{\mu\nu} \qquad (4.10) $$

for some function $\omega$. If $\omega$ = constant, $Y^\mu$ is called a *homothetic Killing vector* since it generates a homothety. For a *timelike geodesic* with tangent vector $T^\alpha = \frac{dx^\alpha}{d\tau}$ where $\tau$ is proper time along the geodesic so that $T^\alpha T_\alpha = -1$, we have instead

$$ (Y_{\alpha} T^{\alpha})_{;\mu} T^{\mu} = -\omega. \qquad (4.11) $$

Thus in general the quantity (4.8) i.e. $Y_\alpha T^\alpha = \frac{1}{m} Y^\mu P_\mu$ (where $P_\mu = m T_\mu$ is the momentum of a particle of mass $m$) is *not* constant along the world line. From the point of view of
the covariant Hamiltonian treatment, $Y^\mu P_\mu$ is the moment map generating the lift to the co-tangent bundle of the conformal transformation of the base manifold.

In the special case of a homothety when $\omega = \omega_0 = \text{const.}$, we find that $\frac{d(Y_\alpha T^\alpha)}{d\tau} = -\omega_0 \Rightarrow Y_\alpha T^\alpha = -\omega_0\tau - \omega_{-1}$. Alternatively, deriving again, we have $d^2(Y_\alpha T^\alpha)/d\tau^2 = 0$, which is a covariant version of the Lagrange-Jacobi identity [48].

Conformal Killing vectors do not generate symmetries for timelike geodesics. However, as observed by Jacobi [48], while $Y_\alpha T^\alpha$ is not in general conserved, the two constants of integration above yield, in modern language, the conserved quantities (4.14) associated with non-relativistic dilations and expansions, respectively [49].

### C. Conserved quantities for null geodesics

By contrast, if one considers an affinely parametrised null geodesic with tangent vector $l^\alpha = dx^\alpha/d\sigma$ that satisfies

$$g_{\mu\nu}l^{\mu}l^{\nu} = 0, \quad l^{\mu}_{;\nu}l^{\nu} = 0, \qquad (4.12)$$

we do obtain a constant of the motion,

$$Q_Y = Y_\mu l^\mu, \quad \frac{dQ_Y}{d\sigma} = 0. \qquad (4.13)$$

• As a first illustration, we re-derive the conserved quantities associated with Schrödinger dilations and expansions. For $\Phi(\mathbf{X}) = 0$ and $\Phi(\mathbf{X}) \propto |\mathbf{X}|^{-2}$ (2.15) describes a free particle and the inverse-square potential respectively. The generators $Y_D$ and $Y_K$ (3.4) are conformal Killing vectors. Following the procedure outlined in sect. 2 C, (4.13) yields the conserved Schrödinger quantities downstairs [45],

$$\mathcal{D} = P_i X^i - 2EU \qquad \text{dilation} \qquad (4.14a)$$

$$\mathcal{K} = -EU^2 + U P_i X^i - \frac{M}{2} X_i X^i \qquad \text{expansion} \qquad (4.14b)$$

These quantities are conserved for null geodesics “upstairs” and project to well-defined conserved quantities for the projected non-relativistic motion. In fact $\mathcal{D}, \mathcal{K}$ close, with the projected Hamiltonian $H_{NR}$ to an $o(2, 1)$ algebra [45].

More generally, for motion along null geodesics eqn. (4.13) associates a conserved quantity to each conformal vector $Y$; if the latter preserves in addition also the “vertical” vector
$\xi = \partial_V$, $L_Y\xi = 0$ (2.23), this quantity (we call of the Schrödinger type) projects to a conserved quantity for the underlying non-relativistic dynamics “downstairs” — this is in fact the original idea of the Bargmann framework [32, 33].

## 5. SCALINGS AS CHRONO-PROJECTIVE TRANSFORMATIONS

Now we present a systematic and detailed discussion of scale transformations in the Bargmann framework. We start with the homothety (1.2) – (2.2). Being a conformal vector for the gravitational wave spacetime, (4.13) provides us with

$$Q_{hom} = X^i P_i + 2VP_V, \qquad (5.1)$$

where $P_V$ is associated with the “vertical” Killing vector $\partial_V$. $Q_{hom}$ is conserved for null (but not for timelike) geodesics, as confirmed also by using the equations of motion (2.5b).

Assuming that the transverse motion $X^i(\sigma)$ had already been determined, the conservation of $Q_{hom}$ allows us to determine the evolution of the “vertical coordinate”,

$$V(\sigma) = \frac{Q_{hom}}{2P_V} - \frac{X^i(\sigma)P_i(\sigma)}{2P_V} = \frac{Q_{hom}}{2P_V} - \frac{1}{4P_V} \frac{d}{d\sigma} (X^i(\sigma)X_i(\sigma)). \qquad (5.2)$$

As explained in sec.2 C, the null dynamics in 4D projects, in the Bargmann framework, to an underlying non-relativistic system in 2 + 1D, whereas $P_V$ becomes the mass, $M$; $U$ becomes the non-relativistic time. The non-relativistic Hamiltonian and Lagrangian are recovered as in (2.19) and (2.20), allowing us to express

$$Q_{hom} = Q_{NR} + 2MV_0, \quad Q_{NR} = X^i P_i - 2 \int^U L_{NR}(u)du. \qquad (5.3)$$

$Q_{NR}$ is thus conserved “downstairs” (as it can be confirmed directly using the eqns of motion). More generally, the anisotropic rescaling

$$U \to \mu^b U, \quad X^i \to \mu^a X^i, \quad V \to \mu^c V, \quad \mu = \text{const.} \qquad (5.4)$$

induces, for the Brinkmann metric (1.1),

$$g_{\mu\nu}dX^\mu dX^\nu \to \mu^{2a} \left( \delta_{ij}dX^i dX^j + \mu^{-2a+b+c} 2dU dV + K_{ij}(\mu^b U)\mu^{2b} X^i X^j dU^2 \right).$$

This is conformal provided $c = 2a - b$ and $K_{ij}(\mu^b U) = \mu^{-2b} K_{ij}(U)$. Then, for any $b$,

$$g_{\mu\nu}dX^\mu dX^\nu \to \Omega^2 g_{\mu\nu}dX^\mu dX^\nu, \quad \Omega = \mu^a. \qquad (5.5)$$
$\mathcal{L}_Y \partial_V = -c \partial_V$ implies that the vector field $Y$ is genuinely chrono-projective whenever $c = 2a - b \neq 0$. The associated conserved quantity

$$Q_{a,b} = aX^i P_i + b U P_U + c V P_V \qquad (5.6)$$

induces a conserved charge downstairs,

$$Q_{a,b} = Q_{a,b} + cV_0 P_V = aX^i P_i - b U E - c \left( \int L_{NR} \right) M. \qquad (5.7)$$

Note here the new “chrono-projective” term proportional to the non-relativistic action.

1. For $a = b = c = 1$ we would get the relativistic (isotropic) dilation $U \to \mu U$, $X^i \to \mu X^i$, $V \to \mu V$; when $b = 2a$ we get Schrödinger dilations.

2. When $b = 0$ we recover, for any profile $K_{ij}(U)$, (5.3).

3. If $K_{ij}$ is $U$-independent (as for Brdička metric (7.3) below), then $b = 0$;

4. $b \neq 0$ could be obtained for the [singular] non-trivial profile [26–29]. (See also class 11 in Table 4 of [31]).

$$K_{ij}(U) = \frac{K_{ij}^0}{U^2}, \quad K_{ij}^0 = \text{const.} \qquad (5.8)$$

Choosing $a = 0$ we obtain a *chrono-projective isometry* – namely our $U-V$ boost $Y_{UV}$, $U \to \mu U$, $X^i \to X^i$, $V \to \mu^{-1}V$. Its conserved charge is “chrono-projective”

$$Q_{UV} = U P_U - V P_V = -UE + \int L_{NR} - V_0 P_V. \qquad (5.9)$$

Choosing instead $b = 0$, we recover (as said above) the homothety (5.3) $^5$. Thus this example has again a maximal i.e. a 7-parameter chrono-projective algebra.

The quotient $z = b/a$ is also called the *dynamical exponent* [50, 51]. The typical relativistic value is $z = 1$; for Schrödinger-type expressions $z = 2$. The “chrono-projective” contribution to the associated conserved quantity $Q$ with an additional action integral term arises when $z \neq 2$ [49–52].

<footnote>$^5$ The profile (5.8) is symmetric also w.r.t. Schrödinger dilations. This is not a surprise, though, because the latter is a combination of an UV boosts and of homothety, as seen before.</footnote>
## 6. CHRONO-PROJECTIVE TRANSFORMATIONS IN BJR COORDINATES

Having reviewed the Bargmannian aspects, now we turn to a systematic study of chrono-projective transformations of the gravitational wave metric (1.1) with a non-trivial profile $K_{ij}(U)$. The conformal transformations of pp-waves have been determined some time ago [26–29]. Below we study them in our case of interest in a novel way. Motivated by their utility to identify the isometries [5, 12, 16], we switch to Baldwin-Jeffery-Rosen (BJR) coordinates $(u, \mathbf{x}, v)$ [42],

$$ U = u, \quad \mathbf{X} = P(u)\mathbf{x}, \quad V = v - \frac{1}{4}\mathbf{x} \cdot \dot{a}(u)\mathbf{x}, \qquad (6.1) $$

where $a(u) = P^{\dagger}(u)P(u)$, the $2 \times 2$ matrix $P = (P_{ij})$ being a solution of the Sturm-Liouville problem [5, 22]

$$ \ddot{P} = KP, \quad P^{\dagger}\dot{P} = \dot{P}^{\dagger}P. \qquad (6.2) $$

Then the metric (1.1) takes the form,

$$ g = a_{ij}(u)dx^i dx^j + 2dudv. \qquad (6.3) $$

The conditions (2.24) determine the form of the general chrono-projective vector field

$$ Y = Y^u(x, u)\partial_u + Y^i(x, u)\partial_i + (b(x, u) - \psi v)\partial_v. \qquad (6.4) $$

The conformal Killing equation (2.24) requires,

$$ \partial_i Y^u = 0, \qquad (6.5a) $$

$$ \partial_u Y^u = 2\omega + \psi, \qquad (6.5b) $$

$$ \partial_u Y^v = 0 \qquad (6.5c) $$

$$ \partial_i Y^v + (\partial_u Y^j)a_{ij} = 0, \qquad (6.5d) $$

$$ Y^u(\partial_u a_{ij}) + a_{kj}(u)\partial_i(Y^k(x, u)) + a_{ki}(\partial_j Y^k(x, u)) = 2\omega(u)a_{ij}. \qquad (6.5e) $$

Eqn. (6.5a) implies that $Y^u = Y^u(u)$ hence $\omega = \omega(u)$; then (6.5b) can be solved as

$$ Y^u(u) = \epsilon + \int_0^u (2\omega(w) + \psi)dw. \qquad (6.6) $$

($\epsilon$ = const.) Eqn. (6.5c) implies that $b$ is $u$-independent, $b = b(x)$. Then $Y^i(x, u)$ can be written as $Y^i(x, u) = K^i(u) + F^i(x) + L^i(x, u)$. Substituting into (6.5d) we get,

$$ Y^k(x, u) = F^k(x) - H^{ki}(u)\partial_i b(x), \qquad (6.7) $$
where $H^{ki}(u)$ is Souriau's $2 \times 2$ matrix [12, 16],

$$H^{ki}(u) = \int_{0}^{u} a^{ki}(w) dw, \qquad (6.8)$$

where $(a^{ij})$ is the inverse matrix, $a^{ij}a_{jk} = \delta_{k}^{j}$. Inserting this into (6.5e), the last condition can be written as

$$-2\omega(u)a_{ij}(u) + Y^{u}(u)(\partial_{u}a_{ij}(u)) + \qquad (6.9)$$

$$a_{kj}(u)(\partial_i F^k(x) - H^{km}(u)\partial_i \partial_m b(x)) + a_{ki}(u)(\partial_j F^k(x) - H^{km}(u)\partial_j \partial_m b(x)) = 0.$$

Collecting our results,

$$Y^{u}(u) = \epsilon + 2 \int_{0}^{u} \omega(w) dw + \psi u, \qquad (6.10a)$$

$$Y^{i}(x, u) = F^{i}(x) - H^{ij}(u)\partial_{j}b(x), \qquad (6.10b)$$

$$Y^{v}(x, v) = b(x) - \psi v. \qquad (6.10c)$$

Thus $\epsilon$ = const. is a time translation; the conformal resp. chrono-projective factors $\omega$ and $\psi$ contribute to time dilations. Although the functions $F^i(x)$ and $b(x)$ are generally profile-dependent and can only be determined from (6.9), we can conclude that $b(x)$ is at most quadratic in $x$,

$$b(x) = b_{ij}x^{i}x^{j} - b_{i}x^{i} + h, \quad b_{i}, h = \text{const.} \qquad (6.11)$$

Thus $F^i(x)$ should be at most of the first order in $x^j$, see our examples in the next section.

A particular transformation in (6.10) is $\psi(u\partial_u - v\partial_v)$, associated with the chrono-projective factor, we called before a $u-v$ boost, cf. (3.1e).

**Hamiltonian structure** The geodesic Lagrangian resp. Hamiltonian are, in BJR coordinates,

$$\mathcal{L} = \frac{1}{2} a_{ij}(u) \dot{x}^i \dot{x}^j + \dot{u}\dot{v}, \qquad \mathcal{H} = \frac{1}{2} a^{ij} p_i p_j + p_u p_v, \qquad (6.12)$$

where the canonical momenta $p_\mu = \partial\mathcal{L}/\partial\dot{x}^\mu$ are $p_u = \dot{u}$, $p_v = \dot{v}$, $p_i = a_{ij}\dot{x}^j \Rightarrow \dot{x}^i = a^{ij}p_j$. By (4.13) the conserved quantity associated with the conformal vectorfield $Y$ is,

$$Q_Y = Y^\mu p_\mu = Y^u(x, u) p_u + Y^i(x, u) p_i + (b(x) - \psi v) p_v. \qquad (6.13)$$
By using the Poisson bracket, {$\mathcal{R}, \mathcal{T}$} = $\frac{\partial \mathcal{R}}{\partial x^{\mu}} \frac{\partial \mathcal{T}}{\partial p_{\mu}} - \frac{\partial \mathcal{R}}{\partial p_{\mu}} \frac{\partial \mathcal{T}}{\partial x^{\mu}}$, the generating vector field is recovered as $Y^{\mu} \partial_{\mu} = \{x^{\mu}, Q_{Y}\} \partial_{\mu}$. Rewriting the Hamiltonian as,

$$ \mathcal{H} = p_{\mu} \dot{x}^{\mu} - \mathcal{L} = \frac{1}{2} (a^{-1})^{ij} p_{i} p_{j} + p_{u} p_{v} = \frac{1}{2} (g^{-1})^{\mu\nu} p_{\mu} p_{\nu}, \qquad (6.14) $$

we have

$$ \{Q, \mathcal{H}\} = -\frac{1}{2} \mathcal{L}_Y g^{\mu\nu} p_\mu p_\nu, \quad \text{and} \quad \{Q, \xi\} = -\mathcal{L}_Y(\xi). \qquad (6.15) $$

By (2.24) a charge which is conserved along null geodesics should satisfy

$$ \{Q, \mathcal{H}\} = 2\omega \mathcal{H}, \qquad \{Q, \xi\} = -\psi \xi \qquad (6.16) $$

with $\omega = \omega(u)$, $\psi = \text{const.}$. These formulas come handy to check whether a given quantity is conserved or not.

**Isometries and homothety:** As seen from (6.9), one can not have a generic profile independent symmetry unless $Y^u = 0$. Setting $Y^u = 0$ correlates the conformal and chrono factors, $\omega = -\psi/2 = \text{const.}$ and $\epsilon = 0$. Then (6.9) simplifies as

$$ \partial_i F^k(x) - H^{km}(u) \partial_i \partial_m h(x) = \omega \delta_i^k. \qquad (6.17) $$

Thus, we conclude that $b(x) = -b_i x^i + h$ and $F^i(x) = \omega x^i + f^i$. Putting all into (6.10), we obtain the combination $Y = Y_{iso} + Y_{hom}$

$$ Y_{iso} = f^i \partial_i + h \partial_v + b_i (H^{ij} \partial_j - x^i \partial_v), \quad f^i = \text{const.}, \qquad (6.18) $$

$$ Y_{hom} = \omega(x^i \partial_i + 2v \partial_v). \qquad (6.19) $$

$Y_{iso}$ contains the 5 standard isometries (namely $x$-translations, $v$-translation and $x$-boosts, respectively), identified as the Carroll group with broken rotations [16–18], see sec.3. In the Hamiltonian framework, they become

$$ T^i = \delta^{ij} p_j, \quad T_v = p_v, \quad B^i = H^{ij} p_j - x^i p_v, \qquad (6.20) $$

and they all commute with the geodesic Hamiltonian $\mathcal{H}$ (6.14); the only non-vanishing brackets are 6

$$ \{T_i, B^j\} = \delta_i^j p_v. \qquad (6.21) $$

<footnote n="6">⁶ Note that this is not a central extension; the generators belong themselves to the algebra.</footnote>
Being proportional to $\omega$, $Y_{hom}$ is the homothetic vector field (2.2) exported to BJR coordinates; it induces the conserved charge for null geodesics

$$Q_{hom} = x^i p_i + 2vp_v, \quad \{Q_{hom}, \mathcal{H}\} = 2\mathcal{H}. \qquad (6.22)$$

We would like to emphasize that the isometries (6.18) and the homothety (6.19) are valid for every profile $a_{ij}(u)$ and do not require any integrability equation. For comparison, we note that the homothetic Killing vector found in [28], their eqn. (48), is subject to an integrability condition.

In addition to the isometries and the homothety other (conformal) symmetries may arise; depending on eqn. (6.9) (see the next section).

We just mention that the Ricci flatness of the Brinkmann metric (1.1), $\mathrm{Tr}(K_{ij}) = 0$, can be exported to BJR coordinates as [12, 42]

$$\mathrm{Tr} \left( \dot{L} + \frac{1}{2} L^2 \right), \quad L = a^{-1} \dot{a}. \qquad (6.23)$$

So far we have not used this condition, therefore our solutions (6.10) should apply for the special conformal Killing vectors of any pp-wave.

# 7. EXAMPLES

Now we illustrate our general theory on selected examples.

## A. Minkowski case

In the flat (Minkowski) case $a_{ij} = \delta_{ij}$ and $H^{ij}(u) = u \delta^{ij}$ the constraint (6.9) requires

$$-2u\partial_i\partial_j b(x) + (\partial_i F^j(x) + \partial_j F^i(x)) = 2\omega(u)\delta_{ij} \qquad (7.1)$$

and a simple calculation yields the BJR form of the free chrono-projective Lie algebra (3.6),

$$Y^u(u) = \epsilon + 2\lambda u + \kappa u^2 + \psi u \qquad (7.2a)$$

$$Y^i(x, u) = \omega_j^i x^j + f^i + ub_i + \lambda x^i + \kappa u x^i, \qquad (7.2b)$$

$$Y^v(x, v) = -\frac{\kappa}{2}\mathbf{x}^2 - \mathbf{b} \cdot \mathbf{x} + h - \psi v, \qquad (7.2c)$$
where $\epsilon, \lambda, \kappa, \psi, \omega_j^i, f^i, b_i, h$ are constants. Thus $\epsilon$ generates non-relativistic time translations, $\lambda$ and $\kappa$ are the parameters of Schrödinger dilations and expansions, $f^i$ and $h$ space and vertical translations, $\omega_j^i$ rotations, $b_i$ as Galilei boosts. $\psi$ generates u-v boosts.

In conclusion and consistently with what we said in sec. 3 we have 7 dimensional isometry group which satisfies $L_Y g_{\mu\nu} = 0$ and $L_Y \xi = 0$ and is identified with the Bargmann group (3.3). The latter is extended to the [extended] Schrödinger group (3.5) by the addition of Schrödinger dilations and expansions, which are conformal and leave the vertical vector invariant,

$$L_Y g_{\mu\nu} = 2\omega g_{\mu\nu} \quad \text{with} \quad \omega = \lambda + \kappa u \quad \text{and} \quad L_Y \xi = 0$$

The 10-parameter chrono-projective group is then obtained by adding one more transformation, for example u-v boosts. Alternatively, by (3.7) u-v boosts can be replaced by the homothety.

As mentioned before, conformal Killing vectors of a plane gravitational wave, eqn. (56) in [28], are chrono-projective. However, flat pp-waves were excluded $^7$, explaining why the full $\mathcal{o}(4, 2)$ algebra (3.1) is not recovered : “antiboosts" (3.1d) are, for example, not chrono-projective transformations.

Now we turn to examples with non-trivial profile.

## B. The Brdička metric

Let us first consider the linearly polarized gravitational wave metric given in Brinkmann coordinates $^8$ [53],

$$dX_1^2 + dX_2^2 + 2dUdV - 2\Phi dU^2, \quad \Phi = \frac{1}{2}\Omega^2(X_1^2 - X_2^2), \quad \Omega = \text{const.} \qquad (7.3)$$

The potential $\Phi(\mathbf{X})$ is attractive in the $X_1$ and repulsive in the $X_2$ sector.

We note that the Brdička profile is $U$-independent, leaving us with the homothety as the only conformal symmetry.

We switch to BJR coordinates using the solution of the Sturm-Liouville equation (6.2)

$$P(u) = \operatorname{diag}(\cos \Omega u, \cosh \Omega u). \qquad (7.4)$$

<footnote>$^7$ The solutions (7.2) can be obtained from (2.11a) by putting $H(u, x^A) = 0$ in eqn. (55) of [28].</footnote>

<footnote>$^8$ See also class 13 in Table 4 of [31].</footnote>
The induced Brinkmann → BJR transformation (6.1) i.e.<fnref n="9" />

$$ \left\{ \begin{array}{l} U = u \qquad X_1 = x^1 \cos(\Omega u) \qquad X_2 = x^2 \cosh(\Omega u) \\ V = v + \frac{\Omega}{4}(x^1)^2 \sin(2\Omega u) - \frac{\Omega}{4}(x^2)^2 \sinh(2\Omega u) \end{array} \right. \qquad (7.5) $$

yields the BJR metric (6.3) resp. Souriau matrix

$$ a(u) = \operatorname{diag}(\cos^2(\Omega u), \cosh^2(\Omega u)), \qquad (7.6a) $$

$$ (H^{ij}(u)) = \Omega^{-1}\operatorname{diag}(\tan(\Omega u), \tanh(\Omega u)). \qquad (7.6b) $$

A “screw-type” isometry<fnref n="10" />. When written in Brinkmann coordinates, the metric (7.3) is *U*-independent, implying that the *U*-translations

$$ U \to U + e \qquad (7.7) $$

add a 6th manifest isometry to the 5 standard ones. It is redundant nonetheless instructive to see how this comes about in BJR coordinates : $u \equiv U$-translations are still symmetries but the implementation becomes distorted, see fig. 1.

<figure><img src="image_2.png" /><figcaption>FIG. 1: The *U*-translation in Brinkmann coordinates (7.7) becomes, in BJR coordinates, “screwed”.</figcaption></figure>

To find all chrono-projective vectorfields we follow the recipe outlined in sec.6 : starting

<footnote n="9">⁹ Indices are lifted by the transverse metric, $x^i = a^{ij}x_i$.</footnote>

<footnote n="10">¹⁰ We borrowed the word from [10], where a broken *U* translation combines with a broken rotation into a symmetry, which acts as a “screw” [11, 15, 16], as illustrated in fig.3 below.</footnote>
with the general equations (6.10), a calculation similar to the one in the free case shows that

$$ \begin{cases} \ddot{\omega} + 2\Omega^2(2\omega + \psi) = 0 \\ \ddot{\omega} - 2\Omega^2(2\omega + \psi) = 0 \end{cases} \qquad (7.8) $$

whose consistency requires $2\omega + \psi = 0$. Therefore $\omega$ is a constant and $Y^u$ is a mere $u$-translation, $Y^u = \epsilon$. Then from (6.9) we deduce that

$$ b(x) = -\frac{\Omega^2}{2}((x^1)^2 - (x^2)^2)\epsilon - b_i x^i + h \quad \text{and} \quad F^i(x) = \omega x^i + f^i, \quad f^i = \text{const.} \qquad (7.9) $$

In conclusion, the most general chrono-projective vectorfield for the Brdička metric has 7 parameters: 6 isometries and one conformal generator (namely the homothety). In BJR coordinates it is,

$$ Y_B = \epsilon \left( \partial_u + \Omega \left( x^1 \tan(\Omega u) \partial_1 - x^2 \tanh(\Omega u) \partial_2 \right) - \frac{\Omega^2}{2} \left( (x^1)^2 - (x^2)^2 \right) \partial_v \right) + f^i \partial_i + h \partial_v + \left( \frac{1}{\Omega} \left( b_1 \tan(\Omega u) \partial_1 + b_2 \tanh(\Omega u) \partial_2 \right) - b_i x^i \partial_v \right) + \omega \left( x^i \partial_i + 2v \partial_v \right). \qquad (7.10) $$

The conformal resp. chrono-projective factors are $\omega = -\psi/2 = \text{const.}$; $f^i$ and $h$ generate space and vertical translations, respectively, and the $b_i$ generate boosts.

The parameter $\epsilon \in \mathbb{R}$ generates the additional isometry induced by $u$-translations. Expressed in Brinkmann coordinates the “screw-charge” that this isometry generates,

$$ Q_{\epsilon}^{B} = p_{u} + \Omega \left( \tan(\Omega u) x^{1} p_{1} - \tanh(\Omega u) x^{2} p_{2} \right) - \frac{\Omega^{2}}{2} \left( (x^{1})^{2} - (x^{2})^{2} \right) \qquad (7.11) $$

turns out to be $P_U$, (minus) the “Brinkmann” energy, as expected.

### C. “Screw” for circularly polarized periodic (CPP) waves

*Circularly polarized periodic waves (2.4) with profile* $^{11}$

$$ A_{+}(U) = A_{0} \cos(\omega U), \quad A_{\times}(U) = A_{0} \sin(\omega U), \quad A_{0} = \text{const.} \qquad (7.12) $$

have, beyond the homothety and the usual 5 isometries also a 6th, “screw” isometry, obtained by combining broken rotations with broken $U$-translations [10, 11, 15, 16],

$$ Y_{CPP}^{scr} = \partial_{U} + \frac{\omega}{2}(X^{1}\partial_{2} - X^{2}\partial_{1}). \qquad (7.13) $$

<footnote n="11">11 See also class 14 with $l = 0$ in Table 4 of [31].</footnote>
We find instructive to outline how this result is recovered using our framework. (We choose $\omega = 2$ for simplicity). After bringing the system to a $U$-independent form by a suitable rotation, (eqn. (5.7) of [10]) we solve the Sturm-Liouville equation (6.2) for the profile (7.12) as

$$P(u) = \begin{pmatrix} \cos \Omega_{-} u & \frac{\sin \Omega_{+} u}{\Omega_{+}} \\ -\frac{\sin \Omega_{-} u}{\Omega_{-}} & \cos \Omega_{+} u \end{pmatrix} \quad \text{where} \quad \Omega_{\pm}^{2} = 1 \pm \frac{A_{0}}{2}. \qquad (7.14)$$

Then (6.1) yields (6.3) with transverse metric

$$(a_{ij}) = \begin{pmatrix} \cos^2 \Omega_{-} u + \frac{\sin^2 \Omega_{-} u}{\Omega_{-}^2} & \frac{\cos \Omega_{-} u \sin \Omega_{+} u}{\Omega_{+}} - \frac{\cos \Omega_{+} u \sin \Omega_{-} u}{\Omega_{-}} \\ \frac{\cos \Omega_{-} u \sin \Omega_{+} u}{\Omega_{+}} - \frac{\cos \Omega_{+} u \sin \Omega_{-} u}{\Omega_{-}} & \cos^2 \Omega_{+} u + \frac{\sin^2 \Omega_{+} u}{\Omega_{+}^2} \end{pmatrix}. \qquad (7.15)$$

The Souriau matrix, calculated using (7.15), is

$$(H^{ij}) = \frac{2}{A_0(\det P)} \times \qquad (7.16)$$

$$\begin{pmatrix} \frac{\cos \Omega_{-} u \sin \Omega_{+} u}{\Omega_{+}} - \Omega_{-} \sin \Omega_{-} u \cos \Omega_{+} u & -1 \\ -1 & \Omega_{+} \sin \Omega_{+} u \cos \Omega_{-} u - \frac{\sin \Omega_{-} u \cos \Omega_{+} u}{\Omega_{-}} \end{pmatrix}$$

where $\det P$ is the determinant of the Sturm-Liouville matrix (7.14),

$$\det P = \cos \Omega_{-} u \cos \Omega_{+} u + 2 \frac{\sin \Omega_{+} u \sin \Omega_{-} u}{\sqrt{4 - A_{0}^{2}}}. \qquad (7.17)$$

A tedious calculation for (6.9) then yields the sixth isometry in BJR coordinates as

$$Y_{CPP}^{scr} = \partial_u - \frac{A_0}{2} \left( (H^{11}x^1 - H^{12}x^2)\partial_1 + (H^{21}x^1 - H^{22}x^2)\partial_2 \right) + \frac{A_0}{4} \left( (x^1)^2 - (x^2)^2 \right) \partial_v, \quad (7.18)$$

Circularly polarized periodic waves have thus 6 isometries and one conformal transformation, namely the homothety.

Figs. 2 and 3 show how trajectories are taken into trajectories by the homothety and by the screw transformation, respectively.

### D. “Screw” with expansion

In [23, 24] Andrzejewski and Prencel investigate the memory effect for the linearly polarized gravitational wave with regular $U$-dependent profile [26–29]<fnref n="12" />

$$K_{ij}(U) = \frac{\epsilon^2}{(U^2 + \epsilon^2)^2} \operatorname{diag}(1, -1), \qquad (7.19)$$

<footnote n="12">¹² See also class 11ii in Table 4 of [31].</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 2: For the circularly polarized periodic profile (2.4) with $A_+(U) = \cos(U)$, $A_-(U) = \sin(U)$ the homothety (1.2) takes the trajectory with initial condition $(U_0, \mathbf{X}_0, V_0)$ [in magenta] into that with initial condition $(U_0, \chi \mathbf{X}_0, \chi^2 V_0)$ [in green].</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 3: Dropping the V-coordinate and unfolding the transverse CPP trajectory by adding U yields spirals. The screw-transformation (7.13) (in blue) carries the trajectory in magenta into another trajectory (in green).</figcaption></figure>

whose explicit U-dependence breaks the U-translation symmetry. On the other hand shows that rescalings are broken with the exception of the homothety. However combining the broken U-translation with a broken Schrödinger expansion (3.1g),

$$Y^{scr} = Y_K + \epsilon^2 \partial_U, \qquad (7.20)$$
we to call here a “screwed expansion”, generates a conformal transformation

$$L_{Y^{scr}} g_{\mu\nu} = 2U g_{\mu\nu}, \quad L_{Y^{scr}} \xi = 0. \qquad (7.21)$$

The conserved quantity associated with (7.20)

$$Q^{scr} = (U^2 + \epsilon^2)P_U + U X^i P_i - \frac{\mathbf{X}^2}{2} P_V \qquad (7.22)$$

satisfies

$$\{Q^{scr}, \mathcal{H}\} = 2U\mathcal{H} \qquad (7.23)$$

and is therefore conserved for null geodesics.

In conclusion, the Brinkmann metric with profile (7.19) provides us with an example with 5 isometries and two conformal generators, namely the homothety and the “screw" (7.20).

### E. “Screw” with U-V boost

Ilderton [11, 54] mentions that for the [singular] profile

$$K_{ij}(U) = \frac{K_{ij}^{0}}{(1+U)^{2}}, \quad K_{ij}^{0} = \text{const.} \qquad (7.24)$$

the manifest breaking of U-translation invariance can be cured by “screw-combining” it with a (broken) boost (3.1e), i.e., $Y_{UV}^{scr} = Y_U + Y_{UV}$. His statement is confirmed by calculating the Poisson bracket of the associated charge with the Hamiltonian,

$$Q_{UV}^{scr} = P_U + Q_{UV}, \quad \{Q_{UV}^{scr}, \mathcal{H}\} = 0. \qquad (7.25)$$

The conformal (resp. chrono-projective) factors are $\omega = 0$ and $\psi = 1$. Adding the homothety, we end up with a chrono-isometry plus a chrono-conformal transformation in addition to the standard 5 isometries. Another way of understanding this is to observe that $U \to U - 1$ carries the profile (7.24) to the form (5.8), whose U-V boost symmetry was established in sec. 5.

## 8. CONCLUSION

Plane gravitational waves have long been known to admit, generically, a 5-parameter isometry group [2, 12–15]. The homothety (1.2) is a universal conformal generator. For
a non-conformally-flat spacetime, the maximum number of conformal Killing vectors is 7 [15, 26–31]. The associated conserved quantities determine the transverse-space trajectory $X(\sigma)$ in (2.5a) [12, 16] and for null geodesics $Q_{hom}$ in (5.1) allows us to find also the vertical motion according to (5.2).

The homothety is a chrono-projective transformation introduced by Duval et al. [36, 37]. The fundamental importance of the latter becomes clear from two, related contexts.

Firstly, the chrono-projective property (1.6) is precisely what we need to derive conserved quantities for null geodesics in the gravitational wave spacetime and – unexpectedly – also in the underlying non-relativistic dynamics [39]. The conserved quantities (5.7) they generate involve a novel term, namely the action integral of the underlying non-relativistic dynamics.

Secondly, for an exact plane gravitational wave, all conformal vectors are chrono-projective [28]. Using the chrono-projective condition makes it simpler to determine the conformal transformations for a given profile, as illustrated in sec. 7. Using BJR coordinates is particularly convenient.

Since we have not made use of Ricci flatness in our calculations in sect. 6, our solutions (6.10) should apply for the special conformal Killing vectors of any pp-wave.

Actually, for a type-N (non-flat) null fluid spacetime [31] arbitrary conformal vector fields satisfy the chrono-projective conditions (1.3) and (1.6). Although different coordinates were used, one can notice the similarity between their eqns. (19) and our (6.10). However, unlike in their case, the form of the unknown functions $F(x)$ and $b(x)$ is readily determined, justifying our preference for BJR coordinates.

As it is illustrated in sect. 7, the isometry group of gravitational waves can, in special instances, be enlarged to 6 parameters [15, 26–29]. This is plainly the case when the profile is $U$-independent so that $U$-translations are isometries; an example is given by the Brdička metric (7.3). When the profile does depend on $U$, $U$-translations are manifestly broken, however they can, under special circumstances, be combined with another broken symmetry generator yielding an additional “screw-type” conformal symmetry [10, 11, 15]. Examples are presented in (5.8) and in sec. 7.

Chrono-projective transformations may extend to Newton-Cartan framework [55, 56] as conformal extensions of Carroll manifolds.

At last, we mention that our investigations have some overlap with those in [20, 58], as we discovered during the final phases of this research.
Acknowledgments

We are grateful to Christian Duval (1947-2018) for his advices during 40 years of friend-
ship and collaboration, and his contribution at the early stages of this project. This paper is
dedicated to his memory. We would like to acknowledge Gary Gibbons for discussions during
the long preparation of this paper. We thank also X. Bekaert, P. Pio Kosiński, Krzysztof
Andrzejewski [25] Anton Ilderton [54] and Nikolaos Dimakis [57] and Thomas Helpin. ME
thanks the *Institute of Modern Physics* of the Chinese Academy of Sciences in Lanzhou and
the *Denis Poisson Institute of Orléans-Tours University*. PH thanks the *Institute of Modern
Physics* of the Chinese Academy of Sciences in Lanzhou for hospitality. This work was par-
tially supported by the Chinese Academy of Sciences President’s International Fellowship
Initiative (No. 2017PM0045), and by the National Natural Science Foundation of China
(Grant No. 11975320).

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