# Anyonic spin-Hall effect on the black hole horizon

Loic Marsot, Peng-Ming Zhang, Peter Horvathy

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Loic Marsot, Peng-Ming Zhang, Peter Horvathy. Anyonic spin-Hall effect on the black hole horizon. Physical Review D, 2022, 106 (12), pp.L121503. 10.1103/PhysRevD.106.L121503. hal-03736537v2

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# Anyonic spin-Hall effect on the Black Hole horizon

L.Marsot<sup>1,2,3*</sup>, P.-M. Zhang<sup>4†</sup>, and P. A. Horvathy<sup>5‡</sup>

<sup>1</sup>School of Science, Sun Yat-sen University,
Shenzhen 518107, China

<sup>2</sup>Aix Marseille Univ,
Université de Toulon, CNRS,
CPT, Marseille, France

<sup>3</sup>Department of Physics,
Babes-Bolyai University,
Kogalniceanu Street,
400084 Cluj-Napoca, Romania

<sup>4</sup>School of Physics and Astronomy,
Sun Yat-sen University,
Zhuhai 519082, China

<sup>5</sup>Institut Denis Poisson CNRS/UMR 7013,
Université de Tours - Université d'Orléans,
Tours 37200 (France)

(Dated: May 15, 2025)

<footnote>* marsot@cpt.univ-mrs.fr</footnote>

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

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

Using the fact that the horizon of black holes is a Carroll manifold, we show that an “exotic photon” i.e. a particle without mass and charge but with anyonic spin, magnetic moment and “exotic” charges associated with the 2-parameter central extension of the 2-dimensional Carroll group moves on the horizon of a Kerr-Newman Black Hole consistently with the Hall law.

Phys. Rev. D **106** (2022) no.12, L121503 doi:10.1103/PhysRevD.106.L121503 [arXiv:2207.06302 [gr-qc]].

Key words: Anyonic spin-Hall effect; motion on the horizon of a Kerr-Newmann Black Hole; centrally extended Carroll particle;
# Contents

<table><tr><td>I. Introduction</td><td>3</td></tr><tr><td>II. Doubly extended Carroll particle</td><td>4</td></tr><tr><td>III. Carroll structure of the Kerr-Newman horizon</td><td>6</td></tr><tr><td>IV. Motion on the Kerr-Newman horizon</td><td>7</td></tr><tr><td>V. Carroll symmetry on the horizon</td><td>9</td></tr><tr><td>VI. Conclusion</td><td>11</td></tr><tr><td>Acknowledgments</td><td>12</td></tr><tr><td>References</td><td>12</td></tr></table>

# I. INTRODUCTION

The horizon of black holes is a genuine Laboratory to explore Gravitational Physics. Several interesting and non-intuitive effects, linked to key questions such as the Information Paradox [1, 2], are expected to take place on it. In this paper we add one more item to the list by showing that by “exotic photons” (to be introduced below) exhibit the *Spin Hall effect* on the horizon.

The clue is *Carroll symmetry*. The Carroll group, a “degenerate cousin” of the Galilei group (as put by Lévy-Leblond [3]) is obtained by contracting, in the Poincaré group, the velocity of light to zero, instead of letting it go to infinity, as in the usual Galilean limit [3, 4]. Alternatively, Carroll symmetry is found by restricting a Lorentzian space-time to a null hypersurface [5–9].

Recent attention in the subject arose when Carroll symmetry was found to be relevant, for instance, for physics on a black hole horizon [10–13] : the celebrated BMS group is indeed conformal Carroll [14–16]. Interest in Carroll dynamics has long been limited, though, by that *Carroll particles* (other than tachyons [16, 17]) were *believed not to move* [3–5, 9].

As it will be explained elsewhere [18], “no-motion” is understood by studying deviations
from null geodesics. The “time” coordinate of Carroll geometries is in fact a null coordinate from the ambient spacetime and “not moving” means following the corresponding ambient null geodesic. Another approach [18, 19] relates the immobility of quasiparticles called fractons [20] to Carrollian boost invariance.

The Anomalous Hall Effect (AHE) observed in ferromagnetic crystals had been attributed to an anomalous current [21]. Later it was argued that spinning particles (including light) exhibit a Spin-Hall effect [22, 23] for which a semiclassical explanation was proposed using a Berry phase-extended framework [24–27]. The clue is the anomalous velocity relation,

$$ \frac{dx}{dt} = \frac{\partial \mathcal{E}(\mathbf{p})}{\partial \mathbf{p}} - e\mathbf{E} \times \mathbf{\Theta}, \qquad (I.1) $$

where $\mathcal{E}(\mathbf{p})$ is the band energy, $\mathbf{E}(\mathbf{x})$ the electric field, and 3-vector $\mathbf{\Theta}(\mathbf{p})$ represents the Berry curvature. The anomalous velocity term here is clearly the mechanical counterpart of the anomalous current. Choosing $\mathbf{E}$ in the $x-y$ plane and $\mathbf{\Theta}$ perpendicular to it, eqn. (I.1) reduces to the “exotic” Galilean model based on a two-parameter central extension of the planar Galilei group [28, 29]. The physical relevance of central extensions was recognized by Bargmann [30], followed by [31]. See [32] for another recent application. Extensions hint at deviations from null geodesics. For instance, the deviations of light from geodesic motions can be attributed to the coupling of photon spin to the gravitational field including gravitational waves [8, 33–40].

It has recently been recognized that in 2 + 1 dimensions the Carroll group admits a 2-parameter central extension [41, 42] inducing an extended dynamics [43].

The aim of this paper is to study the Carrollian analog of the Galilean case, illustrated by motion on a specific Carroll geometry: the Kerr-Newman Black Hole horizon.

## II. DOUBLY EXTENDED CARROLL PARTICLE

The two-parameter non trivial central extension of the planar Carroll algebra [41, 42] is given by,

$$ \begin{align*} [J_3, K_i] &= \epsilon_{ij}K_j, \quad [K_i, K_j] = \epsilon_{ij}A_{exo}, \quad [J_3, P_i] = \epsilon_{ij}P_j, \quad [K_i, P_j] = \delta_{ij}P_0, \\ [J_3, P_0] &= 0, \quad [K_i, P_0] = 0, \quad [P_i, P_j] = \epsilon_{ij}A_{mag}, \quad [P_i, P_0] = 0, \end{align*} $$

where $J_3$ is the rotation, the ($K_i$) are boosts, the ($P_i$) spatial translations, ($P_0$) is time translation, and $A_{exo}$ and $A_{mag}$ are the “exotic” and “magnetic” extensions, respectively.
These extensions allow us to endow planar Carroll particles canonically with two additional central charges we shall call, correspondingly, exotic and magnetic and denote by $\kappa_{exo}$ and $\kappa_{mag}$, respectively [18, 43].

Then we show that massless uncharged particles (we call “exotic Carrollian photon”) move on the horizon of a Kerr-Newman Black Hole by following the Hall law, providing us with a Carrollian version of the Spin-Hall effect for anyons and contradicting the “no-motion” statement [43].

Classical particle models associated with the transitive action of a given symmetry group are conveniently constructed using the Kirillov-Kostant-Souriau (KKS) orbit method [31, 44]. In Souriau’s version that we follow here, the motions are determined by the “Souriau two-form” $\sigma$ which is closed and has constant rank. Then $\sigma = d\varpi$ (locally). The “Cartan form” $\varpi$ could be used for a variational calculus [31, 45]. Splitting the Souriau-form as $\sigma = \Omega - d\mathcal{H} \wedge ds$ provides us, moreover, with a symplectic form $\Omega$ whose inverse define in turn commutation relations [45]. Applied to the doubly-centrally-extended Carroll group, the construction yields for a free particle with mass $m$ the Poisson brackets and Hamiltonian $^1$,

$$ \{x_i, x_j\} = \frac{\kappa_{exo}}{mm^*} \epsilon_{ij}, \quad \{x_i, p_j\} = \frac{m}{m^*} \delta_{ij}, \quad \{p_i, p_j\} = \frac{m}{m^*} \kappa_{mag} \epsilon_{ij}, \quad \mathcal{H}_0 \equiv 0, \quad (II.1) $$

where

$$ m^* = m \left( 1 - \frac{\kappa_{exo}}{m^2} \kappa_{mag} \right) \quad (II.2) $$

is an effective mass, assumed not to vanish [18, 43].

Let us underline that: (i) the coordinates do not commute; (ii) the 2nd extension parameter $\kappa_{mag}$ behaves as an internal magnetic field carried by the particle $^2$.

Note that the free Hamiltonian has no kinetic term. The Hamiltonian equations of motion are thus trivially that a [doubly extended] Carroll particle with non-zero effective mass $m^* \neq 0$ does not move.

The free system (II.1) is by construction invariant w.r.t. the action of the doubly extended

<footnote n="1">¹ The commutation relations (II.1) have an overlap with those of the so-called “Maxwell algebra” [46]. The respective Hamiltonians are substantially different, though : the doubly-extended Carroll and the “Maxwell” or “enlarged” [47] systems are fundamentally different as they are built from different ingredients : constant external electromagnetic fields for “Maxwell” and intrinsic central extension parameters for our doubly-extended Carroll, respectively.</footnote>

<footnote n="2">² In the Galilean theory [29] the mass is part of the moment map.</footnote>
Carroll group

$$x \rightarrow Ax + c \quad s \rightarrow s - b \cdot Ax + f, \quad p \rightarrow Ap + mb \quad (II.3)$$

where $A \in SO(2)$, $c, b \in \mathbb{R}^2$, $f \in \mathbb{R}$, cf. # (3.15) in [43]. Here $s$ is Carrollian time. All this follows from the structure of the Carroll group upon applying the KKS algorithm.

Coupling such a particle to an electromagnetic field modifies both the symplectic structure and the Hamiltonian [31, 45, 48]. For simplicity, we restrict our attention at an uncharged doubly-extended Carroll particle<fnref n="3" /> with magnetic moment $\mu$ and anyonic spin $\chi$ in a static e.m. field ($B, E$). The Poisson brackets are as in (II.1) but the modified Hamiltonian $\mathcal{H} = \mu\chi B$, yields, for $m \neq 0$,

$$(x^i)' = \mu\chi \frac{\kappa_{exo}}{\kappa_{exo}\kappa_{mag} - m^2} \epsilon^{ij} \partial_j B, \quad p_i' = m^2 \frac{\mu\chi\partial_i B}{m^2 - \kappa_{exo}\kappa_{mag}} \quad (II.4)$$

where the prime means $d/ds$.

These equations are both of the first order. The one for $x$ is fully decoupled and can be solved on its own, but the one for $p$ depends on the result for $x$.

Moreover, letting here $m \rightarrow 0 \kappa_{exo}$ drops out as long as it does not vanish, leaving us with

$$(x^i)' = (\mu\chi) \epsilon^{ij} \frac{\partial_j B}{\kappa_{mag}} \quad \text{and} \quad p_i' = 0. \quad (II.5)$$

Note that the usual electromagnetic terms were switched off by choosing $e = 0$, but the magnetic field plays a new role – that of an electric potential.

Motion in a curved Carroll manifold was considered in [9, 43], however the gravitational field does not couple to the Carrollian equations of motion. An intuitive justification is that gravitational minimal coupling impacts, through the covariant derivative, the equation for the momentum, however not that for the position. However (as noticed above) the momentum has no impact on the $x$-motion.

### III. CARROLL STRUCTURE OF THE KERR-NEWMAN HORIZON

A Kerr-Newman Black Hole characterized by its mass $M$, angular momentum $J$, and charge $Q$ can be described by using the Eddington-like coordinates $(u, r, \vartheta, \phi)$ [49]. In these

<footnote n="3">³ Charged particles are studied in [18].</footnote>
coordinates, the metric

$$g = - \frac{\Delta}{\Sigma} \left( du + \frac{\Sigma}{\Delta} dr - a \sin^2 \vartheta d\phi \right)^2 + \frac{\sin^2 \vartheta}{\Sigma} \left( adu - (r^2 + a^2)d\phi \right)^2 + \Sigma d\vartheta^2 + \frac{\Sigma}{\Delta} dr^2 \qquad (III.1a)$$

$$\Sigma = r^2 + a^2 \cos \vartheta, \quad \Delta = r^2 + a^2 + Q^2 - 2Mr, \qquad (III.1b)$$

where $a = J/M$, and its inverse are regular on the (outer) horizon $\mathcal{H}$ of a Kerr-Newman black hole defined by $r = r_+ = M + \sqrt{M^2 - (a^2 + Q^2)} = \text{const.}$ hypersurface defined by $\Delta = 0$. Note that the seemingly problematic $dr^2$ terms in (III.1a) containing $\Delta$ in their denominator cancel one another out. Then we consider the $2+1$ dimensional structure [11, 12] whose ingredients are the induced metric and a vector,

$$\tilde{g} = g|_{\Delta=0} = \frac{\sin^2 \vartheta}{\Sigma} \left( a du - (r^2 + a^2) d\phi \right)^2 + \Sigma d\vartheta^2, \qquad (III.2a)$$

$$\xi = \partial_u + \Omega_H \partial_\phi \quad \text{where} \quad \Omega_H = \frac{a}{r^2 + a^2}, \qquad (III.2b)$$

respectively. Here $\Omega_H$ is the angular velocity of the horizon. The restricted metric (III.2a) is singular as made manifest by the coordinate change $(\vartheta, \phi, u) \mapsto (\vartheta, \phi = \phi - \Omega_H u, u)$, which leads to the metric,

$$\tilde{g} = \frac{(r^2 + a^2) \sin^2 \vartheta}{\Sigma} d\vartheta^2 + \Sigma d\vartheta^2 \quad \& \quad \xi = \partial_u, \qquad (III.3)$$

The kernel is generated by the vector $\xi$, $\tilde{g}(\xi) = 0$. Thus we have a degenerate metric and a vector field in its kernel, allowing us to conclude that the horizon $\mathcal{H}$ of a Kerr-Newman black hole carries a Carroll structure $(\mathbb{S}^2 \times \mathbb{R}, \tilde{g}, \xi)$ [5]. The degenerate “metric” $\tilde{g}$ carries the geometric information of the $\mathbb{S}^2$ part of the black hole, while $\xi$ generates the $\mathbb{R}$ part.

The horizon of a Kerr-Newman Black Hole carries a magnetic field

$$B = (2aQr_+(r_+^2 + a^2)) \frac{\cos \vartheta}{(r_+^2 + a^2 \cos^2 \vartheta)^3}, \qquad (III.4)$$

while in comoving coordinates the electric field vanishes [18].

## IV. MOTION ON THE KERR-NEWMAN HORIZON

A massive particle associated with the unextended Carroll group can stay fixed, but can not move [5, 16]. However the horizon is a $2+1$ dimensional Carroll manifold, therefore the
particle may have an extended dynamics associated with the double central extension with parameters $\kappa_{exo}$ and $\kappa_{mag}$ [18, 41–43].

Now we show that the extended dynamics can lead to motion, namely on the Black Hole horizon. Remember first that geodesics on the horizon are necessarily massless [5]<fnref n="4" />. An “exotic photon” i.e. one with no mass and charge, $m = 0$ and $e = 0$, but with nonvanishing magnetic moment $\mu$, anyonic spin $\chi$ and double central extension can be coupled to the electromagnetic field through a spin-field term $\mathcal{H} = \mu\chi B$ where $B$ is the magnetic field (III.4) on the horizon. Then the equations of motion (II.5) describe an Anomalous Spin-Hall Effect with $\nabla B$ behaving as an effective electric field, $\mu\chi$ as an effective electric charge, and $\kappa_{mag}$ as an effective magnetic field.

Coupling to the gravitational field amounts to replacing the derivative on $p_i$ by a covariant derivative [43]. However, this does not change the velocity equation which is indeed the only relevant one for the poor Carrollian dynamics: the momentum equation remains decoupled.

Having a non-zero gradient for the magnetic field (III.4) requires non-zero electric charge $Q$ and angular momentum $J$ (since $a = J/M$).

Using (comoving) angular coordinates $(\vartheta, \varphi, u)$ we see that the electric field induced on the horizon vanishes. The radial component would survive, but disappears in the $2+1$ restriction. The gradient of $B$ is tangent to the longitudinal great circles $\varphi = \text{const}$. By (II.5), the motion is governed by,

$$ (x^{\vartheta})' = 0, \quad (x^{\varphi})' = \left(2aQ r_{+} (r_{+}^{2} + a^{2}) \frac{\mu\chi}{\kappa_{mag}}\right) \frac{(r_{+}^{2} - 5a^{2} \cos^{2}\vartheta)}{(r_{+}^{2} + a^{2} \cos^{2}\vartheta)^{4}} \sin\vartheta. \quad (IV.1) $$

Thus our “exotic photon” performs azimuthal circular motion with $\vartheta = \text{const.}$, parameterized by $\varphi$. Consistently with the Hall behavior, the motion is perpendicular to the (longitudinal) effective electric field $\nabla B$ (which vanishes at the poles and takes its maximum on the equator). The direction of the rotation is correlated with the angular momentum $J$ and the charge $Q$ which should not vanish — and this is precisely the reason that we consider Kerr-Newman Black Holes. The angular velocity goes smoothly to zero as we approach the poles and depends on the radius of the horizon roughly as $r_{+}^{-3}$, implying that the rotation would be more important for smaller black holes.

<footnote n="4">⁴ Such a photon trajectory could be created by turning on a lamp and then throwing it into a stationary black hole in such a way that when the lamp crosses the horizon, the photons be emitted in the direction of the horizon’s null generator.</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1: On the horizon of a Kerr-Newman Black Hole the **velocity field** (IV.1) is perpendicular to the axis of rotation and obeys the Anomalous Hall law with an **effective electric field** $E^* = \nabla B$ which is tangent to the longitudinal great circles with $\mu\chi$ playing the role of an effective electric charge. The arrows indicate the directions and norms.</figcaption></figure>

The rotation we have just found, although reminiscent of the frame-dragging by a rotating black hole, is however *unrelated* to it: frame-dragging is hidden in the coordinates, which are comoving with the horizon.

V. CARROLL SYMMETRY ON THE HORIZON

We conclude our Letter with a short survey of the symmetries, conveniently studied by looking at the Cartan 1-form [31] $\varpi$ defined by $\sigma = d\varpi$ as mentioned in sec.II. Switching to $v = p/m$ before letting $m \to 0$ yields

$$ \varpi = \frac{\kappa_{exo}}{2} \epsilon_{ij} v^i dv^j + \frac{\kappa_{mag}}{2} \epsilon_{ij} x^i dx^j + \mu\chi Bds. \qquad (V.1) $$

In presymplectic terms, the Noether theorem says: a vector field $X$ is a symmetry of the dynamics, if $L_X\varpi$ vanishes up to a total derivative, $L_X\varpi = df$. Then Cartan's formula implies that

$$ Q_X = i_X \varpi - f \qquad (V.2) $$
is conserved [31].

The isometry group of the Kerr-Newman horizon $\mathcal{H}$ is SO(2) $\times \mathcal{T}$, generated by the vectorfields

$$X = \partial_{\tilde{\varphi}} + \mathcal{T}(\vartheta, \tilde{\varphi})\partial_s, \qquad (V.3)$$

where the “supertranslation” $\mathcal{T}$ is an arbitrary function of the coordinates $(\vartheta, \tilde{\varphi})$ on the horizon. Thus:

* Translations of the black hole horizon generated by $\partial_\varphi$ change $\varpi$ by a surface term, $L_{\partial_\varphi} \varpi = df$ with $f = -(\kappa_{mag}/2)\vartheta$. Thus (V.2) yields the conserved quantity,

$$p_{\varphi} = \kappa_{mag}\vartheta. \qquad (V.4)$$

This unusual expression is consistent with (3.18c) in [43].

* Now look at the zeroth order expansion of a supertranslation, i.e. a (Carrollian) time translation $X = \partial_s$. We readily have $L_X \varpi = 0$, and so

$$i_X \varpi = \mu \chi B \equiv \mathcal{H}, \qquad (V.5)$$

identified with the Carroll Hamiltonian is conserved.

* For a general supertranslation $X = \mathcal{T}(\vartheta, \varphi)\partial_s$ we have, instead,

$$L_X \varpi = (\mu \chi) B \partial_i \mathcal{T} dx^i, \qquad (V.6)$$

which is not a total derivative in general due to $d(L_X \varpi) \propto dB \wedge d\mathcal{T} \neq 0$ unless $\mathcal{T} = \mathcal{T}(\vartheta)$ — for which (V.2) then yields a conserved quantity. If the supertranslation is, for example, induced by the magnetic field, $\mathcal{T} = \mathcal{T}(B)$ e.g. for $\mathcal{T}_n \propto B^n$ for some positive integer $n$, then $L_X \varpi = n d\mathcal{H}^{n+1}$ is a total derivative, providing us with an infinite tower of conserved quantities $Q_n = \mathcal{H}^{n+1}$, — which are however mere powers of the Hamiltonian in (V.5).

* It is instructive to study Carroll boosts in (II.3),

$$\boldsymbol{x} \rightarrow \boldsymbol{x} \qquad s \rightarrow s - \boldsymbol{b} \cdot \boldsymbol{x}, \qquad \boldsymbol{b} \in \mathbb{R}^2, \qquad (V.7)$$

characteristic for the Carroll symmetry. They belong to the isometry “bottom” of BMS supertranslations [15]. “Horizontal” boosts along $\partial_\varphi$ are broken by the magnetic
field, however for “vertical” boosts, $\mathcal{T} = -b_{\vartheta}\vartheta$, (V.2) provides us with <fnref n="5" />

$$Q = (\mu\chi) \left[ -B\vartheta + \int Bd\vartheta \right]. \qquad (V.8)$$

So far we proceeded as follows: first we solved the equations of motion and then checked that the associated Noetherian quantities are indeed conserved along the trajectories. The conservation of $p_{\vartheta}$ in (V.4), of $\mathcal{H}$ in (V.5) or of even of the weird boost-momentum in (V.8) is indeed manifest from that

$$\vartheta = \text{const.} \qquad (V.9)$$

along the trajectories as we had found earlier.

However conservation laws are often used conversely, i.e., to derive the motions. Can we proceed in the reversed direction? Remarkably, the answer is YES: their explicit forms manifestly require (V.9) for being conserved!

## VI. CONCLUSION

The absence of the kinetic term in their Hamiltonian implies that Carroll particles have a purely anomalous velocity relation. Position and momenta are partially decoupled and we end up with first-order equations. The motion of an “exotic photon”, (II.5), is poor but not entirely trivial: it exhibits the Anyonic Spin-Hall effect [23].

The horizon of a Kerr-Newman Black Hole realizes these conditions: its magnetic field $B$ (III.4) induces anomalous Hall motion for our “exotic photon”. Masslessness is mandatory for getting non-trivial dynamics [18].

Particles of the type of our “exotic photons” might actually play a rôle in condensed matter physics as quasiparticles [50]. Here we took them chargeless for simplicity, however they could in principle carry also an electric charge [18].

The double central extension of the Carroll group is a mathematical fact [41, 42]. But is it a physical reality? With no experimental data at hand, we just recall Dirac about his magnetic monopole [51]:

> “This new development . . . is merely a generalisation of the possibilities . . . Under these circumstances one would be surprised if Nature had made no use of it.”

<footnote n="5">⁵ For $B = \text{const.}$ we would get $Q = 0$ consistently with [16].</footnote>
We note also that our gravitational ideas could in principle be tested in Laboratory by
the remarkable analog of a Kerr-Newman Black Hole which could be created in condensed
matter [52, 53].

Note added. After the revised version of our paper has been resubmitted we were informed
by T. R. Perche [54] that they are also considering similar problems and arrived, by following
similar methods, to similar but slightly different conclusions.

Acknowledgments

We are grateful to J. Balog, G. Gibbons, M. Chernodub, M. Stone, P. Chruściel,
M. Plyushchay and Z. Silagadze for advice. LM is thankful to the Physics Department
of Babeş-Bolyai University for hospitality. PMZ was partially supported by the National
Natural Science Foundation of China (Grant No. 11975320).

[1] S. W. Hawking, “The Information Paradox for Black Holes,” [arXiv:1509.01147 [hep-th]].

[2] S. W. Hawking, M. J. Perry and A. Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. **116** (2016) no.23, 231301. doi:10.1103/PhysRevLett.116.231301 [arXiv:1601.00921 [hep-th]].

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

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

[5] C. Duval, G. W. Gibbons, P. A. Horvathy and P. M. Zhang, “Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time,” Class. Quant. Grav. **31** (2014) 085016. [arXiv:1402.0657 [gr-qc]].

[6] C. Duval, G. W. Gibbons and P. Horvathy, “Celestial mechanics, conformal structures and gravitational waves,” Phys. Rev. D **43** (1991), 3907-3922. doi:10.1103/PhysRevD.43.3907 [arXiv:hep-th/0512188 [hep-th]].

[7] X. Bekaert and K. Morand, “Embedding nonrelativistic physics inside a gravitational wave,” Phys. Rev. D **88** (2013) no.6, 063008. doi:10.1103/PhysRevD.88.063008 [arXiv:1307.6263 [hep-th]]. K. Morand, “Embedding Galilean and Carrollian geometries I. Gravitational waves,” J. Math. Phys. **61** (2020) no.8, 082502. doi:10.1063/1.5130907 [arXiv:1811.12681 [hep-th]].
[8] 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]].

[9] E. Bergshoeff, J. Gomis and G. Longhi, “Dynamics of Carroll Particles,” Class. Quant. Grav. **31** (2014) no.20, 205009. doi:10.1088/0264-9381/31/20/205009 [arXiv:1405.2264 [hep-th]].

[10] L. Donnay, G. Giribet, H. A. Gonzalez and M. Pino, “Supertranslations and Superrotations at the Black Hole Horizon,” Phys. Rev. Lett. **116** (2016) no.9, 091101. doi:10.1103/PhysRevLett.116.091101 [arXiv:1511.08687 [hep-th]].

[11] L. Marsot, “Caractérisation géométrique des structures de Bargmann et de Carroll et des groupes de Schrödinger et de Bondi-Metzner-Sachs,” Master thesis. Marseille University 2016. (unpublished).

[12] L. Donnay and C. Marteau, “Carrollian Physics at the Black Hole Horizon,” Class. Quant. Grav. **36** (2019) no.16, 165002. doi:10.1088/1361-6382/ab2fd5 [arXiv:1903.09654 [hep-th]].

[13] L. Freidel and P. Jai-akson, “Carrollian hydrodynamics from symmetries,” [arXiv:2209.03328 [hep-th]]. L. Freidel and P. Jai-akson, “Carrollian hydrodynamics and symplectic structure on stretched horizons,” [arXiv:2211.06415 [gr-qc]].

[14] A. Bagchi, “Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories,” Phys. Rev. Lett. **105** (2010) 171601.

[15] C. Duval, G. W. Gibbons, P. A. Horvathy, “Conformal Carroll groups and BMS symmetry,” Class. Quant. Grav. **31** (2014) 092001. [arXiv:1402.5894 [gr-qc]].

[16] C. Duval, G. W. Gibbons and P. A. Horvathy, “Conformal Carroll groups,” J. Phys. A **47** (2014) no.33, 335204. doi:10.1088/1751-8113/47/33/335204 [arXiv:1403.4213 [hep-th]].

[17] J. de Boer, J. Hartong, N. A. Obers, W. Sybesma and S. Vandoren, “Carroll Symmetry, Dark Energy and Inflation,” Front. in Phys. **10** (2022), 810405. doi:10.3389/fphy.2022.810405 [arXiv:2110.02319 [hep-th]].

[18] L. Marsot, P.-M. Zhang, M. Chernodub, P. A. Horvathy, “Hall effects in Carroll dynamics,” (work in progress).

[19] L. Bidussi, J. Hartong, E. Have, J. Musaeus and S. Prohazka, “Fractons, dipole symmetries and curved spacetime,” SciPost Phys. **12** (2022) no.6, 205. doi:10.21468/SciPostPhys.12.6.205 [arXiv:2111.03668 [hep-th]]. More details will be given in [18] below.

[20] M. Pretko, “The Fracton Gauge Principle,” Phys. Rev. B **98** (2018) no.11, 115134.
doi:10.1103/PhysRevB.98.115134 [arXiv:1807.11479 [cond-mat.str-el]].

[21] R. Karplus, J.M. Luttinger, "Hall Effect in Ferromagnetics," Phys. Rev. 95 (1954) 1154.
DOI:https://doi.org/10.1103/PhysRev.95.1154

[22] M. I. Dyakonov and V. I. Perel, "Possibility of orientating electron spins with current". Sov. Phys. JETP Lett. 13: 467. (1971); J. E. Hirsch (1999). "Spin Hall Effect". Phys. Rev. Lett. 83 (9): 1834-1837. arXiv:cond-mat/9906160. doi:10.1103/PhysRevLett.83.1834.

[23] K. Yu. Bliokh, Yu. P. Bliokh, "Topological spin transport of photons: the optical Magnus effect and Berry Phase," Phys. Lett. **A333**, 181 (2004) [physics/0402110]; M. Onoda, S. Murakami, and N. Nagaosa, "Hall effect for light," Phys. Rev. Lett. **93**, 083901 (2004) [cond-mat/0405129];

[24] M. C. Chang and Q. Niu, "Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands," Phys. Rev. B **53** (1996), 7010-7023.
doi:10.1103/PhysRevB.53.7010 [arXiv:cond-mat/951104 [cond-mat]].

[25] M. C. Chang and Q. Niu, "Berry Phase, Hyperorbits, and the Hofstadter Spectrum," Phys. Rev. Lett. **75** (1995), 1348-1351. doi:10.1103/PhysRevLett.75.1348 [arXiv:cond-mat/9505021 [cond-mat]].

[26] P. A. Horvathy, "Anomalous Hall Effect in non-commutative mechanics," Phys. Lett. **A 359** (2006), 705-706. doi:10.1016/j.physleta.2006.07.033 [arXiv:cond-mat/0606472 [cond-mat]].

[27] C. Duval and P. A. Horvathy, "Chiral fermions as classical massless spinning particles," Phys. Rev. D **91** (2015) no.4, 045013. doi:10.1103/PhysRevD.91.045013 [arXiv:1406.0718 [hep-th]];
P. Zhang and P. A. Horváthy, "Anomalous Hall Effect for semiclassical chiral fermions," Phys. Lett. **A 379** (2014), 507-510. doi:10.1016/j.physleta.2014.12.003 [arXiv:1409.4225 [hep-th]].

[28] J.-M. Lévy-Leblond, "Galilei group and Galilean invariance," in *Group Theory and Applications*, Loebl Ed., II, Acad. Press, New York, p. 222 (1972).

[29] C. Duval and P. A. Horvathy, "The 'Peierls substitution' and the exotic Galilei group," Phys. Lett. **B 479** (2000), 284-290. doi:10.1016/S0370-2693(00)00341-5 [arXiv:hep-th/0002233 [hep-th]]. C. Duval and P. A. Horvathy, "Exotic Galilean symmetry in the noncommutative plane, and the Hall effect," J. Phys. **A 34** (2001), 10097-10108. doi:10.1088/0305-4470/34/47/314 [arXiv:hep-th/0106089 [hep-th]].

[30] V. Bargmann, "On Unitary ray representations of continuous groups," Annals Math. **59** (1954), 1-46. doi:10.2307/1969831
[31] J.-M. Souriau, *Structure des systèmes dynamiques*, Dunod (1970, © 1969); *Structure of Dynamical Systems. A Symplectic View of Physics*, translated by C.H. Cushman-de Vries (R.H. Cushman and G.M. Tuynman, Translation Editors), Birkhäuser, 1997.

[32] L. Inzunza and M. S. Plyushchay, “Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background,” JHEP 01 (2022), 179. doi:10.1007/JHEP01(2022)179 [arXiv:2109.05161 [hep-th]].

[33] C. Duval, Z. Horvath and P. A. Horvathy, “Fermat principle for spinning light,” Phys. Rev. D 74 (2006), 021701(R). doi:10.1103/PhysRevD.74.021701 [arXiv:cond-mat/0509636 [cond-mat]].

[34] P. Gosselin, A. Berard and H. Mohrbach, “Spin Hall effect of photons in a static gravitational field,” Phys. Rev. D 75 (2007), 084035. doi:10.1103/PhysRevD.75.084035 [arXiv:hep-th/0603227 [hep-th]].

[35] M. A. Oancea, J. Joudioux, I. Y. Dodin, D. E. Ruiz, C. F. Paganini and L. Andersson, “Gravitational spin Hall effect of light,” Phys. Rev. D 102 (2020) no.2, 024075. doi:10.1103/PhysRevD.102.024075 [arXiv:2003.04553 [gr-qc]].

[36] C. Duval, L. Marsot and T. Schücker, “Gravitational birefringence of light in Schwarzschild spacetime,” Phys. Rev. D 99 (2019) no.12, 124037. doi:10.1103/PhysRevD.99.124037 [arXiv:1812.03014 [gr-qc]].

[37] V. P. Frolov, “Maxwell equations in a curved spacetime: Spin optics approximation,” Phys. Rev. D 102 (2020) no.8, 084013. doi:10.1103/PhysRevD.102.084013 [arXiv:2007.03743 [gr-qc]].

[38] N. Yamamoto, “Spin Hall effect of gravitational waves,” Phys. Rev. D 98 (2018) no.6, 061701. doi:10.1103/PhysRevD.98.061701 [arXiv:1708.03113 [hep-th]].

[39] L. Andersson, J. Joudioux, M. A. Oancea and A. Raj, “Propagation of polarized gravitational waves,” Phys. Rev. D 103 (2021) no.4, 044053. doi:10.1103/PhysRevD.103.044053 [arXiv:2012.08363 [gr-qc]].

[40] A. I. Harte and M. A. Oancea, “Spin Hall effects and the localization of massless spinning particles,” Phys. Rev. D 105 (2022) no.10, 104061. doi:10.1103/PhysRevD.105.104061 [arXiv:2203.01753 [gr-qc]].

[41] J. A. de Azcarraga, F. J. Herranz, J. C. Perez Bueno and M. Santander, “Central extensions of the quasiorthogonal Lie algebras,” J. Phys. A 31 (1998), 1373-1394. doi:10.1088/0305-4470/31/5/008 [arXiv:q-alg/9612021 [math.QA]].
[42] A. Ngendakumana, “Group Theoretical Construction of Planar Noncommutative Systems,” [arXiv:1401.5213 [math-ph]]. A. Ngendakumana, J. Nzotungicimpaye and L. Todjihounde, “Group theoretical construction of planar Noncommutative Phase Spaces,” J. Math. Phys. **55** (2014), 013508. doi:10.1063/1.4862843 [arXiv:1308.3065 [math-ph]].

[43] L. Marsot, “Planar Carrollian dynamics, and the Carroll quantum equation,” J. Geom. Phys. **179** (2022), 104574. doi:10.1016/j geomphys.2022.104574 [arXiv:2110.08489 [math-ph]].

[44] A. A. Kirillov, “Unitary representations of nilpotent Lie groups,” Russ. Math. Surv. **17**, p.57, (1962) B. Kostant, “Quantization and Unitary Representations,” in *Lectures in Modern Analysis and Applications III*. ed. R.M. Dudley et al, Springer Lecture Notes in Mathematics (170), p.87.

[45] P. A. Horvathy, “The Noncommutative Landau problem and the Peierls substitution,” Annals Phys. **299** (2002), 128-140. doi:10.1006/aphy.2002.6271 [arXiv:hep-th/0201007 [hep-th]].

[46] J. Negro, M. A. del Olmo and J. Tosiek, “Anyons, group theory and planar physics,” J. Math. Phys. **47** (2006), 033508. doi:10.1063/1.2168399 [arXiv:math-ph/0512007 [math-ph]].

[47] P. A. Horvathy, L. Martina and P. C. Stichel, “Enlarged Galilean symmetry of anyons and the Hall effect,” Phys. Lett. B **615** (2005), 87-92. doi:10.1016/j.physletb.2005.04.004 [arXiv:hep-th/0412090 [hep-th]].

[48] S. Sternberg, in: Proc. 2nd Bonn Conf. on Diff. Geom. Methods in Math. Phys., in: Springer Lecture Notes in Mathematics, vol. 676, Springer, Berlin, 1978, pp. 1-80.

[49] E. T. Newman, R. Couch, K. Chinnapared, A. Exton, A. Prakash and R. Torrence, “Metric of a Rotating, Charged Mass,” J. Math. Phys. **6** (1965), 918-919. doi:10.1063/1.1704351

[50] Venema, L., Verberck, B., Georgescu, I. et al. “The quasiparticle zoo,” Nature Phys **12**, 1085-1089 (2016). https://doi.org/10.1038/nphys3977

[51] P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. Roy. Soc. Lond. A **133** (1931) no.821, 60-72. doi:10.1098/rspa.1931.0130

[52] R. A. Tinguely and Andrew P. Turner, “Optical analogues to the equatorial Kerr-Newman black hole,” Commun Phys **3**, 120 (2020). https://doi.org/10.1038/s42005-020-0384-5

[53] P. M. Zhang, C. Duval, G. W. Gibbons and P. A. Horvathy, “Soft gravitons and the memory effect for plane gravitational waves,” Phys. Rev. D **96** (2017) no.6, 064013. doi:10.1103/PhysRevD.96.064013 [arXiv:1705.01378 [gr-qc]].

[54] Finnian Gray, David Kubizňák, T. Rick Perche, and Jaime Redondo-Yuste, “Carrollian Motion
in Magnetized Black Hole Horizons,” draft sent by T. R. Perche.