$$ \mathbf{D} = \sum_a \mathbf{D}_a = \sum_a m_a \mathbf{x}_a = M \mathbf{X}_0, \qquad (\text{IV.25a}) $$

$$ M = \sum_a m_a, \qquad \mathbf{X}_0 = \frac{\sum_a m_a(\mathbf{x}_a)_0}{M}. \qquad (\text{IV.25b}) $$

**Theorem IV.4.** A system composed of non-interacting massive Carroll particles is C-boost symmetric. None of the individual particles which compose the system can move, implying obviously that its center of motion (IV.25b) is fixed, consistently with the conservation of its Carroll momentum (IV.25a).

Similarly, the conserved total linear momentum is the sum of separately conserved particle momenta, $\mathbf{P} = \sum_a m_a \mathbf{v}_a$. This expression contains a curious ambiguity reminiscent of the “internal energy” and “internal angular momentum” of Galilean systems [77, 144, 145]: by (IV.24) each of the $\mathbf{v}_a$ is defined up to an arbitrary constant vector $\mathbf{c}_a$.

Lagrangian (IV.23) of N uncoupled massive particles in d dimensions is formally equivalent to a single particle in the dimension Nd. Therefore, most of the conclusions we made in the previous section for a single Carroll particle still apply.

Moreover, recall the peculiar causality of Carroll systems. The light-cone degenerates into a “vertical line” [2], therefore the particles at different positions are causally disconnected and cannot interact with each other in a way consistent with the Carroll symmetry. Hence, a collection of free Carroll particles is strictly the sum of its parts, not more.

What happens when we put such a system of massive, not interacting particles into an external electromagnetic field? Position and momenta, $\mathbf{x}_a$ and $\mathbf{v}_b$ are uncoupled, and each $\mathbf{x}_a$ remains fixed, as we state in Theorem IV.2. Therefore:

**Corollary IV.2.** The center of mass $\mathbf{X}$ (IV.25b) of N non-interacting massive Carroll particles remains motionless even the system is put into an electromagnetic field.

What happens when an interaction is introduced by adding some, a priori arbitrary, potential? Can such a system move? To see that, we consider coupled Carroll particles through an arbitrary position and momentum-dependent N-body potential $V(\mathbf{x}_a, \mathbf{v}_a, s)^{27}$,

$$ \mathcal{S} = \int \left( \sum_a m \mathbf{v}_a \cdot \mathbf{x}'_a - V \right) ds = \mathcal{S}_0 - \int V ds. \qquad (\text{IV.26}) $$

<footnote n="27">27 Remember that $\mathbf{v}_a$ is (linear momentum)/mass, and is a priori unrelated to the velocity, $\mathbf{x}'_a$.</footnote>
The exotic photon is a massless, electrically-neutral particle which has spin and possesses a nonvanishing magnetic moment. Although this particle carries a “bosonic” name, it can be of a fermionic or even anyonic nature as well.

A distant physical analogue of the exotic photon in relativistic fundamental field theory can be played by Dirac or Majorana neutrinos which, being almost massless and neutral, could possess a sizable magnetic moment in certain exotic scenarios [89, 90]. A non-vanishing neutrino magnetic moment would have important astrophysical and cosmological consequences [91].

In the condensed matter setup, neutral fermionic excitations appear in strongly correlated electronic systems in Kondo lattice materials such as, for example, the newly found compound YbIr$_3$Si$_7$. Neutral fermions cannot, evidently, carry electric charge while they can support thermal energy transfer endowing the charge insulator with metallic thermal conducting properties. This type of exotic materials is sensitive to the background magnetic field implying that neutral fermions couple to the magnetic degrees of freedom [92]. The corresponding electrically-neutral massless excitations are remote condensed-matter analogues of the exotic photons proposed in Ref. [32].

Here we illustrate our theory by a massless particle$^{24}$, $m = 0$, $e = 0$, $\mu\chi = \kappa_{mag} = 1$, put into the electromagnetic field

$$ \mathbf{E}(\mathbf{x}) = f(|\mathbf{x}|) \hat{\mathbf{x}} \quad \text{and} \quad B(\mathbf{x}) = \frac{|\mathbf{x}|}{(1 + |\mathbf{x}|^2)^3}. \qquad (\text{IV.47}) $$

For (IV.45) $\kappa_{exo}$ drops out and (IV.42) reduces to,

$$ (x^i)' = (\mu\chi) \epsilon^{ij} \frac{\partial_j B}{\kappa_{mag}} \quad \text{and} \quad p'_i = 0, \qquad (\text{IV.48}) $$

which is indeed the key formula we used on the horizon used of a black hole [32]. The external electromagnetic fields are switched off by $e = 0$ but there remains an effective electric field, $\mathbf{E}^* = \mu\chi\nabla B$, coming from the spin-field coupling, (II.21), with $\mu\chi$ behaving as an effective charge and $B$ as an effective potential. The various fields and the Hall motion (IV.48) are shown in 4a-b. Fig. 5 prefigures Fig. 10.

An intuitive picture can be obtained by observing that the motion of our Carroll anyon in

<footnote n="24">24 In addition to photons and neutrinos in fundamental particle physics, uncharged and massless quasiparticles also appear in the solid state context as phonon excitations in crystals and fluids.</footnote>
The equations of motion still separate,

$$m_a \boldsymbol{x}'_a = \frac{\partial V}{\partial \boldsymbol{v}_a}, \qquad (\text{IV.27a})$$

$$m_a \boldsymbol{v}'_a = -\frac{\partial V}{\partial \boldsymbol{x}_a}. \qquad (\text{IV.27b})$$

It follows that these particles do move whenever the potential depends on $\boldsymbol{v}_a$, $\frac{\partial V}{\partial \boldsymbol{v}_a} \neq 0$. Their center of mass $\mathbf{X}_0$ in (IV.25b) remains however fixed when the ($\boldsymbol{v}$-dependent) forces on the r.h.s. of (IV.27a) add up to zero,

$$\sum_a \frac{\partial V}{\partial \boldsymbol{v}_a} = 0. \qquad (\text{IV.28})$$

This condition is indeed a sort of “velocity dual” to Newton’s Third law, obtained by changing $\boldsymbol{x}_a$ to $\boldsymbol{v}_a$ $^{28}$.

Condition (IV.28) is satisfied in particular by the curious momentum-dependent potential of the 2-particle system $V = (\frac{1}{2}(\boldsymbol{v}_1 - \boldsymbol{v}_2)^2)$ (we took for simplicity $m_1 = m_2 = 1$) considered in [10], for which (IV.27) requires $\boldsymbol{x}'_1 = -\boldsymbol{x}'_2 = \boldsymbol{v}_1 - \boldsymbol{v}_2$ and $\boldsymbol{v}'_a = 0$ so that $\boldsymbol{v}_a(s) = (\boldsymbol{v}_a)_0 = \text{const.}$ cf. for non-interacting systems in (IV.24). Thus the position coordinates move with constant but opposite Carrollian velocities, whereas their center of mass remains fixed.

More generally, (IV.28) holds whenever the potential is the momentum-equivalent of the usual two-body potential,

$$V = \sum_{a>b=1}^{N} V_{ab}(|\boldsymbol{v}_a - \boldsymbol{v}_b|). \qquad (\text{IV.29})$$

## Relation to the dipole moment

The conservation of the center of mass is reminiscent of that of the dipole moment in fractonic systems, discussed in the next section. Can we incorporate into the multi-Carrollian picture dipole momentum conservation? The answer is positive provided we notice that the quantities $m_a$ play only a formal role of “mass” and can in fact also be viewed as charge. This association is natural, especially if we notice the similarity between the operator of time translations in Carrollian systems which is the Hamiltonian $\mathcal{H}$, which is the counterpart of

<footnote>$^{28}$ The relation (IV.28) should be compared to the discussion in [77] (pp. 141-142 and on pp. 152-153) where Souriau states “Galilean relativity + the Maxwell Principle imply Newton’s 3rd law”. Remember, though, that boost symmetry is broken in the Carroll context as seen above. See also also [5, 144, 145] for related developments.</footnote>
<figure><img src="image_5.png" /><figcaption>FIG. 4: (a) The planar magnetic field $B(|\boldsymbol{x}|)$ in (IV.47) (b) induces a radial effective electric field $\boldsymbol{E}^* = \nabla B$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5: For uncharged particles the radial planar magnetic field $B$ in (IV.47) induces a radial effective electric field $\boldsymbol{E}^*$. An “exotic photon” i.e. a mass and chargeless planar Carroll anyon with nonzero spin $\chi$, magnetic moment $\mu$ and magnetic charge $\kappa_{\text{mag}}$, (IV.45), rotates around the origin along concentric circles, consistently with the generalized Hall Law, (IV.42). The rotation stops on a ring of a fixed radius (marked by full **dots in red**) where the effective electric field vanishes, $\boldsymbol{E}^* = \mathbf{0}$, and its direction is reversed when $\boldsymbol{E}^*$ changes sign.</figcaption></figure>

the effective electric field $\boldsymbol{E}^*$ can be pictured also as motion o,scrobed to the “Mexican hat” potential $-B(|\boldsymbol{x}|)$ unfolded to three dimensions above the plane as shown in Fig. 6.
the charge operator $Q$ in the fracton models (see, for example, [50, 146]). Therefore we view the $m_a$ as a charge, $m_a \to q_a$ and allow the latter to take both positive and negative values, say $q_a = \pm q$. Then the definition of the center of mass (IV.25a) reduces to the familiar expression for the conserved total dipole moment:

$$ \mathbf{D} = \sum_{a} q_{a} \mathbf{x}_{a}. \qquad (\text{IV.30}) $$

**Theorem IV.5.** The motion of interacting Carroll particles conserves the total dipole moment (IV.30) as for fractons, provided the interaction is realized via the momentum-dependent two-particle potential (IV.29).

### C. Massless Carroll particles

Massless Carroll particles constructed along the KKS lines [17] behave differently: unlike their massive counterparts, they can move (as observed on the black hole horizon [35]). The difference is highlighted by their momentum: we readily see from (IV.1c) that when the mass vanishes, $m = 0$, the conserved norm of their momentum, $|\pi|$, is promoted to a Casimir invariant. Massless particles with zero or with non-zero momentum behave differently as we show, below, providing us with two different classes of massless Carroll particles.

Consider first massless particles with non-zero conserved momentum,

$$ \pi \neq 0. \qquad (\text{IV.31}) $$

Then the KKS algorithm yields the spinless “Fermat particles” of geometrical optics [17, 58, 59, 77] — which are both Galilean and relativistic and Carrollian. This comes from the coadjoint orbits for all three of the Galilei, Poincaré and Carroll groups $\mathfrak{G}$, $\mathfrak{P}$, $\mathfrak{C}$ we are interested in are in fact those of their common Euclidean subgroup, $\mathfrak{E}$, obtained when boosts and time translations are dropped [59]. Geometric Optics is Euclidean — both in direct and historical sense [147].

“Spinless light” [alias a Fermat particle] propagates instantaneously along the light rays of geometrical optics, see Theorem IV.6 below. In detail, the geometrical model of Ref. [77] describes a light ray by a pair $(\mathbf{x}, \mathbf{u})$, where $\mathbf{x}$ is an arbitrary point on the ray and $\mathbf{u}$ is a unit vector$^{29}$ such that $\pi = ku$ is oriented along the ray. The norm of the momentum

<footnote>$^{29}$ This is akin to special relativity where the spatial velocity of massless particles also has unit norm, with</footnote>
<figure><img src="image_7.png" /><figcaption>FIG. 6: The motion of an anyon in the effective electric field $\mathbf{E}^*$ can be lifted to “Mexican Hat” potential surface $-B(|\mathbf{x}|)$ above the plane. The trajectories are horizontal “cords” (circles) starting near the “top” with high velocities and, then decreasing progressively until we arrive at $|\mathbf{x}| = |\mathbf{x}_0|$ where the force vanishes, $\mathbf{E}^*(|\mathbf{x}_0|) = 0$, and the rotation stops. The “cord” degenerates to fixed points around the “bottom rim”. Then, as we climb up the outer “flange” by letting $|\mathbf{x}|$ increase, the anyon restarts to move with ever increasing velocity — however in the opposite direction. The 3D picture projects to Fig. 5.</figcaption></figure>

For comparison, we consider also a charged particle,

$$e = 1 \qquad \kappa_{\text{mag}} = \mu\chi = 1, \qquad \kappa_{\text{exo}} \neq 0. \qquad (\text{IV.49})$$

As discussed above, such quasi-particles with might be relevant in Condensed Matter physics (see also [27]).

Then we put both particles into the electric same electric field equal to the effective field in the uncharged case,

$$\mathbf{E}(|\mathbf{x}|) = \nabla B = \frac{1 - 5|\mathbf{x}|^2}{(1 + |\mathbf{x}|^2)^4} \hat{\mathbf{x}}. \qquad (\text{IV.50})$$

Then (IV.42) reduces to the usual Hall motion (I.2) with position-dependent frequency.

The magnetic and electric fields could be, in fact, chosen also independently, see (IV.25). Pairing, for example, a constant electric field $\mathbf{E} = \mathbf{E}_0 = \text{const.}$, e.g., $e\mathbf{E}_0 = eE_0(1, 0)$ with $B = B(|\mathbf{x}|)$ written in (IV.47),

$$\mathbf{E}^* = \mathbf{E}_0 + \mu\chi \frac{1 - 5|\mathbf{x}|^2}{(1 + |\mathbf{x}|^2)^4} \hat{\mathbf{x}} \qquad (\text{IV.51})$$

yields Fig. 7.
$|\pi| = k$ is a Casimir invariant which is assumed not to vanish.

### Coadjoint action of the euclidean group $\mathfrak{E}$.

An element $(A, c)$ of the euclidean group $\mathfrak{E}$ acts on $\mu_0 = (\ell, \pi)$ in the dual euclidean algebra $\mathfrak{e}^*$ labeled as,

$$ \ell \to A\ell - c \times A\pi, \quad \pi \to A\pi. \qquad (IV.32) $$

We immediately have that the above Euclidean orbits lie within the Carroll orbits (IV.1) with $m = 0$ and $D = 0$. The Casimir invariants are,

$$ |\pi| = k, \quad \ell \cdot \frac{\pi}{k} = j, \qquad (IV.33) $$

where the non-zero constant $k$ is Souriau's "color" [77], which corresponds to the absolute value of the conserved momentum. $j$ is the spin of the particle. Choosing the basepoint $\mu_0 = (0, \pi) \in \mathfrak{e}^*$, the KKS algorithm [77, 86, 87] endows $\mathcal{E}$ with Cartan resp. Souriau forms

$$ \varpi_0 = k \mathbf{u} \cdot d\mathbf{x}, \quad \sigma_0 = k d\mathbf{u} \wedge d\mathbf{x}. \qquad (IV.34) $$

Equivalently, the dynamics is given by the Lagrangian

$$ \mathcal{L}_0 ds = \pi \cdot x' ds = k \mathbf{u} \cdot x' ds = \pi \cdot d\mathbf{x} \qquad (IV.35) $$

cf. (IV.9) in the massive case, which makes the Carroll symmetry manifest, since C-boost (I.1) act only (by reparametrization) on $s$. In fact, $s$ can even be dropped from the evolution space (IV.4), leaving us with

$$ \mathcal{E} = \mathbb{R}^3 \times \mathbb{S}^2 = \{(x, u)\} \qquad (IV.36) $$

which is thus 5-dimensional. While the dynamics can be found within the Carroll group, $s$ has effectively dropped out from the Cartan 1-form (IV.34) and thus also from the evolution space, (IV.36). Indeed, the contribution proportional to $ds$ to $\varpi_0$ in (IV.3a) valid in the massive case is suppressed by letting the mass to go to zero.

We emphasise that while $s$ is not needed to describe the evolution of the system, we could (trivially) add it by hand and return to the Carroll the evolution space (IV.4). This would

<footnote>units where $c = 1$. This highlights, once again, that these orbits lie in Poincaré orbits.</footnote>
<figure><img src="image_8.png" /><figcaption>FIG. 7: Combining a constant external electric field $\mathbf{E} = \mathbf{E}_0$ with $\nabla B$ induced by the spin-field term (II.21) yields effective electric field $\mathbf{E}^*$ (IV.41). Velocity and force are perpendicular, as required by the Hall law. The number of equilibrium points has been considerably reduced.</figcaption></figure>

V. CONTRACTION FROM RELATIVISTIC ANYONS

An insight into the physical origin of exotic Galilean symmetry has been provided by its derivation from relativistic anyons [80] by a tricky contraction [81, 82]. Now we study the Carrollian counterpart [2]: one defines the timelike coordinate $x^0 = s/c$ and then sends $c \to \infty$. We start with a massive and spinless particle, described by the Cartan 1-form [59, 82],

$$ \varpi = p_{\mu}dx^{\mu} = p_{i}dx^{i} - m\sqrt{1 + \frac{p^2}{m^2c^2}} ds. \qquad (V.1) $$

Taking the limit $c \to \infty$, we readily recover the Cartan 1-form (III.4) for Carroll,

$$ \varpi_0 = p_i dx^i - mds. \qquad (V.2) $$

Notice that the Carroll limit is in fact more straightforward than the Galilean one, for which the limit $c \to \infty$ of $mc^2\sqrt{1 + \frac{p^2}{m^2c^2}} dt$ requires regularization, achieved in [82] by adding a central extension i.e., working with the Bargmann model.

Now consider a massive relativistic model with spin. Generalizing the Jackiw-Nair ansatz $s = \kappa c^2$ [81], we set $s = \chi + \kappa c^2$, whose Cartan form is (V.2) plus a term describing the spin
restore the action of Carroll boosts on (IV.4) — but its action would have no effect on the dynamics.

A light ray is determined by the characteristic foliation of $\sigma_0$ in (IV.34), $\delta u = 0$ and $\delta x \propto u$, which is indeed a straight line oriented along $u$. The description by $(u, x)$ is however redundant: $x_1$ and $x_2$ lie on same ray if they differ by a multiple of $u$. A light ray is thus labeled ultimately by the two orthogonal vectors,

$$ u \text{ and } q = x - (u \cdot x)u, \quad u \cdot q = 0. \qquad (IV.37) $$

The “space of motions” of a Fermat particle is thus $\mathcal{M} = TS^2$, the tangent bundle of the 2-sphere, depicted in Fig. 2. In conclusion:

<figure><img src="image_2.png" /><figcaption>FIG. 2: (a) A Fermat particle in flat space moves along oriented straight lines. The unit vector $u$ gives its direction and the vector $q$ points to its closest point to a (chosen) origin. $x$ is any point on the ray. (b) The space of motions is thus the tangent bundle of the unit 2-sphere, endowed with $k$-times its canonical symplectic form.</figcaption></figure>

**Theorem IV.6.** A free, mass, spin and charge-less Carroll particle is equivalent 30 to the euclidean “Fermat particle” in refs. [58, 59, 77] and moves instantaneously, along the oriented straight lines of geometrical optics.

<footnote n="30">30 Up to interchanging Galilean and Carrollian times, $t \leftrightarrow s$.</footnote>
state of the particle, $\varpi = \varpi_0 + \varpi_s$. The rather complicated term $\varpi_s$ can be read off from \# (4.1) in [82]. The result is seen not depend on the time coordinate. The *Carrollian limit* of $\varpi_s$ is *therefore the same as in the Galilean case*, and we end up with the Cartan form,

$$ \varpi_{any} \equiv \varpi = p_i dx^i - mds + \kappa c^2 d\phi + \chi d\phi + \frac{\theta}{2} \epsilon^{ij} p_i dp_j. \qquad (V.3) $$

The term $\kappa c^2 d\phi$ here diverges when $c \to \infty$, however its exterior derivative behaves regularly, yielding the exotic part of the symplectic form (IV.10) with $\kappa_{exo} = \kappa$,

$$ \Omega_{exo} = dp_i \wedge dx^i + \frac{\kappa}{2m^2} \epsilon^{ij} dp_i \wedge dp_j. \qquad (V.4) $$

Thus the exotic Carroll charge $\kappa_{exo}$ is deduced from the relativistic anyon model, just as in the Galilean case. The remarkably difference is however that we *do not recover the second charge*, $\kappa_{mag}$ by our contraction.

Some more insight can be gained by regularizing $\varpi$. We follow [82]. Consider a *trivial* $\mathbb{R}$-central extension to the Poincaré group. Its Cartan 1-form will be $\tilde{\varpi} = \varpi + \alpha d\theta$. Now, we choose $\alpha = -\kappa c^{225}$ and $\theta = \phi - w/c^2$ for some $w \in \mathbb{R}$. This implies that $\alpha d\theta = -\kappa c^2 d\phi + \kappa dw$, and thus,

$$ \tilde{\varpi} = p_i dx^i - mds + \chi d\phi + \frac{\kappa}{2m^2} \epsilon^{ij} p_i dp_j + \kappa dw. \qquad (V.5) $$

This expression is to be compared to $\varpi_{exo}$ in (IV.8). The presymplectic structure has gained one dimension, spanned by the parameter $w$. Such a situation arises in the “non-exotic” Galilean case, where the infinite energy is regularized similarly [60, 82].

In conclusion, the exotic term is recovered (up to a total derivative), but the magnetic term is missing, as it does also from the presymplectic 2-form $d\varpi$, cf. (IV.8).

## VI. HALL ON THE HOLE (MOTION ON THE HORIZON)

A genuine physical application of (possibly doubly-extended) Carroll dynamics is provided by motion on the *horizon of a black hole* [15–19]. We first recall the following general properties:

* A black hole is characterized by its mass $M$, angular momentum $J$ and charge $Q$:

<footnote n="25">25 The quantity $\alpha$ has to be a constant (or depend only on $\theta$) for $\alpha d\theta$ to be exact.</footnote>
An intuitive explanation is that there is no canonical quantity with the dimension of velocity in that theory, so either they do not move at all, or they move instantaneously. Since we know that they do move (the trajectory is a ray), it has to be instantaneous.

Recalling that Carroll boosts, (I.1), leave the free dynamics (IV.34) - (IV.35) invariant, the Noether theorem yields the conserved bost momentum,

$$ \mathbf{D} \equiv 0, \qquad (IV.38) $$

cf. eqn. # (VI.17) of [17], which is again consistent with letting $m \to 0$ in the massive boost momentum (IV.8c). However, unlike in the massive case, its conservation is does not allow us to infer any conclusion about the dynamics — consistently with those instantaneous motions we had found.

Extension to spin in $d \ge 3$ space dimensions (which allows us to study the “spin-Hall effect for light”, see [57–59]) is not considered in this paper. Anyonic spin in $d = 2$ will be studied in sec. VI. The description of Carroll particles can be extended to curved space [10].

## Coupling to a background electromagnetic field.

A free particle constructed by the KKS method can be coupled to an electromagnetic field by the rule (II.48). The Carrollian version can be obtained by taking the Carrollian limit [148]. One starts with the relativistic electromagnetic tensor

$$ \mathbb{F} = F_{ij}dx^i \wedge dx^j + \frac{E_i}{c}dx^i \wedge dx^0. \qquad (IV.39) $$

However, rescaling the electric field as $E_i \mapsto \tilde{E}_i = E_i/(cC)$ one gets, after introducing Carrollian time by $x^0 = s/C$ (so that $C \to \infty$ is the Carrollian limit),

$$ \mathbb{F} = F_{ij}dx^i \wedge dx^j + \tilde{E}_i dx^i \wedge ds \qquad (IV.40) $$

where Carroll time, $s$, has replaced the usual time coordinate$^{31}$.

The electric and magnetic fields defined above satisfy, in the vacuum, the Carroll-Maxwell laws [7], either in their electric version,

$$ \left\{ \begin{array}{ll} \epsilon_{ijk} \partial_j E^k + \partial_s B^i = 0, & \partial_i B^i = 0 \\ \partial_s E^i = 0, & \partial_i E^i = 0 \end{array} \right. \qquad \text{electric} \qquad (IV.41) $$

<footnote>$^{31}$ The same result is obtained [7, 92] by switching to light-cone coordinates $u$ and $v$. Then the restriction to the $u = \text{const.}$ null hypersurface is again (IV.40) with $\tilde{E}_i = E_i - F_{iz}$.</footnote>
• The horizon $\mathcal{H}$ is a Carroll manifold [15–17];

• The horizon is a null hypersurface, therefore the geodesics constrained to it are necessarily null [94],

$$ \epsilon = \frac{1}{2} g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} = 0, \qquad (\text{VI.1}) $$

see chapters 34.3 and 34.4 in [93].

• The geodesic dynamics can be extended by adding both exotic & spin-field (commonly called spin-orbit) terms discussed in sec.IV.

This section is devoted to the study of an exotic photon defined in (IV.45) whose key dynamical ingredients are, by (IV.43),

$$ \bar{\omega}_{\text{spec}} = \frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} v^i dv^j + \frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} x^i dx^j + \mu \chi B dx^i ds, \qquad (\text{VI.2a}) $$

$$ \sigma_{\text{spec}} = \frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} dv^i \wedge dv^j + \frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} dx^i \wedge dx^j + \mu \chi \partial_i B dx^i \wedge ds, \qquad (\text{VI.2b}) $$

$$ \mathcal{L}_{\text{spec}} = \frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} v^i \frac{dv^j}{ds} + \frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} x^i \frac{dx^j}{ds} + \mu \chi B. \qquad (\text{VI.2c}) $$

## A. Schwarzschild Horizons

We first study the simplest black hole, which is the static and spherically symmetric Schwarzschild metric, described by using (ingoing) Eddington-Finkelstein coordinates $\{v, r, \vartheta, \varphi\}$ and natural units with the line element

$$ g \equiv g_{\mu\nu} dx^\mu dx^\nu = - \left(1 - \frac{2M}{r}\right) dv^2 - 2dvd\bar{r} + r^2(d\vartheta^2 + \sin^2\vartheta d\varphi^2), \qquad (\text{VI.3}) $$

where $M$ is the mass of the Schwarzschild black hole. The Schwarzschild horizon, $\mathcal{H}$, is the hypersurface $r = 2M$, i.e.,

$$ \mathcal{H} = \{r = 2M, \vartheta, \varphi, v\} \cong \mathbb{S}^2 \times \mathbb{R}. \qquad (\text{VI.4}) $$

The $dv^2$ component of the metric (VI.3) induced on $\mathcal{H}$ vanishes, whereas the radial component disappears on the fixed-$r$ hypersurface $\mathcal{H}$, leaving us with the clearly degenerate “metric”,

$$ \tilde{g} = g|_{r=2M} = 0 \cdot dv^2 + 4M^2(d\vartheta^2 + \sin^2\vartheta d\varphi^2). \qquad (\text{VI.5}) $$
or in their magnetic version,

$$ \left\{ \begin{array}{ll} \partial_s B^i = 0, & \partial_i B^i = 0 \\ \epsilon_{ijk} \partial_j B^k - \partial_s E^i = 0, & \partial_i E^i = 0 \end{array} \right. \qquad \text{magnetic} \qquad (\text{IV.42}) $$

respectively $^{32}$.

One may wonder then what would happen when we couple a massless Carroll particle with non-zero “color” $k \neq 0$ in (IV.33), endowed also with an electric charge to electromagnetism. Such particles would be described by the Souriau 2-form

$$ \sigma = k du_i \wedge dx^i + \frac{e}{2} F_{ij} dx^i \wedge dx^j + e E_i dx^i \wedge ds. \qquad (\text{IV.43}) $$

The model looks similar to a massive, non-relativistic charged particle in a background electromagnetic field, — but with the color $k$ playing the role of a mass, and $u$ playing that of momentum (except that $u$ is a unit vector). The motions are the integral leaves of $\ker \sigma$, given by,

$$ \mathbf{E} \cdot \mathbf{u} = 0, \qquad \frac{d\mathbf{x}}{ds} \times \mathbf{u} = 0, \qquad k \frac{d\mathbf{u}}{ds} - e\mathbf{E} + e\mathbf{B} \times \frac{d\mathbf{x}}{ds} = 0. \qquad (\text{IV.44}) $$

Note that the third equation forbids the existence of massless Carroll particle with color $k$ in a pure electric field background. However, if the magnetic field does not vanish, the system interestingly admits Hall-type solutions. Assume indeed for simplicity that electric and magnetic fields are $\mathbf{E} = (E^1, E^2, 0)$ and $\mathbf{B} = (0, 0, B)$ with $E^i = \text{const.} \neq 0$, $B = \text{const.} \neq 0$. Then some calculation yields,

$$ \frac{d\mathbf{x}}{ds} = \left( \frac{|\mathbf{E}|^2}{|\mathbf{E} \times \mathbf{B}|} \right) \mathbf{u}, \qquad \vec{\mathbf{u}} = \frac{\mathbf{E} \times \mathbf{B}}{|\mathbf{E} \times \mathbf{B}|}. \qquad (\text{IV.45}) $$

When $\mathbf{E} = 0$, we find instead helical motions.

## D. “No-particle” motion

A curious degenerate case arises, though, for

$$ \pi = 0, \qquad (\text{IV.46}) $$

<footnote n="32">32 When compared to Galilean electromagnetism [7, 148] the words “electric” and “magnetic” are interchanged, underlining the dual nature of Carrollian vs Galilean physics. These equations are written in $d = 3$ spatial dimensions.</footnote>
On the other hand,

$$ \tilde{\xi} = \partial_v \equiv \partial_s \qquad (VI.6) $$

is a nowhere vanishing vector which belongs to the kernel of $\tilde{g}$ where, to anticipate the role of Carrollian time that it will play, we renamed $v$ to $s$.

We conclude that the Schwarzschild horizon ($\mathbb{S}^2 \times \mathbb{R}, \tilde{g}, \tilde{\xi}$) has a Carroll structure [16]$^{26}$, as defined in sec. III.

The vector $\tilde{\xi}$ (called the generator of the horizon) is also a Killing vector field for the Carroll metric in that $L_{\tilde{\xi}}\tilde{g} = 0$. It is in fact a Killing vector field of the original Schwarzschild spacetime, $L_{\tilde{\xi}}g = 0$, which becomes a null vector on the horizon,

$$ g(\tilde{\xi}, \tilde{\xi})|_{r=2M} = 0. \qquad (VI.7) $$

The Schwarzschild horizon provides us with an insight into the peculiarities of Carrollian dynamics. One of them is that the horizon of (stationary) black holes can “trap” photons i.e., a photon may remain on the horizon forever. As a stationary horizon is always a null hypersurface, and since photons travel in null directions, a photon could move along the null direction defined by the ($\mathbb{R}$ component of the) horizon, thus staying on it forever. In the Schwarzschild case for instance, this amounts to emit a photon right on the horizon and radially outward, see Fig. 8. For details, see e.g. [93, §33.6].

“Photon trapping” is a nice thought experiment to understand Carroll geometries. Consider a photon on a Schwarzschild horizon. We can look at its motion (or lack of motion) intrinsically, on the horizon itself. On this surface, the photon is created at some point $(x_0, s_0)$ in (Carroll) time (where “Carroll time” is represented by the $\mathbb{R}$-axis generated by the null vector $\xi = \partial_s$, and $x_0 \in \mathbb{S}^2$). In order to stay on the horizon, the photon must have its momentum directed along the generator $\xi$ of the horizon. Its momentum is then zero in the angular directions, and therefore the photon “stays in place” on $\mathbb{S}^2$. Its “trajectory” is precisely the Carrollian “no-motion”, $x(s) = x_0$.

The situation is illustrated by the Eddington-Finkelstein diagram on Fig. 8, cf. [93].

While this intuitive picture on the Schwarzschild horizon seems to imply no motion on this instance of a Carroll structure, the horizon is 2+1 dimensional and thus might accommodate doubly-extended (and potentially non-trivial) dynamics associated with the double central

<footnote>$^{26}$ Any null hypersurface in a Lorentzian spacetime is Carroll [13, 14, 73, 74].</footnote>
<table><caption>TABLE I: Motion of an unextended Carroll particle with vanishing spin, s = 0.</caption>
  <thead>
    <tr>
      <th></th>
      <th>free Carroll no charge e = 0</th>
      <th>charged e ≠ 0 in EM field</th>
      <th>in curved metric</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>m ≠ 0</th>
      <td>no motion</td>
      <td>no motion</td>
      <td>no motion</td>
    </tr>
    <tr>
      <th>m = 0, k = 0</th>
      <td>no motion</td>
      <td>Hall motions</td>
      <td>no motion</td>
    </tr>
    <tr>
      <th>m = 0, k ≠ 0</th>
      <td>instantaneous motion</td>
      <td>helical/Hall motions</td>
      <td>geometric optics</td>
    </tr>
    <tr>
      <th>Fermat particle</th>
      <td>straight lines</td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>

for which the KKS algorithm would yield identically zero Cartan (and thus Souriau) forms,

$$
\varpi_{m=0} \equiv 0 \quad \text{and} \quad \sigma_{m=0} \equiv 0, \qquad (\text{IV.47})
$$

respectively, — and thus *no dynamics* at all! However coupling “it” naively to electro-
magnetism by the rule (II.48) yields . . . a *Carroll dynamics with a “purely electromagnetic*
*Souriau form* ,

$$
\sigma^{(0)} = \frac{e}{2} F_{ij} dx^i \wedge dx^j + e E_i dx^i \wedge ds, \qquad (\text{IV.48})
$$

whose characteristic foliation yields nevertheless non-trivial equations of motion (in 2 + 1 dimensions).

The weird-looking “naked” electromagnetic Souriau form (IV.48) can actually be derived
from the massive 2-form (IV.15). Replacing **p** by *mv* we get,

$$
\sigma^{(m)} = m d\boldsymbol{v} \wedge d\boldsymbol{x} + \frac{e}{2} F_{ij} dx^i \wedge dx^j + e E_i dx^i \wedge ds. \quad (\text{IV.49})
$$

Then letting here $m \to 0$ switches off the first term, leaving us just with (IV.48). Remarkably,
the electric charge drops out from the equations of motion (as long as it does not vanish),
leaving us with:

**Corollary IV.3.** Coupling minimally a charged degenerate “no-particle” with vanishing conserved momentum, $\pi = 0$, having therefore no free dynamics, (IV.47), the system will move by following the Hall law (I.2) i.e. ,

$$
(x^i)' = \epsilon_{ij} \frac{E^j}{B}. \qquad (\text{IV.50})
$$

How can a “no-particle” have non-trivial motion? The mystery is clarified by observ-
ing that belonging to the kernel of $\sigma^{(0)}$ means that the combined forces vanish along the
trajectories. Perpendicularly to the electric direction, the motion is thus “free”!
<figure><img src="image_9.png" /></figure>

FIG. 8: Eddington-Finkelstein diagram of a Schwarzschild spacetime, with outgoing light rays $u = \text{const.}$ in green, highlighting key features such as the singularity, the horizon, and different trajectories of outgoing photons. Infinitely far away from the black hole, space is Minkowskian. However, the closer we get, the more the gravitational field bends the light cones inward, resulting in outgoing photons taking an ever longer time to escape from the black hole region. Past the horizon, the light cone is completely bent inward such that the only future-pointing directions are toward the singularity. An outgoing photon emitted right on the horizon would have a vertical trajectory, thus staying on the horizon.

extension discussed in sec.IV. However, it has been shown in [31] and recalled in section IV C that the two charges $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$ only couple to the electromagnetic field but not the gravitational field. But the Schwarzschild spacetime carries no electromagnetic field. Hence,

**Theorem VI.1.** An uncharged massless (exotic or not) particle with purely outgoing momentum on the horizon of a Schwarzschild black hole stays fixed, but does not move.

This conclusion is valid also for spin-field interaction (IV.35) since $B = 0$

## B. Motion on the Kerr-Newman horizon

Eqn. (IV.42) suggests that non-trivial motion may be possible on the horizon of a black hole for a massless particle with magnetic moment $\mu \neq 0$, non-zero anyonic spin $\chi \neq 0$ and
## Partially broken C-boost symmetry

From the symmetry point of view, Carroll boosts are generated by the 2-parameter vector field $X = \beta_i \partial_{v_i} - \beta_i x^i \partial_s$, (IV.18) with $\beta = (\beta_1, \beta_2)$, implemented on the Souriau 2-form (IV.48) as,

$$L_X \sigma^{(0)} = -(\mathbf{E} \times \boldsymbol{\beta}) dx^1 \wedge dx^2. \qquad (IV.51)$$

Since the above should vanish for a symmetry of the system, we see that the Carroll boosts symmetry requires $\boldsymbol{\beta}$ to be parallel to the electric field, $\mathbf{E}$. Thus the electromagnetic field breaks the full Carroll boost invariance, only the boost in the direction of $\mathbf{E}$ remains. However, no C-boost symmetry in the direction perpendicular to $\mathbf{E}$ implies no obstruction against motion. Thus we get:

**Theorem IV.7.** The electromagnetic field breaks partially the C-boost symmetry: C-boosts perpendicular to the direction of the electric field $\mathbf{E}$ are broken, while those parallel to $\mathbf{E}$ remain symmetries. C-boost symmetry is consistent with the Hall law (IV.50): motion is possible in perpendicularly to the electric field but not parallel to it.

C-boost symmetry thus implies immobility however not conversely: $\mathbf{D}$ is conserved in the direction parallel to the electric field $\mathbf{E}$. However its momentum vanishes because the particle is massless. No conclusion can be drawn from its conservation therefore. In the broken (perpendicular) direction we have in turn no conserved momentum at all – and thus neither garantie nor obstruction against motion. In conclusion, the immobility derived from the conserved boost momentum does not work for massless particles.

Using a more conventional language, putting $\mathbf{p} = m\mathbf{v}$ into the massive Carroll Lagrangian with EM coupling (IV.12) to get

$$\mathcal{L}_{em} = (mv - eA) \cdot \mathbf{x}' - eV. \qquad (IV.52)$$

This Lagrangian is gauge-invariant in that it changes by a time derivative under $\mathbf{A} \to \nabla f$, $\mathcal{L}_m \to \mathcal{L}_m - \partial_s (e\partial_i f x^i)$. Its variational equations are, instead of (IV.14)

$$m(x^i)' = 0 \quad \text{and} \quad mv_i' = eE_i + eB\epsilon_{ij}(x^j)' \qquad (IV.53)$$

Then letting $m \to 0$ switches off the first equation leaving $(x^i)'$ undetermined, while the second one in (IV.53) reduces, when $e \neq 0$, to the no-force-along-the-trajectory condition

$$E_i + B\epsilon_{ij}(x^j)' = 0 \quad \text{i.e.} \quad \mathbb{F}_{\mu\nu}(x^\nu)' = 0, \qquad (IV.54)$$
non-zero magnetic charge $\kappa_{\text{mag}} \neq 0$, when the black hole has a non-uniform magnetic field $B \neq 0$ on the horizon.

Below we show that these conditions are met for a Kerr-Newman black hole, described by its mass $M$, its angular momentum $J$, and its charge $Q$. In terms of Eddington-like coordinates $(v, r, \vartheta, \varphi)$ [95], a Kerr-Newman metric can be written in the form $^{27}$,

$$g = -\frac{\Delta}{\Sigma} \left( -dv + \frac{\Sigma}{\Delta} dr + a \sin^2 \vartheta d\varphi \right)^2 + \frac{\sin^2 \vartheta}{\Sigma} \left( adv - (r^2 + a^2)d\varphi \right)^2 + \Sigma d\vartheta^2 + \frac{\Sigma}{\Delta} dr^2 \quad (\text{VI.8a})$$

$$\Sigma = r^2 + a^2 \cos^2 \vartheta, \quad \Delta = r^2 + a^2 + Q^2 - 2Mr, \quad (\text{VI.8b})$$

where $a = J/M$. The (outer) horizon of a Kerr-Newman black hole is the $r = r_+ = \text{const.}$ hypersurface $\mathcal{H}$ defined by requiring$^{28}$

$$\Delta = 0 \quad \text{i.e.} \quad r = M + \sqrt{M^2 - (a^2 + Q^2)} = r_+. \quad (\text{VI.9})$$

Then we consider again the $2+1$ dimensional structure [16, 17] whose key ingredients are the induced metric and a vector,

$$\tilde{g} = g|_{\Delta=0} = \frac{\sin^2 \vartheta}{\Sigma} (a \, dv - (r_+^2 + a^2) d\varphi)^2 + \Sigma d\vartheta^2, \quad (\text{VI.10a})$$

$$\xi = \partial_v + \Omega_H \partial_\varphi \quad \text{where} \quad \Omega_H = \frac{a}{r_+^2 + a^2}. \quad (\text{VI.10b})$$

Here $\Omega_H$ is the angular velocity of the horizon.

The metric (VI.10a) is again singular, as seen from shifting the coordinates as $(\vartheta, \varphi, v) \mapsto (\vartheta, \tilde{\varphi} = \varphi - \Omega_H v, v)$,

$$\tilde{g} = \frac{(r_+^2 + a^2)^2 \sin^2 \vartheta}{\Sigma} d\tilde{\varphi}^2 + \Sigma d\vartheta^2 \quad \& \quad \xi = \partial_v. \quad (\text{VI.11})$$

The kernel is generated by the vector $\xi$ which takes now the familiar form, $\tilde{g}(\xi) = 0$.

Thus we have a degenerate metric and a vector field in its kernel allowing us to conclude that the horizon of a Kerr-Newman black hole carries indeed a Carroll structure $(\mathbb{S}^2 \times \mathbb{R}, \tilde{g}, \tilde{\xi})$. Viewed in the original spacetime, $\xi$ is again null on the horizon and generates the $(\mathbb{R}$ component of the) horizon.

<footnote>$^{27}$ The metric and its inverse are regular at $\Delta = 0$, as the $dr^2$ terms cancel each other out.</footnote>

<footnote>$^{28}$ In fact, there are two hypersurfaces which satisfying $\Delta = 0$. We are only interested in the behavior at the outer radius $r_+ .</footnote>
which is the Hall law, (IV.50). Note that the electric charge has dropped out. We note for completeness that this first-order equation comes from the first-order purely electromagnetic Lagrangian

$$ \mathcal{L}_{em}^{0} = -\mathbf{A} \cdot \mathbf{x}' - V, \qquad (IV.55) $$

which is indeed the $m \to 0$ limit of (IV.52).

The behaviour of a “no-particle” under C-boosts can conveniently be studied by looking at the classical action,

$$ \mathcal{A} = \int \mathcal{L}_{em}^{0} ds = \int (\mathbf{A} \cdot d\mathbf{x} + Vds). \qquad (IV.56) $$

Assuming for simplicity that the fields are constant, the action changes under a C-boost (I.1) as,

$$ \mathcal{A} \to \mathcal{A} + \int (\text{surface term}) - \int \Delta \quad \text{where} \quad \Delta = (E_1 b_2 dx^1 x^2 + E_2 b_1 dx^1 x^2). $$

A C-boost is a symmetry if $\Delta$ is also a surface term. Having assumed $\mathbf{E} = \text{const.}$, we must have $d\Delta = -(\mathbf{E} \times \mathbf{b}) dx^1 \wedge dx^2 = 0$, which implies :

**Theorem IV.8.** A C-boost (IV.18) oriented along the electric field, $\mathbf{E}$, is a symmetry for a no-particle in a constant planar electromagnetic field ($\mathbf{E}, \mathbf{B}$) but is broken otherwise.

For $\mathbf{E} = (0, E_y)$, for example, boosts are symmetries in the $y$ direction but broken in that of $x$. The first one induces the conserved Noether quantity $D_y = yV(y)$, whose conservation implies, for $E_y = -\partial_y V \neq 0$, no motion in the $y$ direction, but does not exclude motion in the direction perpendicular to the electric field.

The correlation of “allowed” and “forbidden” directions with the C-boost symmetry is analogous that of dipole symmetry for fractons, whose immobility can be concentrated to lines or even surfaces [41].

## E. Carroll “no motion”: interpretation through General Relativity

Carroll structures implying Carroll dynamics are plentiful in General Relativity: in fact, any null hypersurface of a Lorentzian spacetime carries a Carroll structure [7, 92, 93].

If one wants to induce the motion of a particle on a light-like hypersurface, then the class of particles to consider is that of massless particles. Let us look at such particles in flat
Now if we were to study the naïve photon trapping on the horizon of a KN black hole as we did for Schwarzschild, we would arrive at the same negative conclusion: we would find no motion. However the Carroll manifold $\mathcal{H}$ has 2 + 1 dimensions and therefore can accommodate the 2-parameter extension yielding an “exotic photon” (IV.45), which may again be coupled to the electromagnetic field and which can move, as we will see it below.

In Eddington-like coordinates $(v, r, \vartheta, \varphi)$ [95]$^{29}$ the electromagnetic tensor in the Kerr-Newman spacetime is given by,

$$F = \frac{2aQr \sin\vartheta \cos\vartheta}{(r^2 + a^2 \cos^2\vartheta)^2} (adv - (r^2 + a^2)d\varphi) \wedge d\vartheta + \frac{Q(r^2 - a^2 \cos^2\vartheta)}{(r_+^2 + a^2 \cos^2\vartheta)^2} (a \sin^2\vartheta d\varphi - dv) \wedge dr. \qquad (VI.12)$$

The magnetic field is conveniently derived by using the Hodge operator which interchanges the electric and magnetic fields, $\star F = B_i dx^i \wedge dv + \dots$. We then identify the magnetic field induced on the horizon ($r = r_+$) as $B = B^r|_{\Delta=0} = (\star F)^{rv}$, which yields, in our case,

$$B = \frac{F_{\theta\varphi}}{\sqrt{|\det g|}} \Big|_{\Delta=0} = (2aQr_+(r_+^2 + a^2)) \frac{\cos\vartheta}{(r_+^2 + a^2 \cos^2\vartheta)^3}. \qquad (VI.13)$$

We note for later reference that the magnetic field is a function of $\vartheta$ only, $B = B(\vartheta)$, as it follows from the axial symmetry of the Kerr-Newman spacetime. Non-vanishing $B$ is obtained for non-zero angular momentum $J = aM$ and charge $Q$.

The electromagnetic field induced on the horizon,

$$\tilde{F} = F|_{\Delta=0} = \frac{aQr_+ \sin 2\vartheta}{(r_+^2 + a^2 \cos^2 \vartheta)^2} (adv \wedge d\vartheta + (r_+^2 + a^2) d\vartheta \wedge d\varphi), \qquad (VI.14)$$

seems to have a non-vanishing electric field component,

$$E_{\vartheta} = F_{u\vartheta} = (a^2 Q r_+) \frac{\sin 2\vartheta}{(r_+^2 + a^2 \cos^2 \vartheta)^2}. \qquad (VI.15)$$

However switching to co-moving coordinates, (VI.11), the coordinate transformation $\varphi \to \tilde{\varphi} = \varphi - \Omega_H v$ with $\Omega_H$ given in (VI.10b),

$$\tilde{F} \to \hat{F} = (aQr_+(r_+^2 + a^2)) \frac{\sin 2\vartheta}{(r_+^2 + a^2 \cos^2 \vartheta)^2} d\vartheta \wedge d\tilde{\varphi}, \qquad (VI.16)$$

eliminates the electric field (VI.15): $E = 0$ on the horizon. Thus we are left with the pure magnetic field (VI.13), which is indeed left invariant under this coordinate transformation.

<footnote>$^{29}$ $0 \le \vartheta \le \pi$ is oriented with $\vartheta = 0$ at the north pole.</footnote>
spacetime. Their geodesic motion is given by the set of first order equations,

$$ \dot{X}^{\mu} = P^{\mu}, \qquad (IV.57) $$

$$ \dot{P}^{\mu} = 0, \qquad (IV.58) $$

where the dot denotes the derivative with respect to some affine parameter. Pick coordinates $(x, y, z, t)$ such that the momentum for outgoing massless particles be $P = (0, 0, 1, 1)$, yielding the spacetime diagram in figure 3.

<figure><img src="image_3.png" /><figcaption>FIG. 3: Spacetime diagram. The light-cone is in **blue**. Photon trajectories are either **outgoing** ($\dot{z} > 0$) depicted in **green** or **ingoing** depicted in **red**.</figcaption></figure>

In- and outgoing null geodesics define new coordinates that are well suited for the study of null trajectories. Let us then introduce null coordinates, $u = \frac{ct+z}{2}$, $v = \frac{ct-z}{2}$ and compute the geodesic equation of motion (IV.57) for a massless particle, for an “outgoing” particle with momentum $P$ chosen as above, for instance. We readily see that $\dot{u} = 1$ and $\dot{v} = 0$, meaning that outgoing massless particles have their trajectories along a $v = \text{const.}$ hypersurface, while $u$ is a natural evolution parameter. Conversely, an ingoing massless particle would
### C. Dynamics on the Kerr-Newman black hole horizon

Turning to the dynamics of an “exotic photon” (IV.45) we recall that it has no electric charge, and therefore the terms $eE$ and $eB$ are switched off, leaving us with the effective fields,

$$B^* = \kappa_{\text{mag}}, \qquad (\text{VI.17a})$$

$$E_{\vartheta}^* = \mu\chi \partial_{\vartheta} B = -\mu\chi (2aQr_+(r_+^2 + a^2)) \frac{(r_+^2 - 5a^2 \cos^2 \vartheta)}{(r_+^2 + a^2 \cos^2 \vartheta)^4} \sin \vartheta, \qquad (\text{VI.17b})$$

respectively. The fields $B$ and $E^* = \nabla B$ are shown on Fig. 9.

<figure><img src="image_10.png" /><figcaption>FIG. 9: (a) The magnetic and (b) the effective electric field, $B = B(\vartheta)$ and $E^* = \nabla B$, respectively, on the horizon of a Kerr-Newman black hole.</figcaption></figure>

Thus the general (Hall) equations of motion (IV.42) reduce to$^{30}$

$$(x^{\vartheta})' = 0, \quad (x^{\varphi})' = \underbrace{(\mu\chi(2aQr_+)(r_+^2 + a^2)) \frac{(r_+^2 - 5a^2 \cos^2 \vartheta)}{(r_+^2 + a^2 \cos^2 \vartheta)^4}}_{E_{\vartheta}^*} \sin \vartheta \cdot \underbrace{\frac{1}{\kappa_{\text{mag}}} \quad (B^*)^{-1}}_{\kappa_{\text{mag}}} \qquad (\text{VI.18})$$

which is the general Hall law with the cast (VI.17). The forces and velocities are depicted in Fig. 10. The trajectories are horizontal ($\vartheta = \text{const.}$) circles; the velocity is determined by $E_{\vartheta}^*$, shown in Fig. 9b. The angular velocity is maximal on the equator and goes smoothly to zero as we approach the poles. Curiously, the angular velocity depends on the radius of the horizon roughly as $r_+^{-3}$, which implies higher speed for a smaller black hole.

In conclusion, we get:

<footnote>$^{30}$ Here the coefficient $\mu\chi$ is included into $E^*$ for convenience, cf. (IV.41).</footnote>
follow $u = \text{const. hypersurfaces, with } v$ the evolution parameter. The situation is depicted in figure 3. The other equations of an outgoing particle are, from (IV.57), given by,

$$ \frac{dx}{du} = 0, \quad \frac{dy}{du} = 0, \qquad (IV.59) $$

where we used $u$ as the evolution parameter.

What is said above allows us to understand why the “no-motion” statement for Carroll particles is so natural: the $v = \text{const. hypersurfaces define the Carroll structure. Such a structure would be described by 2 spatial coordinates } x \text{ and } y \text{ and by another one “along” the null direction, } u$. Geometrically, $u$ would be what we called the “Carroll time”. The outgoing massless particles stay on these Carroll structures, and their equations of motion are thus induced from the ambient spacetime (IV.59). Thus there is “no motion” in the $(x, y)$ plane, as the induced momentum vanishes in the spatial directions, as expected for Carroll dynamics.

For the Carroll structures that are found as null hypersurfaces in lorentzian spacetimes, the intrinsic statement that Carroll particles do not move mean simply that particles move in the null direction that defines the hypersurface, i.e., that it follows a null geodesic in the ambient spacetime. The massless particle has all its momentum directed toward the null direction, and has no spare momentum left to deviate in the other directions. In that sense, the statement of the Red Queen<fnref n="33" /> makes perfect sense for Carrollian physics.

So in a sense, we can think of Carroll structures/dynamics as an effective description of phenomena which follow null geodesics. This also implies that finding Carrollian motion means finding potential deviations from null geodesics.

Comparison of the behaviors in the massive/massless cases yields further insight: eqn. (IV.13) implies that the mass $m \neq 0$ “glues” the particle to a fixed position. However in the massless limit the particle is “liberated” and moves as in (IV.50). This non-analytical behavior of the mass-induced spatial “confinement” of the Carroll particle has similar features with the conventional Hall effect (I.2) in two spatial dimensions.

The motion of a particle with a nonvanishing (even infinitesimally small, but nonzero) electric charge $e \neq 0$ is radically different from that of a particle with an exactly zero charge $e = 0$. In the first case, the electromagnetic field generates the Hall motion of the

<footnote n="33">³³ According to [127]: “A slow sort of country! ... it takes all the running you can do, to keep in the same place.”</footnote>
<figure><img src="image_11.png" /><figcaption>FIG. 10: The horizon of a Kerr-Newman black hole (topologically a two-sphere) rotates around its vertical axis. For an uncharged particle the usual EM fields are switched off but the spin-field term induces an **effective electric field E*** $\propto \nabla B$ indicated by vertical **blue arrows**. The horizontal **red arrows** depict the **velocity**, (VI.18), of a massless, uncharged doubly-extended anyon on the horizon. The **force** and the **velocity** follow the Hall law (I.2) which, in the case of the Kerr-Newman black hole, has the form of (VI.18).</figcaption></figure>

**Theorem VI.2.** A massless uncharged anyon with non-vanishing exotic charges $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$ moves on the horizon of a Kerr-Newman black hole by following the Hall law (VI.18).

The rotation we found is reminiscent of the known frame-dragging by a rotating black hole. It is however *unrelated* to it: frame-dragging is hidden in the coordinates we used in (VI.18). Our “Carroll time” coordinate $s$ is defined indeed through $\partial_s = \partial_v + \Omega_H \partial_\varphi$, see (VI.10b), with $\Omega_H$ the angular velocity of the horizon: our coordinates are comoving with the horizon. Equations (VI.13)-(VI.18) and Figs. 9-10 should be compared with those in
particle (I.2), while in the second case, the background field does not affect the motion at all. Moreover, despite the paramount requirement of the nonvanishing electric charge $e \neq 0$, the Hall motion does not depend explicitly on the actual value of the electric charge $e$ (I.2). In the Hall conditions (with perpendicular electric and magnetic fields), the motion is non-analytic in $e$ as $e = 0$ and $e \neq 0$ cases are qualitatively different.

The “no-particle” motion of the Carroll particle is a non-analytical phenomenon but now in the particle mass rather than in the electric charge: the motion is radically different for massive ($m \neq 0$) and massless ($m = 0$) cases. The second similarity appears because the actual value of the mass does not enter the solution of equations of motion for the Carroll particle, similarly to the electric charge in the Hall effect.

## F. Carroll field theories

Carrollian field theory can also be considered. As already pointed out some time ago by Henneaux [12], such theories involve only time derivatives; spatial derivatives are absent. They will be useful to understand fractons, discussed in the following section V.

As explained in section III B, the Lévy-Leblond equations (III.22) can be obtained from the relativistic Dirac theory by letting the velocity of light go to infinity [128]. Now we show that letting the velocity of light go instead to zero yields Carroll field theory [34].

The Carrollian limit of the Dirac equation is obtained by writing it in terms of $s = x^0/C$ coordinate [7, 34] (instead of $x^0 = ct$ as for Galilean contraction),

$$C \left( i\gamma^0 \partial_s + \frac{i}{C} \gamma^j \partial_j - m \right) \Psi = 0. \qquad (IV.60)$$

Choosing $\gamma^0 \propto \sigma^3$ we get, in the $C \to \infty$ limit $^{34}$,

$$\gamma^0 \partial_s \Psi = -im\Psi, \qquad \Psi = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}. \qquad (IV.61)$$

The Dirac matrix $\gamma^0$ has been choosen diagonal, therefore it does not mix the spinor components and the first order field equations split into two uncoupled equations for the com-

<footnote>$^{34}$ Carroll contraction can also be seen as the limit $c \to 0$, where $c$ is the usual speed of light, — a fact which lead to Lévy-Leblond to his initial *clin d'oeil* [2].</footnote>
flat space, discussed in sec.IV.

One could, theoretically, consider massless particles with electric charge, extending our flat-case study in sect.IV to the curved spaces. Combining the effective fields would yield a distorted version. Massless charged (quasi)particles are considered in condensed matter physics [27].

The general Carrollian equations of motion used to derive (VI.18) depend on the spin-orbit Hamiltonian (II.21), inspired from its flat spacetime form. However we cannot be sure of its the precise expression in curved spacetime until 2+1 dimensional dynamics are derived from an established theory in the ambient spacetime — which does not exists yet, to our knowledge.

One could consider a general Hamiltonian of the form $\mathcal{H} = -\mu\chi\mathcal{B}(\vartheta)$, for some yet unknown $\mathcal{B}(\vartheta)$. It is straightforward to see that the conclusion of this section would not change: there would still be motion on the horizon of the black hole, of the same qualitative form: particles would still display an azimuthal motion.

## D. Partially broken Carroll and BMS on the Kerr-Newman horizon

Carroll manifolds have been discussed in connection with BMS symmetry [11] – for the good reason that the BMS group is the conformal extension of the Carroll group [12–14]. Let us briefly summarize how the relation goes. For more details, the reader is advised to consult [13, 14]. The conformal symmetries of a (general) Carroll structure ($\mathcal{C} = \Sigma \times \mathbb{R}, g, \xi$) are vectorfields on $\mathcal{C}$ s.t.,

$$L_X g = \lambda g, \quad L_X \xi = \mu \xi \quad \text{with} \quad \lambda + 2\mu = 0. \qquad (\text{VI.19})$$

Such an $X$ has the form,

$$X = Y + \left( \frac{\lambda}{2}s + \mathcal{T}(x) \right) \partial_s \qquad (\text{VI.20})$$

where $(x, s)$ are coordinates on $\mathcal{C}$ and $Y$ is a conformal vector field of $(\Sigma, g_\Sigma)$. The $\mathcal{T}(x)$ (called “the super translations”) are arbitrary functions of the coordinate $x$. So the symmetry group of $(\mathcal{C} = \Sigma \times \mathbb{R}, g, \xi)$ is the semi-direct product $\text{Conf}(\Sigma, g_\Sigma) \times \mathcal{T}$ with $\mathcal{T}$ the group of supertranslations $\mathcal{T}$. When $\lambda$ and thus also $\mu$ vanish, such a transformation is a Carrollian isometry, whose form is

$$X = Y + \mathcal{T}(x)\partial_s \qquad (\text{VI.21})$$
ponents,

$$ \partial_s \psi_{\pm} = \mp im\psi_{\pm}. \qquad (IV.62) $$

This equation is the Carroll counterpart to the Lévy-Leblond equation (III.22). It can also be derived from its Lagrangian which is Carroll limit of the Dirac Lagrangian,

$$ \mathcal{L}_C = \frac{i}{2} (\Psi \gamma^0 \partial_s \Psi^* - \Psi^* \gamma^0 \partial_s \Psi) - m |\Psi|^2. \qquad (IV.63) $$

While the above first-order Carroll field equations (IV.61) describe spinors, it is also possible to have the same equation but for scalar fields. For instance, quantization of the Carroll model (IV.3) yields the same equation (IV.62) but implemented on a scalar field $\Phi$ [34],

$$ \partial_s \Phi = im\Phi. \qquad (IV.64) $$

That the same equation holds for spinors and scalar fields comes from their rather trivial form. Indeed, differences between spinor and scalar field equations typically involve terms with spatial derivatives, such as $\sigma \cdot D$ in (III.22), — but spatial derivatives are missing in Carroll theories. While in the free case these equations coincide, their behavior may be different, though, when coupled to a background field. For instance, in Galilean quantum mechanics, the free Schrödinger and the Pauli equations are formally identical, despite one acting on scalar fields, and the other one acting on spinors but they become different when coupled to an electromagnetic field.

Similar statements holds for the second-order system. Let us indeed consider the Klein-Gordon equation for a free scalar field of mass $m$ in $1 + d$ dimensions with coordinates $x^0, x^1 \dots x^d$ and a metric of signature $(-1, 1, \dots, 1)$,

$$ \partial_{\mu} \partial^{\mu} \Phi - m^2 C^2 \Phi = 0, \qquad (IV.65) $$

where the parameter $C$ has the dimension of velocity and is, in general, different from the velocity of light $c$. Introducing Carroll time, $s$, by putting $x^0 = s/C$, (IV.65) becomes,

$$ \left[ \frac{1}{C^2} \Delta - \partial_s^2 - m^2 \right] \Phi = 0. $$

Then letting $C \to \infty$ we end up with the second-order-in-$s$ equation with no spatial derivatives,

$$ \partial_s^2 \Phi + m^2 \Phi = 0. \qquad (IV.66) $$
where Y is a Killing vector of the singular Carroll “metric” g on Σ.

For the round sphere Σ = S² with its usual metric, for example, one has that
Conf(S², dΩ²) = SL(2, C), and thus we get the BMS group: SL(2, C) × T.

For the Kerr-Newman horizon ($\mathbb{S}^2$, $\tilde{g}$) with $\tilde{g}$ as in (VI.11), the conformal isometries are
found by solving the system of PDE,

$$
2a^2 Y^\vartheta \sin\vartheta \cos\vartheta + (r^2 + a^2 \cos^2\vartheta)(\lambda - 2\partial_\vartheta Y^\vartheta) = 0, \quad (\text{VI.22})
$$

$$
2(r^2 + a^2) Y^\vartheta \cos\vartheta + (r^2 + a^2 \cos^2\vartheta) \sin\vartheta (2\partial_{\tilde{\varphi}} Y^{\tilde{\varphi}} - \lambda) = 0, \quad (\text{VI.23})
$$

$$
(r^2 + a^2)^2 \sin^2 \vartheta \partial_\vartheta Y^{\tilde{\varphi}} + (r^2 + a^2 \cos^2 \vartheta)^2 \partial_{\tilde{\varphi}} Y^\vartheta = 0 \quad (\text{VI.24})
$$

with $Y^{\vartheta}$, $Y^{\tilde{\varphi}}$ and $\lambda$ functions of $(\vartheta, \tilde{\varphi})$. For isometries $\lambda$ (and thus $\mu$) vanish. The Killing vector of the (spatial part of the) horizon is a rotation around the axis of the black hole [93],

$$
Y = \partial_{\tilde{\varphi}}, \tag{VI.25}
$$

The spatial isometries of the Kerr-Newman horizon $\mathcal{H}$ are thus the same as that of the full
Kerr-Newmann spacetime, $\text{SO}(2) \times \mathcal{T}$, generated by the

$$
X = \partial_{\tilde{\varphi}} + \mathcal{T}(\vartheta, \tilde{\varphi}) \partial_s . \qquad (\text{VI.26})
$$

The symmetries of the equations of motion (IV.42), $(x^i)' = \epsilon^{ij} (\frac{E_j}{B^*})$, and $p_i' = 0$, spelled
out on the horizon are conveniently encoded in the Lagrangian (VI.2c) [or equivalently, in
the Cartan form (VI.2a)]:

$$
\mathcal{L}_{\text{spec}} = \frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} v^i \frac{dv^j}{ds} + \frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} x^i \frac{dx^j}{ds} + \mu \chi B \, dx^i . \qquad (\text{VI.27})
$$

where $\boldsymbol{x} = (\vartheta, \varphi)$. The magnetic field is a function of $\vartheta$ only, $B = B(\vartheta)$ as seen from (VI.13).

Recall now the mechanical version of the Noether theorem : a vector field X is a symmetry
if it changes the Cartan form / Lagrangian by a surface term, $\delta \mathcal{L} = df_X$. Then

$$
Q_X = \frac{\partial \mathcal{L}}{\partial (x^i)'} X^i - f_X \qquad (\text{VI.28})
$$

is a conserved quantity$^{31}$.

<footnote n="31">³¹ For aficionados of the Souriau approach [59], $L_X \varpi = df_X$ implies that $Q_X = i_X \varpi - f_X$ is conserved.</footnote>
In the same spirit, the Klein-Gordon Lagrangian is, in terms of Carroll time s,

$$ \mathcal{L}_{KG} = C^2 \left( \partial_s \Phi \partial_s \Phi^* + \frac{1}{C^2} |\nabla \Phi|^2 - m^2 \Phi \Phi^* \right). \qquad (IV.67) $$

The overall factor $C^2$ can be dropped and the $C \to \infty$ limit of the bracketed quantity is

$$ \mathcal{L}_C^{(2)} = |\partial_s \Phi|^2 - m^2 |\Phi|^2. \qquad (IV.68) $$

We underline here a characteristic property of Carrollian field theories: space derivatives are missing from the equations of motion and Lagrangians.

Let us then look at the symmetries. While these theories are built using the Carroll limit, or directly, by quantizing the Carroll group [34], their symmetry groups turn out to be much larger than the Carroll group alone: due to the absence of spatial derivatives they are invariant under the transformation

$$ \Phi(\mathbf{x}, s) \to \hat{\Phi}(\mathbf{x}, s) = e^{imT(\mathbf{x})} \Phi(\mathbf{x}, s) \qquad (IV.69) $$

for an arbitrary function $T(\mathbf{x})$. The absence of spatial derivatives implies also that the solutions of the free first-order theory (IV.62) are of the form,

$$ \Phi(\mathbf{x}, s) = e^{ims\phi(\mathbf{x})}, \qquad (IV.70) $$

where $\phi(\mathbf{x})$ is an arbitrary function of $\mathbf{x}$ alone. The symmetry (IV.69) then corresponds to shifting the time coordinate by an arbitrary function $T(\mathbf{x})$ of the position,

$$ s \mapsto s + T(\mathbf{x}) \qquad (IV.71) $$

called, in the BMS context [14] a *supertranslation*. Augmented with rotations and spatial translations, we then get an infinite dimensional symmetry,

$$ \Phi(\mathbf{x}, s) \to \hat{\Phi}(\mathbf{x}, s) = e^{imT(\mathbf{x})} \Phi(R\mathbf{x} + \mathbf{c}, s), \qquad (IV.72) $$

where we recognize the “super-Aristotle” group $\mathfrak{A}^\infty$ (item # 9. in sec.II A), implemented as in (II.18). Note that, unlike for Galilean field theories III B, in the present, Carroll context, all symmetries of the field equations are isometries of the underlying spacetime, as $\mathfrak{A}^\infty$ is actually the isometry group of flat weak Carroll structures, see def. II.3. We recall that Aristotle $\mathfrak{A}$ is a subgroup of Carroll $\mathfrak{C}$, itself being a subgroup of “super-Aristotle” $\mathfrak{A}^\infty$, cf.
* The only isometries of the black hole spacetime are “horizontal” rotations generated by $X = \partial_{\varphi}$ for which we find $\delta\mathcal{L} = -d(\kappa_{\text{mag}}\vartheta/2)$, which combines to yield

$$ \pi_{\varphi} = \kappa_{\text{mag}}\vartheta. \qquad (\text{VI.29}) $$

Its conservation is manifest: trajectories of (VI.18) are “horizontal” ($\vartheta = \text{const.}$). This unusual form agrees with the (3.18c) in [31].

* The zeroth order term of super translations, i.e. a (Carrollian) time-translation $X = \partial_s$ leaves the Lagrangian / Cartan form invariant, providing us with the conserved quantity

$$ Q_0 = \mu\chi B = -\mathcal{H}, \qquad (\text{VI.30}) $$

which is indeed the Hamiltonian (IV.46) of a Carroll anyon.

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

$$ L_X \varpi = \mu\chi B \partial_i \mathcal{T} dx^i, \qquad (\text{VI.31}) $$

which is not closed in general,

$$ d(B d\mathcal{T}) = dB \wedge d\mathcal{T} \neq 0. $$

Thus the magnetic field breaks the supertranslation symmetry in general. However remembering that $B = B(\vartheta)$ on the horizon, cf. (VI.13), the obstruction vanishes when the supertranslation is “vertical”,

$$ \mathcal{T} = \mathcal{T}(\vartheta). \qquad (\text{VI.32}) $$

For a C-boost for example, the $\varphi$-component must vanish, $\mathcal{T}(\vartheta) = -\beta_1^\vartheta\vartheta$. The associated Noether quantity

$$ Q_\vartheta = \mu\chi \left( -B\vartheta + \int B d\vartheta \right) \qquad (\text{VI.33}) $$

# (V.8) in [32], is a combination of the Carroll Hamiltonian $\mathcal{H}$ (VI.30) and the integral of $B$. Complicated as it is, $Q_\vartheta$ depends on $\vartheta$ only, therefore its conservation implies, once again, that the trajectories must have $\vartheta = \text{const.}$ (as we had found it). On the contrary, $\varphi$-boost are broken and thus do not obstruct motion in the $\varphi$ direction.
(II.19). Choosing $T(x) = b_0$ yields the Aristotle group $\mathfrak{A}$ in $d+1$ dimensions, composed of the Euclidean group $\mathfrak{E}$ augmented by translations of Carroll time, $s$. The next (i.e. first order in $x$) term, $T(x) = -\mathbf{b}_1 \cdot x$, yields Carroll (C-) boosts.

An irreducible representation of the Carroll group on the general solutions (IV.70) can be defined [34]. One has, for a Carroll transformation $C$ in (II.14), $\Phi \to \Phi \circ C^{-1}$, i.e.,

$$ \Phi(\boldsymbol{x}, s) \to e^{i[-b_0 + \mathbf{b}_1 \cdot (\boldsymbol{x} - \mathbf{c})]} \Phi(R^{-1}(\boldsymbol{x} - \mathbf{c}), s). \qquad (IV.73) $$

We underline that, even though Carroll boosts (and time translation) are kinematical in that it comes from the action of the group on a spacetime – more precisely, on *Carroll spacetime* with coordinates $x$ and $s$.

Additional insight into Carroll symmetry can be gained by looking at the Lagrangian for the Klein-Gordon equation (IV.65), which is invariant under Lorentz boosts,

$$ \boldsymbol{x} \to \boldsymbol{x}' = \boldsymbol{x} + \frac{\gamma^2}{\gamma+1}(\boldsymbol{\beta} \cdot \boldsymbol{x})\boldsymbol{\beta} + \gamma\boldsymbol{\beta}x_0, \quad x_0 \to x_0' = \gamma(x_0 + \boldsymbol{\beta} \cdot \boldsymbol{x}), \qquad (IV.74) $$

where $\gamma = (1 - \boldsymbol{\beta}^2)^{-1/2}$. Then putting $s = Cx_0$, $\mathbf{b} = -C\boldsymbol{\beta}$ and letting $C \to \infty$ (IV.74) becomes the C-boost [2]

$$ \boldsymbol{x}' = \boldsymbol{x} \quad \text{and} \quad s' = s - \mathbf{b} \cdot \boldsymbol{x}, \qquad (IV.75) $$

which is precisely the Carroll boost (I.1). The conserved quantity

$$ \mathbf{D} = m \int \Phi \Phi^{\dagger} \boldsymbol{x} \, d^2 \boldsymbol{x} \qquad (IV.76) $$

generated by C-boosts we call Carroll field boost momentum will reappear again in the fraction section as dipole momentum, (V.12b). Comparison with the Galilean boost momentum $\mathbf{D}^{\text{Gal}}$ in (III.29) indicates the absence of the momentum term $-t\mathbf{P}$. The conserved angular momentum for the first-order system (IV.61) is in turn

$$ \ell = \int \left\{ \boldsymbol{x} \times \frac{1}{2i} (\boldsymbol{\Phi}^{\dagger} \boldsymbol{\nabla} \boldsymbol{\Phi} - (\boldsymbol{\nabla} \boldsymbol{\Phi})^{\dagger} \boldsymbol{\Phi}) \right\} d^2 \boldsymbol{x} + \frac{1}{2} \int \left\{ |\psi_+|^2 - |\psi_-|^2 \right\} d^2 \boldsymbol{x}, \qquad (IV.77) $$

consistently with having spins $\pm 1/2$, cf. the Lévy-Leblond expression in (III.29).

# V. FRACTIONALS AND CARROLL PARTICLES

## A. The fracton model

Hypothetical quasiparticles in condensed matter systems with limited mobility called *fractons* attract much current attention [36–44]. As it is understood now, the existence of
• If the supertranslation happens to be induced by the magnetic field itself, e.g.

$$ \mathcal{T} \equiv \mathcal{T}_n \propto B^n \qquad (VI.34) $$

for some $n = 0, \dots$, then, we get a conserved quantity for each $n$,

$$ Q_n \propto (\mu\chi B)^{n+1} = \mathcal{H}^{n+1}. \qquad (VI.35) $$

## VII. CONCLUSION

Our investigations are devoted to the study of particles with Carroll symmetry, and to their rôle in Hall-type effects.

From the very beginning, a fundamental question has concerned the *mobility* of such a particle [5, 30, 32]. In the massive case, the “no-motion” statement follows either directly from the eqns of motion, (III.7) or alternatively, from the conservation of the quantity (III.10b), associated with Carroll boosts.

After having long limited interest in the subject, the “no-motion property” of Carroll particles [2–4] is now, conversely, *attracting new attention*, namely for quasiparticles called *fractons* (sect. III B), studied in condensed matter physics.

Putting a Carroll particle into an external EM field, the C-boost symmetry (I.1) may be broken in one, but remain unbroken in the perpendicular direction and implying “no-motion” in one, but allowing motion in the perpendicular direction — and realizing the Hall scenario in eqn. (I.2).

The “no-motion” property changes radically for *massless particles* (sec.III C). The conservation of the boost momentum in the free case, (III.46), does not prevent motion any more.

A remarkably fact is that the Carroll group has, in $d = 2$ dimensions, a double central extension [6, 31, 63] with additional parameters we denote by $\kappa_{exo}$ and $\kappa_{mag}$, respectively. Taking them into account, and also that in $d = 2$ we can have anyons with spin $\chi$ and magnetic moment $\mu$, provides us with additional terms in their dynamics which may exhibit, under certain circumstances, quite surprising behavior, which is however consistent with the general formula (IV.42) in Theorem IV.7. The motion of extended Carroll particle is subjected to a generalized (anomalous) Hall law which includes a Zeeman force acting on the particle in inhomogeneous fields.
restricted-mobility quasiparticles is ultimately a consequences of exotic spacetime symmetries. The Carroll particles constitute an important class of fractons as we show below.

The fracton phase of matter can be described by complex scalar field theories. A Lagrangian has been put forward (e.g. in [41], eqn. # (2.50)),

$$ \mathcal{L} = \frac{i}{2} (\Phi \partial_t \Phi^* - \Phi^* \partial_t \Phi) - m |\Phi|^2 - \lambda \underbrace{\left( \partial_i \Phi \partial_j \Phi - \Phi \partial_i \partial_j \Phi \right) \left( \partial_i \Phi^* \partial_j \Phi^* - \Phi^* \partial_i \partial_j \Phi^* \right)}_{\text{self-interaction } \mathcal{L}_{\text{int}}} \quad (\text{V.1}) $$

where $\Phi(x, t)$ is a complex scalar field. A second-order Lagrangian could also be considered by replacing the first-order-in-time derivative terms by $|\partial_t \Phi|^2$ [41, 42],

$$ \mathcal{L}^{(2)} = \underbrace{|\partial_t \Phi|^2 - m^2 |\Phi|^2}_{\mathcal{L}_0^{(2)}} - \lambda \left( \partial_i \Phi \partial_j \Phi - \Phi \partial_i \partial_j \Phi \right) \left( \partial_i \Phi^* \partial_j \Phi^* - \Phi^* \partial_i \partial_j \Phi^* \right) . \quad (\text{V.2}) $$

The simple model (V.2) incorporates a very nontrivial non-perturbative physics that features strongly coupled phases. This fact could be guessed from a nontrivial dimensionality of the coupling $\lambda$ in Eq. (V.2). In the absence of mass, $m=0$, the coupling $\lambda$ sets a single spatial scale and thus can be absorbed into the redefinition of spatial coordinates. The resulting theory contains no small parameters and, thus, is intrinsically non-perturbative. (The physical situation is very similar to the three-dimensional Yang-Mills theory, which has a single dimensionful coupling constant $g$ that can also be absorbed by a scaling transformation to a redefinition of coordinates. This fact has important physical consequences implying, for example, that quark-gluon plasma in four spacetime dimensions does not allow a perturbative treatment even at high temperatures where the gauge coupling should become small due to asymptotic freedom. The reason is that the system in thermal equilibrium experiences dimensional reduction to its three-dimensional counterpart, which has no perturbative treatment [149]).

In the scalar models of type (V.1), fractons can be treated in a large-$N$ limit, where the field space is extended to multiple fields $\Phi \to \Phi^a$ with $a = 1, \dots, N$. Treating $1/N$ as a small parameter, the system can be considered in a standard large-$N$ perturbation theory. The solution for the ground state reveals that this class of models is indeed characterized by strongly correlated phases [150]. Moreover, the classical dipole symmetry of the model (V.1), which we discuss later, is spontaneously broken in the whole phase diagram of these models even in the absence of a scalar potential $V(\Phi)$. As a result, the ground state of such a model is characterized by dipole superfluids of various types [150]. Despite the fact that
After presenting intuitively appealing toy models in sec. IV F in sec. VI we illustrated our
theory in the gravitational context.

Note added. During the final stage to complete this study we were informed about related investigations on magnetized black hole horizons [96].

Acknowledgments

The authors are grateful to P. Chruściel, G. Gibbons and M. Stone for discussions. LM
is grateful to Prof. T. Harko for hospitality at Babeş-Bolyai University. PMZ was partially
supported by the National Natural Science Foundation of China (Grant No. 11975320).

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the true ground state cannot be revealed in the classical approach, the classical symmetries of the fracton model (V.1) are important for the quantum behavior through a low-energy Goldstone description.

We study first the free classical dynamics given by the Lagrangian $\mathcal{L}_0$, given by the first term in Eq. (V.1). It is reminiscent of the usual Galilei Lagrangian in that it has a first order derivative in time $t$. However, it is also different from the Galilean expression as it has no space derivatives: it is, in fact Carroll field theory.

Variation of $\mathcal{L}_0$ in (V.1) yields the truncated first-order wave equation

$$ \partial_t \Phi(\mathbf{x}, t) = im\Phi(\mathbf{x}, t) \qquad (V.3) $$

whose solutions are,

$$ \Phi(\mathbf{x}, t) = e^{imt}\phi(\mathbf{x}), \qquad (V.4) $$

where $\phi(\mathbf{x})$ is an arbitrary function of $\mathbf{x}$ alone. The second-order Lagrangian $\mathcal{L}_0^{(2)}$ in (V.2) yields instead a truncated Klein-Gordon equation with no space derivatives,

$$ \partial_t^2 \Phi(\mathbf{x}, t) + m^2 \Phi(\mathbf{x}, t) = 0, \qquad (V.5) $$

which is indeed the “square” (iteration) of (V.3). The missing of spatial derivative terms from both equations (V.3) and (V.5) is analogous to the absence of the kinetic term $p^2/2m$ in the classical Carroll dynamics, studied in sec.IV A.

We note for further reference that eqn. (V.3) is indeed identical (up to $s \leftrightarrow t$) to eqn. (IV.64) satisfied by a Carroll scalar field. It is also reminiscent of a Dirac equation except for missing spatial derivatives. The second order equation, (V.5), is in turn the Carroll field equation (IV.66). Thus we conclude: free fracton field theory is a Carroll field theory.

We highlight that this statement works only at the classical level, as the interaction term in (V.2) drives towards the non-perturbative domain with strongly interacting phases and broken symmetries [150]. While strongly coupled effects cannot be captured by a classical theory (especially by a free classical theory), our analysis below provides us with a relation between the classical models which shows that the fractonic and Carrollian models are indeed related.

The theory given by the Lagrangian (V.1) with $\lambda = 0$ has a trivial energy spectrum:

$$ \varepsilon_k = m, \qquad (V.6) $$
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implying that the group velocity of the excitations is zero:

$$v_k = \frac{\partial \varepsilon_k}{\partial k} = 0, \qquad (V.7)$$

confirming that the quasiparticles described by the free Lagrangian $\mathcal{L}_0$ do not move [36–43] — which is indeed their “raison d’être”. This hints at the similarity of fractons with massive Carrollian particles we studied in sec. IV A.

The independence of the particle energy level from its momentum is characteristic of the so-called flat bands, the subject which is under staring interest in condensed matter physics. Flat bands appear in the magic-angle bilayer graphene [151] which hosts unconventional superconductivity at low temperatures [152]. Fermionic models and the relation to Carroll physics of flat bands has been discussed very recently Ref. [153].

Let us now see how Carrollian concepts provide a key to study fractons.

## B. Dipole vs Carroll boost symmetry

Condensed matter physicists argue that the manifest Aristotle symmetry of fractons can be augmented by a somewhat mysterious dipole (or even multipole) symmetry implemented through a position-dependent but time independent phase factor,

$$\Phi(\mathbf{x}, t) \rightarrow \hat{\Phi}(\mathbf{x}, t) = e^{im(b_0 - \mathbf{b}_1 \cdot \mathbf{x})} \Phi(\mathbf{x}, t), \qquad (V.8)$$

where $\mathbf{b}_1 \in \mathbb{R}^2$ and where a constant phase rotation by $b_0$ has also been added [37–42, 45]. Infinitesimally we have,

$$\delta\Phi(\mathbf{x}, t) = i\beta_0 \Phi(\mathbf{x}, t), \qquad (V.9a)$$

$$\delta\Phi(\mathbf{x}, t) = -i(\beta_1 \cdot \mathbf{x}) \Phi(\mathbf{x}, t) \qquad (V.9b)$$

($\beta_0 = \delta b_0$, $\beta = \delta \mathbf{b}$). Dipole symmetry is reminiscent of Carroll symmetry: it implies immobility — the characteristic feature of fractons [40–44].

Fractons and Carroll particles are in fact closely related $^{35}$. Below we show that their dipole symmetry is in fact the remnant of a curious double Carroll boosts symmetry of the free system broken to a single, “internal” symmetry by the interaction.

<footnote>$^{35}$ Remarkably, the word “Carroll” does not even appear in the cond-mat literature (with the notable exception of [41]).</footnote>
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We start with the free first-order Lagrangian $\mathcal{L}_0$ in (V.1), which is manifestly invariant under Aristotle transformations (see sec.II A). However the absence of spatial derivatives implies that it is also invariant by the “internal phase shift” in (V.8) [40–42]. We note also that $\mathcal{L}_0$ is in fact invariant under

$$ \Phi(\mathbf{x}, t) \rightarrow \hat{\Phi}(\mathbf{x}, t) = e^{imT(\mathbf{x})} \Phi(\mathbf{x}, t) \qquad (V.10) $$

for any function $T(\mathbf{x})$, called, in cond-mat, a multipole transformation [41]. This follows from (V.4) and the absence of spatial derivatives by insertion into (V.3).

The expressions (V.10) (and (V.8) in particular) are reminiscent of a Galilei boost acting on a scalar field, eqn. (III.25) — except that it does not shift the position $\mathbf{x}$, hinting rather at an intriguing similarity with Carroll field theory (sec.IV F). Bidussi et al. [41] recognized indeed that dipole (resp. multipole) symmetry is in fact a Carroll transformation which shifts the time $t$ but not the position,

$$ \mathbf{x} \rightarrow \mathbf{x}, \quad t \mapsto t + T(\mathbf{x}), \qquad (V.11) $$

called, in the BMS context, a supertranslation [14].

The dipole (multipole) symmetries of free fractons belong to the “super-Aristotle” group $\mathfrak{A}^{\infty}$ (sec. II A) implemented as in (IV.72) after interchanging $s$ and $t$. Now the Carroll group $\mathfrak{C}$ is a subgroup of $\mathfrak{A}^{\infty}$. Carroll time translations and Carroll boosts are just the first two lowest-order supertranslations. Dipole symmetry (V.8) is thus recovered as Carroll-boost (C-boost) symmetry built on shifting the time, $t$. The Noether theorem (sec.II.2) applied to the free Lagrangian $\mathcal{L}_0$ in (V.1) then yields the conserved quantities,

$$ q = \int |\Phi(\mathbf{x}, t)|^2 d^3\mathbf{x}, \qquad (V.12a) $$

$$ \mathbf{D} = -\int |\Phi(\mathbf{x}, t)|^2 \mathbf{x} d^3\mathbf{x}, \qquad (V.12b) $$

interpreted as conserved electric charge$^{36}$ and dipole moment$^{37}$.

While the conservation of charge (V.12a) does not affect how the excitations move, the immobility of a single particle follows from the conservation of the dipole moment (V.12b). Indeed, the motion of a particle would immediately change the dipole moment of the system

<footnote>$^{36}$ The charge and mass densities are assumed to be up to sign proportional.</footnote>

<footnote>$^{37}$ To be compared to (IV.76)</footnote>
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so that the particle cannot move if the dipole moment is conserved. However, the motion of two oppositely charged particles is allowed because such a motion conserves not only the electric charge (V.12a) but also the dipole moment (V.12b), provided the distance between the particles stays unchanged.

For any solution of the wave equations (V.3) (or (V.5)), the particle density is manifestly independent of time [31, 34],

$$|\Phi(\boldsymbol{x}, t)|^2 = |\phi(\boldsymbol{x})|^2, \qquad (V.13)$$

providing us with the field-theoretical analog of immobility: the conserved charges (V.12),

$$q = -m \int |\phi(\boldsymbol{x})|^2 d\boldsymbol{x} \quad \text{and} \quad \mathbf{D} = m \int |\phi(\boldsymbol{x})|^2 \boldsymbol{x} \, d^2\boldsymbol{x}, \qquad (V.14)$$

are manifestly t-independent. Thus:

**Theorem V.1.** The free fracton Lagrangian $\mathcal{L}_0$ and thus the equation of motion (V.3) are identical to the free Carroll expressions (IV.62) and (IV.68), up to interchanging $t$ and $s$. The dipole (resp. multipole) symmetry of free fractons can be understood as symmetry under Carroll boosts (IV.75) (resp. supertranslations). The conserved dipole moment in (V.14) and the conserved C-boost moment in (IV.76) are the same, $\mathbf{D} = \mathbf{D}^{\text{Car}}$.

Free fractons can thus be seen as Carroll field theories. However they can be seen also from another, complementary point of view.

Recall first certain similarities shared with the boost symmetry in Galilean quantum mechanics, (III.25) [89, 90]$^{38}$. Galilean boosts are represented projectively, i.e., up to a phase on the wavefunctions. The phase freedom can be incorporated by lifting the latter from Galilean (or Newton-Cartan) spacetime to one dimension higher, to Bargmann spacetime $\mathcal{B}$ [89, 90] with coordinates $(\boldsymbol{x}, t, s)$, where the variable $s$ corresponds to the phase. The lifted wavefunctions are equivariant,

$$\Psi(\boldsymbol{x}, t, s) = e^{ims}\Phi(\boldsymbol{x}, t), \qquad (V.15)$$

<footnote>$^{38}$ To incorporate the phase degree of freedom by lifting the wave function to a fiber bundle with a new, “internal” coordinate has first appeared in Geometric Quantization [155] and then in “monopole without strings” discussions [156–158]. The latter are discussed also in the Prequantization Chap. V. written around 1975 for the planned but never completed revised edition of Souriau’s book [77].</footnote>
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for some function $\Phi(\boldsymbol{x}, t)$. The 1-parameter central extension of the Galilei group (called the Bargmann group) acts on $\mathcal{B}$ and the phase factor in the Galilean phase symmetry, (III.25) is recovered by equivariance (V.15) with respect to the “vertical” variable, $s$.

Now we argue that the same procedure could be applied also for fractons. We could indeed consider fractons as wavefunctions lifted at some (for now unknown), fiber bundle $\mathcal{Q}$ with coordinates $(\boldsymbol{x}, t, s)$ with a “vertical” coordinate $s$. $\mathcal{Q}$ is analogous to Bargmann space — except that its base is an Aristotle, instead of Newton-Cartan manifold. Such wavefunctions would still be (V.15). Given the form (V.4) of free fractons, the latter lift to $\mathcal{Q}$ by equivariance, as,

$$ \Psi_{\text{free}}(\boldsymbol{x}, t, s) = e^{ims} (e^{imt}\phi(\boldsymbol{x})) . \qquad (\text{V.16}) $$

Let us emphasise that for free fractons the time, $t$, and the internal variable, $s$, play an equivalent role: what previously looked as a Carroll transformations acting on the time $t$, could be seen as Carroll (or even BMS-type) transformations acting on the submanifold defined by fixing $t$, shifting on the internal (phase) coordinate $s$, $s \mapsto s + T(\boldsymbol{x})$ cf. (IV.71) implying, by (V.16),

$$ \Phi(\boldsymbol{x}, t) \rightarrow \hat{\Phi}(\boldsymbol{x}, t) = e^{imT(\boldsymbol{x})} \Phi(\boldsymbol{x}, t) . \qquad (\text{V.17}) $$

The remarkable result is that C-boosts involving $t$ or $s$ yield the same implementation, (V.8), and thus the same conserved quantity. This follows from the missing of both $s$ and $t$ and having just $\boldsymbol{x}$ in the exponent. The clue is the double role of (V.4): both equivariance and solution.

Thus conservation laws can be derived in two different ways: either “horizontally”, by identifying fracton wavefunctions (V.4) as defined on a Carroll structure, or “vertically” by lifting the wavefunctions through (V.15) to a fiber-bundle in one dimension higher.

Free fractons can be coupled to a background electromagnetic field using the standard minimal coupling procedure, $\partial_{\mu} \rightarrow \partial_{\mu} - ieA_{\mu}(\boldsymbol{x})$. But the free Lagrangian $\mathcal{L}_0$ has no spatial derivatives $\nabla$, therefore the vector part of the gauge potential, $A$, does not enter into the Lagrangian.

The electric potential $V = V(\boldsymbol{x}, t)$ does appear via $\partial_t \rightarrow \partial_t - ieV(\boldsymbol{x}, t)$, but it only modifies the phase of the solution,

$$ \Phi_V(\boldsymbol{x}, t) = e^{i[mt + \int^t eV(\boldsymbol{x}, t)dt]} \phi(\boldsymbol{x}), \qquad (\text{V.18}) $$
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cf. in (V.4), without affecting the particle density (V.13), $|\Phi(\boldsymbol{x}, t)| = |\phi(\boldsymbol{x})|$. Therefore free fractons do not move even in an e.m. field: the immobility statement is extended to fractons in external fields. The same conclusion can be drawn also from dipole conservation: $\Phi_V$ in (V.18) solves the coupled field equation

$$ (\partial_t - i e V(\boldsymbol{x}, t)) \Phi_V = i m \Phi_V, \qquad (V.19) $$

which implies in turn the conservation of $\mathbf{D}$ in (V.12b), which boils down again to (V.14).

That Aristotle symmetry generates a Carrollian model is not surprising: the same happens at the classical level. Aristotelian particles can be constructed, for instance, by the KKS algorithm [9]. Denoting an element of the dual by $\mu = (\ell, \boldsymbol{p}, \mathcal{E})$, where we have as usual $\ell$ the (anyonic) spin of the particle, $\boldsymbol{p}$ its momentum and $\mathcal{E}$ its energy, the coadjoint action is

$$ \mathrm{Coad}(g)\mu = (\ell - \boldsymbol{c} \times A\boldsymbol{p}, A\boldsymbol{p}, \mathcal{E}). \qquad (V.20) $$

Thus the coadjoint orbits are just the Euclidean orbits shifted by $\mathcal{E}$. The Casimirs are the norm of the momentum, $|\boldsymbol{p}|$, and the energy $\mathcal{E}^{39}$. For $|\boldsymbol{p}| \neq 0$ and $\mathcal{E} \neq 0$, the Cartan 1-form is $^{40}$,

$$ \varpi_{\mathrm{Aris}} = \boldsymbol{p} \cdot d\boldsymbol{x} + \mathcal{E}dt. \qquad (V.21) $$

However in (V.21) we recognize that we formally have the same model as the Carroll one in (IV.3), in the interchange $s \leftrightarrow t$ and $m \leftrightarrow \mathcal{E}$.

The Aristotle group $\mathfrak{A}$ is by construction (and also manifestly) a symmetry of (V.21). However, owing to the fact that the time translation in the Aristotle group appears as a trivial central extension of the Euclidean group, the coupling between space and time coordinates are uncoupled explaining why the Aristotle models may have position-dependent time translations, i.e., supertranslations. Then, as mentioned above, the Carroll group is a subgroup of Aristotle + supertranslations.

Turning now to the full Lagrangian (V.1), we note first the self-interaction term $\mathcal{L}_{\mathrm{int}}$ is invariant under the phase transformations (V.8),

$$ \mathcal{L}_{\mathrm{int}}(\Phi) = \mathcal{L}_{\mathrm{int}}(\hat{\Phi}) \qquad (V.22) $$

<footnote>$^{39}$ In 3 dimensions, we would have also the scalar spin $\ell \cdot \boldsymbol{p}$.</footnote>

<footnote>$^{40}$ The Lagrangian resp. Souriau form are, accordingly, $\mathcal{L}_{\mathrm{Arist}} = \boldsymbol{p} \cdot \frac{d\boldsymbol{x}}{dt} + \mathcal{E}$ (notice, however, that $\mathcal{E}$ is merely a constant here) and $\sigma = d\boldsymbol{p} \wedge d\boldsymbol{x}$.</footnote>
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which are thus symmetries for the interacting system (V.1) [41]. Then the Noether theorem implies that the free expressions (V.12) remain correct for the full dynamics.

What about the Carroll approaches? One sees readily that while the lifting condition (III.26) remains valid, the field equations for interacting fractons do change. Thus the “horizontal” Carroll transformation involving $t$, (V.11) with $T(x) = -b_1 \cdot x$, is not more a symmetry for the full expression (V.1). However the “vertical” one with $s$, (I.1), is still a symmetry, confirming the dipole symmetry for (V.1). Their curious accidental identity is not more valid for $\lambda \neq 0$.

One can wonder if the BMS-type higher symmetry survives the addition of the interaction term $\mathcal{L}_{int}$ in (V.1). Plugging the Ansatz (V.10) into (V.1), we readily see that

$$ \mathcal{L} \to \tilde{\mathcal{L}} = \mathcal{L} + \lambda \Phi^4 (\partial_i \partial_j T) (\partial_i \partial_j T). \qquad (V.23) $$

Not being a total derivative, the shift breaks the symmetry, unless

$$ \partial_i \partial_j T = 0 \quad \text{for all } i, j. \qquad (V.24) $$

It follows that only terms up to linear order in $x$, $T(x) = \beta_0 - \vec{\beta}_1 \cdot \vec{x}$ survive in general, as in (V.8).

Similarly, a dilation $\Phi \to \hat{\Phi} = \Lambda\Phi$, $\Lambda = \text{const.}$ breaks the scale invariance of the free Lagrangian,

$$ \mathcal{L}_0 \to \tilde{\mathcal{L}}_0 = \Lambda^2 \mathcal{L}_0 \quad \text{but} \quad \mathcal{L}_{int} \to \Lambda^4 \mathcal{L}_{int}. \qquad (V.25) $$

In conclusion,

**Theorem V.2.** The BMS-type infinite multipole symmetry (V.10) of the free Lagrangian $\mathcal{L}_0$ is broken by the interaction term $\mathcal{L}_{int}$ to the first two (constant and linear) terms, as in eqns (V.8). The first of these generates conserved mass/electric charge, while the linear one, which is indeed an internal Carroll boost (I.1), generates the conserved dipole moment $\mathbf{D}$ in (V.14) which implies immobility even for interactions considered in (V.1).

With hindsight to the Galilean case, it would be expected that the boost symmetry is broken by the interaction. It is therefore remarkable that one of two Carroll boost — namely the $s$-based “internal one” — does survive the addition of $\mathcal{L}_{int}$ which thus keeps its dipole symmetry $^{41}$.

<footnote>$^{41}$ Bidussi et al. [41] call the “horizontal” ($t$-based) transformation “Carroll boost” and the “vertical” one based on $s$ “internal”.</footnote>
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The link between Carroll symmetry and fractons is also investigated in [154].

# VI. DOUBLY-EXTENDED CARROLL DYNAMICS

## A. Free massive doubly-extended particle

A fact which escaped attention until recently is that in $d = 2$ space dimensions the Carroll group admits a two-parameter central extension [8, 34, 81]. Its central charges shall be denoted by $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$, respectively, whose non-vanishing will henceforth be assumed. Such additional charges are allowed by the theory and they may play an important role for planar systems with Carroll symmetry.

The doubly-extended group, $\hat{\mathcal{C}}$, can be represented by $6 \times 6$ matrices (Eqn # (2.9) in [34]) to be compared with (II.14),

$$ \left( \begin{array}{ccccc} A_j^i & 0 & c^i & 0 & \epsilon_k^i b^k \\ -b_k A_j^k & 1 & f & 0 & A_{\text{exo}} \\ 0 & 0 & 1 & 0 & 0 \\ c_k \epsilon_l^k A_j^l & 0 & A_{\text{mag}} & 1 & -(f + \mathbf{b} \cdot \mathbf{c}) \\ 0 & 0 & 0 & 0 & 1 \end{array} \right), \qquad (\text{VI.1}) $$

where $A_{\text{mag}}$ and $A_{\text{exo}}$ are new parameters associated with the double extension. The extended algebra has a 8-parameter moment map, $\mu(\ell, \mathbf{D}, \pi, m, \kappa_{\text{mag}}, \kappa_{\text{exo}})$ with commutators,

$$ \begin{align} [J, K_i] &= \epsilon_{ij} K_j, & [K_i, K_j] &= \epsilon_{ij} A_{\text{exo}}, & [J, P_i] &= \epsilon_{ij} P_j, & [K_i, P_j] &= \delta_{ij} P_0, \tag{VI.2} \\ [J, P_0] &= 0, & [K_i, P_0] &= 0, & [P_i, P_j] &= \epsilon_{ij} A_{\text{mag}}, & [P_i, P_0] &= 0, \nonumber \end{align} $$

where $J$ is the rotation, the $(K_i)$ are boosts, the $(P_i)$ spatial translations, $(P_0)$ is time translation, and $A_{\text{exo}}$ and $A_{\text{mag}}$ are the exotic and magnetic extensions, respectively.

More insight into the extended algebra structure is obtained by comparison with previously considered extensions. The one with $A_{\text{exo}}$ is plainly the same as the “exotic” extension considered for the planar Galilean algebra [68–70]. The extension $A_{\text{mag}}$ in the doubly extended Carroll algebra (VI.2) implies in turn the non-commutativity of spatial translations, consistently with its interpretation as an additional, “internal” magnetic field see sec.II.
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The extended Carroll group acts on these moments by coadjoint action,

$$ \ell' = \ell + \mathbf{b} \times A\mathbf{D} - \mathbf{c} \times A\boldsymbol{\pi} + m\mathbf{b} \times \mathbf{c} - \frac{1}{2}\kappa_{\text{mag}}\mathbf{c}^2 - \frac{1}{2}\kappa_{\text{exo}}\mathbf{b}^2 \quad (\text{VI.3a}) $$

$$ \mathbf{D}' = A\mathbf{D} + m\mathbf{c} + \frac{1}{2}\kappa_{\text{exo}}\epsilon\mathbf{b} \quad (\text{VI.3b}) $$

$$ \boldsymbol{\pi}' = A\boldsymbol{p} + m\mathbf{b} - \frac{1}{2}\kappa_{\text{mag}}\epsilon\mathbf{c} \quad (\text{VI.3c}) $$

$$ m' = m \quad (\text{VI.3d}) $$

$$ \kappa'_{\text{mag}} = \kappa_{\text{mag}} \quad (\text{VI.3e}) $$

$$ \kappa'_{\text{exo}} = \kappa_{\text{exo}} \quad (\text{VI.3f}) $$

The free doubly-extended model obtained by the KKS algorithm is, by construction, symmetric w.r.t. the doubly-extended Carroll group $\hat{\mathfrak{C}}$, with associated conserved quantities,

$$ \ell = \boldsymbol{x} \times \boldsymbol{p} + \frac{1}{2}\theta\boldsymbol{p}^2 + \frac{1}{2}\kappa_{\text{mag}}\boldsymbol{x}^2 + \chi, \qquad \text{angular momentum} \quad (\text{VI.4a}) $$

$$ \pi_i = p_i - \kappa_{\text{mag}} \epsilon_{ij} x^j, \qquad \text{“impulsion”} \quad (\text{VI.4b}) $$

$$ D^i = m(x^i + \theta \epsilon^{ij} p_j), \qquad \text{C-boost momentum} \quad (\text{VI.4c}) $$

completed with $m, \kappa_{\text{exo}}, \kappa_{\text{mag}}$, cf. \# (3.18) in [34] and sec.IV for the terminology.

These quantities are analogous to the “exotic” Galilean expressions in a constant magnetic field [69, 70]. The real constant $\chi$ in (VI.4a) is Carrollian/anyonic spin, defined in (VI.5). The boost momentum $\mathbf{D}$ (VI.4c) plays a particularly important role, as it will be explained below.

The unusual form of the “impulsion” $\boldsymbol{\pi}$ in (VI.4b) deserves a comment. First we note that $A_i = -\frac{1}{2}(\kappa_{\text{mag}}/e)\epsilon_{ij}x^j$ is a vector potential for $\kappa_{\text{mag}}$ viewed as a (constant) “internal magnetic field”. Then $\boldsymbol{\pi}$ has the unexpected form $\pi_i = p_i + 2eA_i$, where $\mathbf{p}$ is just a coordinate. However in terms of the canonical momentum $P_i = p_i - \frac{1}{2}\kappa_{\text{mag}}\epsilon_{ij}x^j$ obtained from the Lagrangian (VI.13), $\pi_i = P_i - \frac{1}{2}\kappa_{\text{mag}}\epsilon_{ij}x^j = P_i + eA_i$.

**Casimirs.** When the mass does not vanish, $m \neq 0$, we have four Casimirs [34]: $\kappa_{\text{mag}}, \kappa_{\text{exo}}$, the mass, $m$, and the anyonic spin, $\chi$, defined by,

$$ m^*\chi = m^*\ell - \mathbf{D} \times \boldsymbol{\pi} + \frac{\kappa_{\text{exo}}}{2m}\boldsymbol{\pi}^2 + \frac{\kappa_{\text{mag}}}{2m}\mathbf{D}^2, \quad (\text{VI.5}) $$

where

$$ m^* = m \left( 1 - \frac{\kappa_{\text{exo}}}{m^2} \kappa_{\text{mag}} \right) \quad (\text{VI.6}) $$
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is an effective mass, analogous to $m_G^*$ in (III.10) in the Galilean case.

Eqn. (VI.5) is valid also for $m^* = 0$ when both the spin $\chi$ and the total angular momentum $\ell$ drop out. In this case, realized for a special relation between the Casimir invariants $\kappa_{\text{mag}} \cdot \kappa_{\text{exo}} = m^2$, we get:

$$ \frac{1}{m} \mathbf{D} \times \boldsymbol{\pi} - \frac{1}{2\kappa_{\text{mag}}} \boldsymbol{\pi}^2 - \frac{1}{2\kappa_{\text{exo}}} \mathbf{D}^2 = 0. \qquad (\text{VI.7}) $$

Multiplying (VI.5) by $m^2 \neq 0$ and then letting $m \to 0$ yields in turn the expression valid in the massless case

$$ \chi = \ell - \frac{\boldsymbol{\pi}^2}{2\kappa_{\text{mag}}} - \frac{\mathbf{D}^2}{2\kappa_{\text{exo}}}, \qquad (\text{VI.8}) $$

where the Casimir invariants $\kappa_{\text{mag}}$ and $\kappa_{\text{exo}}$ are assumed not to vanish. Equation (VI.8) implies that the Carrollian anyonic spin $\chi$ is given by the angular momentum $\ell$ with contributions coming from the impulsion $\boldsymbol{\pi}$ and the C-boost (alias generalized dipole) momentum.

The KKS construction applied to $\hat{\mathcal{C}}$ yields, for a particle with mass $m \neq 0$, the Cartan resp. Souriau-forms with two additional terms [34],

$$ \bar{\omega}_{\text{exo}} = \boldsymbol{p} \cdot d\boldsymbol{x} + \frac{\theta}{2} \boldsymbol{p} \times d\boldsymbol{p} + \frac{\kappa_{\text{mag}}}{2} \boldsymbol{x} \times d\boldsymbol{x}, \qquad (\text{VI.9a}) $$

$$ \sigma_{\text{exo}} = d\boldsymbol{p} \wedge d\boldsymbol{x} + \underbrace{\frac{1}{2}\theta \epsilon_{ij} dp^i \wedge dp^j}_{\text{exotic}} + \underbrace{\frac{\kappa_{\text{mag}}}{2} \epsilon_{ij} dx^i \wedge dx^j}_{\text{"internal magnetic"}}, \qquad (\text{VI.9b}) $$

defined on Carroll evolution space $\mathcal{E}$ (IV.4), cf. (3.10) in [34]. The constant

$$ \theta = \frac{\kappa_{\text{exo}}}{m^2} \qquad (\text{VI.10}) $$

here is interpreted as the non-commutativity parameter, see (VI.12) below.

The terms with $\theta \sim \kappa_{\text{exo}}$ in (VI.9) are those of exotic Galilean dynamics, (I.3), and are in fact present also for *relativistic anyons* from which the Galilean model can be deduced as it will be shown in sec. VII. The terms with $\kappa_{\text{mag}}$ are reminiscent of a (constant) “internal magnetic” field, as mentioned above. Cf. also (III.8).

The Souriau form $\sigma_{\text{exo}}$ in (VI.9) is formally $\Omega_{\text{exo}} - d\mathcal{H}_0 \wedge ds$ with zero Hamiltonian,

$$ \Omega_{\text{exo}} = d\boldsymbol{p} \wedge d\boldsymbol{x} + \frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j + \frac{\kappa_{\text{mag}}}{2} \epsilon_{ij} dx^i \wedge dx^j \quad \text{and} \quad \mathcal{H}_0 \equiv 0, \qquad (\text{VI.11}) $$

cf. (IV.5). $\Omega_{\text{exo}}$ is a closed and regular (i.e. symplectic) 2-form on the $s = s_0 = \text{const.}$ submanifold $\mathbb{R}^2 \times \mathbb{R}^2 = \{\boldsymbol{x}, \boldsymbol{p}, s_0\}$ of the Carroll evolution space $\mathcal{E}$.
When the effective mass $m^*$ is non-zero, $m^* \neq 0$, i.e. $\kappa_{\text{exo}} \cdot \kappa_{\text{mag}} \neq m^2$, the associated Poisson brackets [8]

$$ \begin{aligned} \{x^i, x^j\} &= \left(\frac{1}{1-\theta\kappa_{\text{mag}}}\right) \theta\epsilon^{ij}, \\ \{x^j, p_i\} &= \left(\frac{1}{1-\theta\kappa_{\text{mag}}}\right) \delta_i^j, \\ \{p_i, p_j\} &= \left(\frac{1}{1-\theta\kappa_{\text{mag}}}\right) \kappa_{\text{mag}} \epsilon_{ij} \end{aligned} \qquad (\text{VI.12}) $$

are identical with eqn. (III.12) of the “exotic” Galilei case under the replacement $\kappa_{\text{mag}} \sim eB$. The first relation implies that the coordinates do not commute when $\kappa_{\text{exo}} \neq 0$ and the last one implies noncommuting momenta for $\kappa_{\text{mag}} \neq 0$.

Alternatively (and equivalently when $m^* \neq 0$) we can consider the doubly-extended Carroll Lagrangian defined on the evolution space $\mathcal{E}$ in (IV.4) which has two more terms added to the free unextended Carroll Lagrangian $\mathcal{L}_0$ (IV.9),

$$ \mathcal{L}_{\text{exo}} = \underbrace{\boldsymbol{p} \cdot \boldsymbol{x}'}_{\mathcal{L}_0} + \underbrace{\frac{1}{2}\theta \epsilon_{ij} p^i (p^j)'}_{\text{exotic}} + \underbrace{\frac{1}{2}\kappa_{\text{mag}} \epsilon_{ij} x^i (x^j)'}_{\text{magnetic}}. \qquad (\text{VI.13}) $$

The associated variational equations (which are valid without the assumption $m^* \neq 0$)

$$ (x^k)' + \theta \epsilon^{kl}(p_l)' = 0, \qquad (p_i)' - \kappa_{\text{mag}} \epsilon_{ij}(x^j)' = 0, \qquad (\text{VI.14}) $$

are integrated at once to yield the conserved vectors

$$ x^k + \theta \epsilon^{kl} p_l = \text{const.} = Q^k, \qquad (\text{VI.15a}) $$

$$ p_i - \kappa_{\text{mag}} \epsilon_{ij} x^j = \text{const.} = \pi_i. \qquad (\text{VI.15b}) $$

In particular, the 2-vector $Q = (Q^k)$ in (VI.15a) is identified as the guiding center [54, 119].

The conservation of the boost momentum $D$ in (VI.4) alone does not now imply immobility for $x$ – however it does imply that of $Q$ in (VI.15a). Combining with the conservation of $\pi$ (VI.15b) then yields the pair of decoupled equations which involve the effective mass (VI.6),

$$ m^*(x_i)' = 0, \qquad m^*(p_i)' = 0 \qquad (\text{VI.16}) $$

which allow us to infer:
**Theorem VI.1.** A free doubly-extended Carroll particle with non-vanishing effective mass, $m^* \neq 0$ in (VI.6), does not move.

However when the effective mass vanishes,

$$m^* = 0 \quad \text{i.e.} \quad \theta \kappa_{\text{mag}} = 1 \quad \text{i.e.} \quad \kappa_{\text{exo}}\kappa_{\text{mag}} = m^2, \qquad (\text{VI.17})$$

then no conclusion can be drawn from (VI.16). For $m^* = 0$ the determinant of the symplectic form$^{42}$,

$$\det(\Omega_{ij}) = \left(\frac{m^*}{m}\right)^2 = (1 - \theta\kappa_{\text{mag}})^2 \qquad (\text{VI.18})$$

becomes indeed zero, highlighting the fact that the variational system is singular, requiring “Faddeev-Jackiw” reduction [69, 70, 85]. The dimension of the phase space drops from 4 to 2. The variables $x$ and $p$ become redundant however their (fixed) combination $Q = (Q^k)$ in (VI.15a) is still a physical quantity. In its terms the reduced free symplectic form is simply

$$\Omega_{\text{red}} = \kappa_{\text{mag}} dQ^1 \wedge dQ^2 = \frac{1}{\theta} dQ^1 \wedge dQ^2. \qquad (\text{VI.19})$$

The reduced Poisson brackets and variational forms are thus

$$\{Q_1, Q_2\}_{\text{red}} = -\theta, \quad \varpi_{\text{red}} = \frac{1}{2\theta} \epsilon_{ij} Q^i dQ^j, \quad \mathcal{L}_{\text{red}} = \frac{1}{2\theta} \mathbf{Q} \times \mathbf{Q}'. \qquad (\text{VI.20})$$

The Hamiltonian $\mathcal{H}$ is identically zero and the equations of motion, $Q'_i = \{Q^i, \mathcal{H}\}$, are therefore,

$$Q'_i = 0 \Rightarrow \mathbf{Q} = \mathbf{Q}_0 = \text{const.} \qquad (\text{VI.21})$$

consistently with (VI.15a). In conclusion,

**Theorem VI.2.** When the effective mass of a massive system vanishes, $m^* = 0$, the free massive doubly-extended particle model becomes singular. The dynamics of $x$ and of $p$ can not be separately determined, however the guiding center $Q$ in (VI.15a), remains fixed. The conserved quantities in (VI.4) are expressed in terms of the guiding center alone,

$$\ell = \frac{1}{2\theta} \mathbf{Q}^2 \qquad \text{angular momentum} \qquad (\text{VI.22a})$$

$$\pi_i = -\frac{1}{\theta} \epsilon_{ik} Q^k \qquad \text{“impulsion”} \qquad (\text{VI.22b})$$

$$\mathbf{D} = m\mathbf{Q} \qquad \text{C-boost momentum} \qquad (\text{VI.22c})$$

consistently with the loss of two physical degrees of freedom.

<footnote>$^{42}$ Compare with (III.10).</footnote>
For $m^* = 0$ the (no-)motion of the guiding center, (VI.21), follows directly from the conservation of $\mathbf{D}$ or of $\pi$, themselves proportional to $Q^k$.

We conclude that extended Carroll symmetry plays for the extended model a role analogous to that of Carroll in the unextended case, with the effective mass $m^*$ (VI.6) replacing the “naked” mass, $m$.

## B. Massless doubly-extended particles

Let us now study the double extension of the massless Carroll particles discussed in section IV C. The coadjoint action of the Euclidean group (IV.32) extends to one of the doubly extended Carroll group. The group element $(A, \mathbf{b}, \mathbf{c}, f, A_{\text{exo}}, A_{\text{mag}}) \in \tilde{C}$ is implemented on $\tilde{\mu}_0 = (\ell, \mathbf{D}, \pi, m = 0, \kappa_{\text{exo}}, \kappa_{\text{mag}}) \in \tilde{\mathcal{C}}^*$ as,

$$ \ell \rightarrow \ell - \mathbf{c} \times A\pi + \mathbf{b} \times AD - \frac{1}{2}\kappa_{\text{exo}}\mathbf{b}^2 - \frac{1}{2}\kappa_{\text{mag}}\mathbf{c}^2, \qquad (\text{VI.23a}) $$

$$ \pi_i \rightarrow (A\pi)_i - \kappa_{\text{mag}} \epsilon_{ij} c^j, \qquad (\text{VI.23b}) $$

$$ D^i \rightarrow (AD)^i + \kappa_{\text{exo}} \epsilon_{ij} b^j \qquad (\text{VI.23c}) $$

while $m = 0, \kappa_{\text{exo}}, \kappa_{\text{mag}}$ are left invariant, [34]. Having $\kappa_{\text{mag}} \neq 0$ in (VI.23c) then implies that the norm of the momentum is not a Casimir invariant anymore [34]. We thus have only one kind of massless doubly-extended particle — namely the trivial one associated with $\pi = 0$. Put in another way, the “color” $k$ in a chosen basepoint can be eliminated by a translation which carries $\tilde{\mu}_0$ to the origin of the dual algebra, whose contribution to the KKS symplectic form is therefore obviously zero. Thus the first term in (VI.9a) disappears, leaving us with the “truncated” or “purely exotic” Cartan and Souriau forms,

$$ \bar{\omega}_{\text{exo}} = \frac{\kappa_{\text{exo}}}{2} \mathbf{v} \times d\mathbf{v} + \frac{1}{2} \kappa_{\text{mag}} \mathbf{x} \times d\mathbf{x}, \qquad (\text{VI.24a}) $$

$$ \sigma_{\text{exo}} = \underbrace{\frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} dv^i \wedge dv^j}_{\text{exotic}} + \underbrace{\frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} dx^i \wedge dx^j}_{\text{magnetic}}, \qquad (\text{VI.24b}) $$

where we reverted to the $\mathbf{v}$-notation since $\mathbf{p} = mv$ is not well-defined anymore$^{43}$ which further extends the “no-particle” model discussed in subsec. IV D. Had we set the extension parameters to zero we would find vanishing forms and thus no particle trajectories, as we

<footnote>$^{43}$ The same result is obtained alternatively by putting $\mathbf{p} = mv$ in (VI.9a) and then letting $m \to 0$.</footnote>
had observed before in section IV C. However keeping the terms induced by the central extension, we do find “trajectories”: the equations of motion become trivial and yield fixed positions$^{44}$,

$$ (x^i)' = 0, \quad (p_i)' = 0 \quad \Rightarrow \quad \boldsymbol{x}(s) = \boldsymbol{x}_0 = \text{const.}, \quad \boldsymbol{p}(s) = \boldsymbol{p}_0 = \text{const.} \qquad (\text{VI.25}) $$

**Theorem VI.3.** A “purely exotic” free massless doubly-extended Carroll particle (VI.24) which has $\pi = 0$ does not move: the double extension eliminates the free motions we studied in sec. IV C.

This no-motion conclusion will be modified when the particle is coupled to external fields, see the section below.

### C. Coupling to a gauge field: eppur si muove

The free model can be coupled to an em field by the rule (II.48). The external and “internal” magnetic fields combine into the generalized magnetic field:

$$ B^* = eB + \kappa_{\text{mag}}, \qquad (\text{VI.26}) $$

and yield non-commuting coordinates and momenta [34],

$$ \sigma_{\text{exo}} = dp_i \wedge dx_i + \frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j + \frac{1}{2} B^* \epsilon_{ij} dx^i \wedge dx^j + eE_i dx_i \wedge ds, \qquad (\text{VI.27a}) $$

$$ \{x_i, x_j\} = \frac{m}{m^*} \theta \epsilon_{ij}, \quad \{x_i, p_j\} = \frac{m}{m^*} \delta_{ij}, \quad \{p_i, p_j\} = \frac{m}{m^*} B^* \epsilon_{ij}, \qquad (\text{VI.27b}) $$

where the Carrollian effective mass which generalizes (VI.6) now involves the combined field (VI.26),

$$ m^* = m(1 - \theta B^*), \qquad (\text{VI.28}) $$

assumed not to vanish$^{45}$. For a static scalar potential $V = V(\boldsymbol{x})$, $\boldsymbol{E} = -\nabla V$, for example, the Hamilton equations are [34]),

$$ (x^i)' = - \frac{\theta}{1 - \theta B^*} \epsilon^{ij} eE_j, \qquad (\text{VI.29a}) $$

$$ (p_i)' = \frac{1}{1 - \theta B^*} eE_i. \qquad (\text{VI.29b}) $$

<footnote>$^{44}$ The vector $\boldsymbol{u}$ in the Fermat model of sec. IV C is now ill-defined: null vectors have no direction.</footnote>

<footnote>$^{45}$ Compare with (III.8), (III.10) and (III.12) in the Galilean case.</footnote>
For completeness, we mention that the symplectic form associated with (VI.27a) is

$$ \Omega_{\alpha\beta} = \begin{pmatrix} 0 & B^* & -1 & 0 \\ -B^* & 0 & 0 & -1 \\ 1 & 0 & 0 & \theta \\ 0 & 1 & -\theta & 0 \end{pmatrix} \Rightarrow \Omega^{\alpha\beta} = \frac{1}{(1 - \theta B^*)} \begin{pmatrix} 0 & \theta & 1 & 0 \\ -\theta & 0 & 0 & 1 \\ -1 & 0 & 0 & B^* \\ 0 & -1 & -B^* & 0 \end{pmatrix} \quad (VI.30) $$

so that the Hamilton equations for $\mathcal{H} = eV$ are indeed (VI.29). The position $\boldsymbol{x}$ splits off from $\boldsymbol{p}$ which is coupled to $\boldsymbol{x}$ through the electric field, — but its dynamics does not effect the motion of $\boldsymbol{x}$.

The equations (VI.29) are reminiscent of but different from their Galilean counterparts in (III.13): $p_i$ is missing from the “velocity” relation (VI.29a) which is purely anomalous, whereas the Lorentz force is missing from the purely electric $\boldsymbol{p}$-equation (VI.29b). The external magnetic field is hidden in $m^*$ through $B^*$.

**Theorem VI.4.** The position $\boldsymbol{x}$ of a massive doubly-extended Carroll particle in a static EM field with non-vanishing effective mass $m^* \neq 0$ moves by following the (anomalous) Hall law, (VI.29a).

However combining the equations (VI.29) implies,

**Theorem VI.5.** The guiding center $Q^i = x^i + \theta \epsilon^{ij} p_j$ in (VI.15a) of a doubly-extended particle with $m^* \neq 0$, coupled to an electromagnetic field does not move,

$$ Q'_i = 0 \Rightarrow \mathbf{Q} = \mathbf{Q}_0 = \text{const.} \quad (VI.31) $$

When $m^* \neq 0$ the real-space motion is recovered by solving the $\boldsymbol{p}$ equation and using (VI.15a),

$$ x^i(s) = Q_0^i - \theta \epsilon^{ij} p_j(s). \quad (VI.32) $$

For constant EM fields s.t. $m^* \neq 0$, for example, the motion is perpendicular to the electric field,

$$ x^i(s) = Q_0^i - \epsilon^{ij} \left( \frac{e\theta}{1 - \theta B^*} \right) E_j s. \quad (VI.33) $$

For vanishing effective mass, $m^* = 0$ the $\boldsymbol{x}$-motion can not be recovered as said before (and consistently with (VI.29) and (VI.33)).
Another example is obtained by a radial electric field $\mathbf{E} = f(|\mathbf{x}|) \hat{\mathbf{x}}$, where $f(|\mathbf{x}|)$ is a function of the radius in the $d=2$ plane. Multiplying (VI.29a) by $\hat{\mathbf{x}}$ shows that the motion which is circular. In terms of the complex variables $\zeta = x^1 + ix^2$ and $\Pi = p_1 + ip_2$,

$$ \zeta(s) = |\mathbf{x}_0| \exp[-i\Omega s] \quad \text{where} \quad \Omega = \frac{e\theta}{1 - \theta B^*(|\mathbf{x}_0|)} \frac{f(|\mathbf{x}_0|)}{|\mathbf{x}_0|}, \qquad (\text{VI.34a}) $$

$$ \Pi(s) = i \frac{|\mathbf{x}_0|}{\theta} \exp[-i\Omega s] + \Pi_0 = \frac{i}{\theta} \zeta(s) + \Pi_0. \qquad (\text{VI.34b}) $$

Thus the momentum rotates around $\Pi_0 = \text{const.}$ (which can be absorbed into the guiding center) with the same angular velocity $\Omega$ as $\zeta$, but with a phase advance by $\pi/2$.

For $m^* = 0$ but $\mathbf{E} \neq 0$ the eqns (VI.29) are ill-defined, however their guiding center, i.e. their combination $\mathbf{Q}$ in (VI.15a) remains fixed when $m^*$ changes sign by sweeping through zero,

$$ \frac{m^*}{m} (Q^i)' = \frac{m^*}{m} ((x^i)' + \theta \epsilon^{ij} p_j') = 0 \qquad (\text{VI.35}) $$

by (VI.31), including at $m^* = 0$. See Fig. 4 for an insight into the “phase transition”. The external forces vary the frequency $\Omega = \Omega(|\mathbf{x}|)$ but the trajectories remain circular with fixed radius $|\mathbf{x}_0|/\theta$ for all values of $m^*$, cf. [119].

<figure><img src="image_4.png" /><figcaption>FIG. 4: Trajectories for unit charge and exotic parameter in a radial electric field $\mathbf{E} = \mathbf{x}/|\mathbf{x}|^3$ in the plane for effective masses passing from negative to positive, unfolded in Carroll time $s$. The rotation around the fixed guiding center $\mathbf{Q}(s) = \mathbf{Q}_0$ speeds up when the effective mass increases from $m^* < 0$. The motion is instantaneous for $m^* = 0$ and changes orientation for $m^* > 0$.</figcaption></figure>
## D. Extension to vanishing Carroll mass, m=0

The above results can be extended from massive to massless particles. The clue is to rewrite (VI.29) for $m \neq 0$ as,

$$ (x^i)' = \frac{1}{B^* - m^2/\kappa_{\text{exo}}} \epsilon^{ij} e E_j, \quad p_i' = m^2 \frac{eE_i}{m^2 - \kappa_{\text{exo}}B^*}. \qquad (\text{VI.36}) $$

Then letting $m \to 0$ the exotic charge $\kappa_{\text{exo}} \neq 0$ drops out and we get:

**Theorem VI.6.** A massless, charged doubly-extended Carroll particle in combined electric and effective magnetic fields $E$ and $B^*$ moves according to the generalized Hall law,

$$ (x^i)' = \epsilon^{ij} \frac{eE_j}{B^*} = \epsilon^{ij} \frac{eE_j}{eB + \kappa_{\text{mag}}} \quad \text{and} \quad p_i' = 0, \qquad (\text{VI.37}) $$

The uncoupled first-order eqns. (VI.37) and the “interesting” one for $x$ can in fact be derived from the first-order Lagrangian resp. Souriau form defined on the 2+1 dimensional Carroll manifold itself,

$$ \mathcal{L}_{m=0}^{\text{mag}} = \frac{1}{2} B^* \epsilon_{ij} x^i (x^j)' - eV(\boldsymbol{x}), \qquad (\text{VI.38a}) $$

$$ \sigma_{m=0}^{\text{mag}} = \frac{1}{2} B^* \epsilon_{ij} dx^i \wedge dx^j + eE_i dx^i \wedge ds. \qquad (\text{VI.38b}) $$

These expressions do not involve the exotic charge $\kappa_{\text{exo}}$. Its magnetic counterpart, $\kappa_{\text{mag}}$, is harmless, as it merely shifts the magnetic field by a constant and can even be turned off if we wish and then (VI.37) reduces to the usual Hall law (I.2). The expressions (VI.38) are consistent also with the degenerate “no-particle” expressions (IV.55) in sec. IV D associated with $\pi = 0$, written in the gauge $A_i = -\frac{B}{2} \epsilon_{ij} x^j$.

Returning to the Souriau framework, coupling the free massless particle with “purely exotic dynamics”, (VI.24), to an external EM field by the rule (II.48) yields,

$$ \sigma_{m=0} = \underbrace{\frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} dv^i \wedge dv^j + \frac{1}{2} B^* \epsilon_{ij} dx^i \wedge dx^j}_{\Omega_{m=0}} + \underbrace{eE_i dx^i \wedge ds}_{-d\mathcal{H}_{m=0} \wedge ds}. \qquad (\text{VI.39}) $$

Rewriting the Souriau form $\sigma_{\text{exo}}$ (VI.27a) in terms of $v = p/m$ and then letting $m \to 0$ suppresses the first term and yields once again (VI.39). The determinant of the symplectic form $\Omega_{m=0}$ in (VI.39) is,

$$ \det(\Omega_{m=0}) = (\kappa_{\text{exo}})^2 (B^*)^2. \qquad (\text{VI.40}) $$
Thus the conditions $\kappa_{\text{exo}} \neq 0$ and $B^* \neq 0$ guarantee that $\Omega_{m=0}$ is a regular symplectic form. However position $\boldsymbol{x}$ and $\boldsymbol{v}$ are decoupled and the dynamics projects to Carroll spacetime $C = \{(\boldsymbol{x}, s)\}$: inuitively, the first term in (VI.39) is switched off, and “forgetting” about the “irrelevant” dynamics $\boldsymbol{v}' = 0$ we are left with (VI.38).

For $\Omega_{m=0} = 0$ the system becomes singular and requires regularization called “Faddeev-Jackiw” [or Hamiltonian] reduction [85]. The rigorous statement is that in the singular case the 5-dimensional evolution space $\mathcal{E} = \{(\boldsymbol{v}, \boldsymbol{x}, s)\}$ is reduced to 2+1-dimensional Carroll space $C$, endowed with the reduced dynamics (VI.38). The exotic charge $\kappa_{\text{exo}}$ drops out in the process. The reduced symplectic form $\sigma_{m=0}^{\text{mag}}$ in (VI.38b) is regular provided $B^* \neq 0$.

The reduced dynamics in eqns (VI.37) – (VI.38) yields Hall motion also for a massive, charged Carroll particle with no central extension, $\kappa_{\text{exo}} = \kappa_{\text{mag}} = 0$ as long as $B \neq 0$.

What can be the physical realization of these particles? Rigorously speaking, there are no strictly massless charged fields in the Standard Model of fundamental interactions in its low-temperature phase, which is relevant to present-day phenomenology [159]. While genuinely massive constituents of the model can be treated as effectively massless particles at very high energies (such as leptons and quarks in heavy ion collisions [160]), they propagate in three spatial dimensions and, therefore, are not relevant for the peculiar features of two-dimensional physics considered in this section.

However, one cannot exclude that distant analogs could be found in the condensed matter context. As a promising example of quasiparticles possessing massless and, simultaneously, electrically-charged excitations in two spatial dimensions has been realized in graphene, a two-dimensional monolayer crystal made of carbon atoms, discovered experimentally in 2004 [161]. In this material, the dynamics of the low-energy electronic excitations is described by the 2 + 1 dimensional massless Dirac Hamiltonian,

$$ \mathcal{H} = \bar{\psi} i \boldsymbol{\alpha} (\boldsymbol{\nabla} - i e \boldsymbol{A}) \psi + eV, \qquad (\text{VI.41}) $$

where $\boldsymbol{\alpha} = (\gamma^0 \gamma^1, \gamma^0 \gamma^2)$ and the $\gamma^\mu$ are spatial Dirac matrices in 2 + 1 dimensions. Being charged, the massless electronic quasiparticles interact with the background electromagnetic field $A^\mu = (V, \boldsymbol{A})$, thus providing us with a system that exhibits equivalences with quantum electrodynamics, in particular in terms of the (quantum analogue of) Hall effect [162].

In the absence of background electromagnetic fields, the Dirac particle described by the Hamiltonian (VI.41) possesses the ‘relativistic’ Dirac particle-like dispersion, $\varepsilon_{\boldsymbol{p}} = v_F |\boldsymbol{p}|$,
where $p = (p_1, p_2)$ corresponds to a quasimomentum of the particle with the origin at the tip of a Dirac cone. The particles propagate relativistically with the Fermi velocity, $v_F$, which is typically hundreds of times smaller than the speed of light. The massless nature of the excitations in free-standing graphene is protected by the symmetries of the underlying carbon crystal lattice, thus making the massless nature ($m = 0$) of these charged excitations ($e \neq 0$) a robust feature of the material [163].

Strictly speaking, the quasiparticles in graphene should be distinguished from the Carroll particles because the former do propagate in the absence of background electromagnetic field contrary to the latter. Still, this example – which has no analogues in terms of fundamental particles – gives us some hope to find charged massless (quasi)particles in two spatial dimensions behaving similarly to charged, doubly-extended Carroll particles which, by Theorem VI.6, move, according to the generalized Hall law, only in a combined electric and effective magnetic field.

## E. Carrollian anyons

Carroll particles with anyonic spin, $\chi \neq 0$ in (VI.4), can also be studied along the same lines. The Souriau 2-form $\sigma_{\text{exo}}$ (VI.27a) is generalized consistently with the coupling rule (II.48) but by replacing the e.m. fields by their effective values,

$$ \sigma_{\text{any}} = dp_i \wedge dx_i + \frac{\theta}{2} \epsilon_{ij} dp^i \wedge dp^j + \frac{1}{2} eB^* \epsilon_{ij} dx^i \wedge dx^j + E_i^* dx^i \wedge ds, \qquad (\text{VI.42}) $$

where the electric field and the Stern-Gerlach type gradient of the magnetic field are again combined into an effective electric term$^{46}$,

$$ e\mathbf{E} \rightarrow \mathbf{E}^* = e\mathbf{E} + \mu\chi\nabla B \qquad (\text{VI.43a}) $$

$$ eB \rightarrow B^* = eB + \kappa_{\text{mag}}, \qquad (\text{VI.43b}) $$

The equations of motion take the form (VI.29) with $e\mathbf{E}$ replaced by $\mathbf{E}^*$.

The extension to zero mass is obtained along the same lines as above, yielding,

<footnote>$^{46}$ cf. (III.19) in the Galilean theory. A constant shift $B \rightarrow B + \text{const.}$ produces no effect because of the gradient.</footnote>
**Theorem VI.7.** Massless Carrollian anyons move by following the generalization of the Hall law (1.2) with the electric and magnetic fields replaced by the effective values $\mathbf{E}^*$ and $\mathbf{B}^*$ in (VI.43a),

$$ (x^i)' = \epsilon^{ij} \left( \frac{eE_j + \mu\chi\partial_j B}{eB + \kappa_{\text{mag}}} \right) = \epsilon^{ij} \left( \frac{E_j^*}{B^*} \right) \quad \text{and} \quad p'_i = 0. \qquad (\text{VI.44}) $$

Note here the double role played by the magnetic field $\mathbf{B}$.

The massless limit can also be discussed in the Souriau framework. We start with (VI.42) with $m \neq 0$ and eliminate $\mathbf{p}$ in favor of $\mathbf{v} = \mathbf{p}/m$. Then letting $m \to 0$ we get a regular Souriau form which, compared to (VI.24b), has an additional “effective electric” term,

$$ \sigma_{\text{any}} = \frac{1}{2}\kappa_{\text{exo}} \epsilon_{ij} dv^i \wedge dv^j + \frac{1}{2}B^* \epsilon_{ij} dx^i \wedge dx^j + E_i^* dx^i \wedge ds. \qquad (\text{VI.45}) $$

The kernel of $\sigma_{\text{any}}$ yields the equation of motion (VI.37) [with $p_i$ replaced by $v_i$].

<table><caption>TABLE II: Motion of doubly-extended Carroll particles in $d = 2$ spatial dimensions.</caption><thead><tr><th></th><th>free extended Carroll</th><th>in EM field</th><th>in curved metric</th></tr></thead><tbody><tr><td>$m \neq 0$</td><td>no motion</td><td>generalized Hall</td><td>no motion</td></tr><tr><td>$m = 0, e \neq 0$</td><td>no motion</td><td>generalized Hall</td><td>no motion</td></tr><tr><td>$m = 0, e = 0, \mu\chi \neq 0$</td><td>no motion</td><td>generalized Hall</td><td>no motion</td></tr><tr><td>$m^* = 0$</td><td>Q fixed, x motion undetermined</td><td></td><td></td></tr></tbody></table>

The generalized Hall law (VI.44) also contains an “anomalous” anyonic contribution activated by the inhomogeneities of the magnetic field (VI.43a). For massive particles, the anyonic term can be interpreted as a conventional Zeeman force

$$ \mathbf{F} = g\mu_B s \nabla \mathbf{B}, \qquad (\text{VI.46}) $$

exerted by the spatial gradient of the transverse magnetic field $B \equiv B_z$ on the particle propagating in the plane. Here $g$ is the Landé factor, and $\mu_B$ is the Bohr magneton. A comparison of Eqs. (VI.43a) and (VI.46) provides us with the identification $e\mu\chi \equiv g\mu_B s$.

In spintronic devices, the electrons with different orientations of their spins $s \equiv s_z$ are split by the Zeeman force (VI.46). This phenomenon lies at the heart of a mesoscopic Stern-Gerlach effect which provides an efficient spin filter for a current traversing a specially fabricated device that incorporates an inhomogeneous magnetic field $B = B_z$ [164].

Thus, two-dimensional Carrollian anyons can be interpreted as spin-polarized charged particles with a nonzero Landé factor.
## F. A toy model

An “exotic photon” is defined by being a massless, uncharged particle, carrying magnetic moment, anyonic spin <fnref n="47" />, and having non-vanishing “exotic charges”. It has a pure Stern-Gerlach-type Carroll Hamiltonian of a spinning anyon with no kinetic term,

$$ \mathcal{H}_{SG} = -\mu\chi B. \qquad (VI.47) $$

cf. (III.17). Although this particle carries a “bosonic” name, it can be of a fermionic or even anyonic nature as well since $\chi$ can be any real number.

A distant physical analogue of the exotic photon in relativistic fundamental field theory can be played by Dirac or Majorana neutrinos which, being neutral and almost massless, could possess a sizable magnetic moment in certain exotic scenarios [165, 166]. A non-vanishing neutrino magnetic moment would have important astrophysical and cosmological consequences [167].

In the condensed matter setup, neutral fermionic excitations appear in strongly correlated electronic systems in Kondo lattice materials such as, for example, the newly found compound YbIr₃Si₇. Neutral fermions cannot, evidently, carry electric charge while they can support thermal energy transfer endowing the charge insulator with metallic thermal conducting properties. This type of exotic materials is sensitive to the background magnetic field implying that neutral fermions couple to the magnetic degrees of freedom [168]. The corresponding electrically-neutral massless excitations are remote condensed-matter analogues of the exotic photons proposed in Ref. [35].

Now we illustrate this theory by choosing a massless uncharged particle with magnetic moment and anyonic spin with exotic charges,

$$ m = 0, \quad e = 0, \quad \mu\chi = 1, \quad \kappa_{\text{mag}} = 1, \quad \kappa_{\text{exo}} \neq 0, \qquad (\text{VI.48}) $$

put into the electromagnetic field

$$ \mathbf{E}(\mathbf{x}) = f(|\mathbf{x}|) \hat{\mathbf{x}} \quad \text{and} \quad B(\mathbf{x}) = \frac{|\mathbf{x}|}{(1 + |\mathbf{x}|^2)^3}, \qquad (\text{VI.49}) $$

where $f$ is some function of the radius. The electromagnetic fields are switched off by $e = 0$ but there remains an effective Stern-Gerlach-type electric field $\mathbf{E}^* = \mu\chi\nabla B$ coming from

<footnote n="47">⁴⁷ In addition to photons and neutrinos in particle physics, uncharged and massless quasiparticles also appear in the solid state context as phonon excitations in crystals and fluids.</footnote>
(VI.47) [cf. (III.17)], with $\mu\chi$ behaving as an effective charge and $B$ as an effective potential. $\kappa_{\text{exo}}$ (assumed not to vanish) drops out and the general formula (VI.44) reduces to,

$$ (x^i)' = (\mu\chi) \epsilon^{ij} \frac{\partial_j B}{\kappa_{\text{mag}}} \quad \text{and} \quad p_i' = 0, \qquad (\text{VI.50}) $$

which is indeed the formula we used on the horizon of a black hole [35].

The various fields and the Hall motion (VI.50) are shown in 5a-b. Fig.6 prefigures Fig.13.

<figure><img src="image_5.png" /><figcaption>FIG. 5: (a) The planar magnetic field $B(|\mathbf{x}|)$ in (VI.49) (b) induces a Stern-Gerlach type radial effective electric field $\mathbf{E}^* = \nabla B$.</figcaption></figure>

An intuitive picture can be obtained by observing that the motion of a Carroll anyon in the effective electric field $\mathbf{E}^*$ can be pictured also as motion unfolded to three dimensions and lifted to “Mexican hat” potential surface $-B(|\mathbf{x}|)$ above the plane, as shown in Fig. 7.

For the sake of comparison, we consider also a charged massless particle with no spin,

$$ m = 0, \quad e = 1, \quad \chi = 0, \quad \kappa_{\text{mag}} = 0, \quad \kappa_{\text{exo}} \neq 0, \qquad (\text{VI.51}) $$

where we put $\kappa_{\text{mag}}$ to zero with no loss of generality (VI.43b). Such quasi-particles might be relevant in Condensed Matter physics [36].

Putting both particles (VI.48) and (VI.51) into the effective field considered in the uncharged case, (VI.48),

$$ \mathbf{E}(|\mathbf{x}|) = \nabla B = \frac{1 - 5|\mathbf{x}|^2}{(1 + |\mathbf{x}|^2)^4}, \quad \hat{\mathbf{x}} \qquad (\text{VI.52}) $$

the general formula (VI.44) reduces to$^{48}$,

$$ (x^i)' = \epsilon^{ij} \frac{E_j}{B}, \qquad (\text{VI.53}) $$

<footnote>$^{48}$ For our choice (VI.52) the r.h.s. of (VI.53) is $\epsilon^{ij} \partial_j \ln B$.</footnote>
<figure><img src="image_6.png" /><figcaption>FIG. 6: For uncharged particles the radial planar magnetic field B in (VI.49) induces a radial effective electric field E*. An “exotic photon” i.e. a mass and chargeless planar Carroll anyon with nonzero spin $\chi$, magnetic moment $\mu$ and magnetic charge $\kappa_{mag}$, (VI.48), rotates around the origin along concentric circles, consistently with the generalized Hall Law, (VI.44). The rotation stops on a ring (marked by full dots in red) where the effective electric field vanishes, $E^* = 0$. The direction of the rotation is reversed after $E^*$ changes sign.</figcaption></figure>

which means usual Hall motion (I.2). The only difference with (VI.50) is that B in the denominator is now position-dependent instead of being a constant.

We conclude that an uncharged spinning anyon with magnetic moment moves essentially as a charged scalar particle. What is surprising is that while the first case (VI.48) involves a Stern-Gerlach dynamics, the second (VI.51) is of the spin-Hall type.

$$e \sim \mu\chi \quad \text{and} \quad \mathbf{E} \sim \nabla B. \qquad (\text{VI.54})$$

Plotting (VI.53) yields a figure similar to fig.6 and is not reproduced here therefore.

The magnetic and electric fields could plainly be chosen also independently, see (VI.26). Pairing, for example, a constant electric field $\mathbf{E} = \mathbf{E}_0 = \text{const.}$, e.g., $\mathbf{E}_0 = (E_0, 0)$,

$$\mathbf{E}^* = \mathbf{E}_0 + \mu\chi \frac{1 - 5|\mathbf{x}|^2}{(1 + |\mathbf{x}|^2)^4} \hat{\mathbf{x}} \qquad (\text{VI.55})$$

which with $B = B(|\mathbf{x}|)$ in (VI.49) yields Fig. 8.
<figure><img src="image_7.png" /><figcaption>FIG. 7: The motion of an anyon in the effective electric field $\mathbf{E}^*$ can be lifted to “Mexican Hat” potential surface $-B(|\mathbf{x}|)$ above the plane. The trajectories are horizontal “cords” (circles) starting near the “top” with high velocities and, then decreasing progressively until we arrive at $|\mathbf{x}| = |\mathbf{x}_0|$ where the force vanishes, $\mathbf{E}^*(|\mathbf{x}_0|) = 0$, and the rotation stops. The “cord” degenerates to fixed points around the “bottom rim”. Then, as we climb up the outer “flange” by letting $|\mathbf{x}|$ increase, the anyon restarts to move with ever increasing velocity — however in the opposite direction. The 3D picture projects to Fig. 6.</figcaption></figure>

## VII. CONTRACTION FROM RELATIVISTIC ANYONS

An insight into the physical origin of exotic Galilean symmetry has been provided by its derivation from relativistic anyons [120] by a tricky contraction [121, 122]. Now we study the Carrollian counterpart [2]: one defines the timelike coordinate $x^0 = s/c$ and then sends $c \to \infty$. We start with a massive and spinless particle, described by the Cartan 1-form [77, 122],

$$ \varpi = p_{\mu}dx^{\mu} = p_{i}dx^{i} - m\sqrt{1 + \frac{p^2}{m^2c^2}} ds. \qquad (VII.1) $$

Taking the limit $c \to \infty$, we readily recover the Cartan 1-form (IV.3) for Carroll,

$$ \varpi_0 = p_i dx^i - mds. \qquad (VII.2) $$

Notice that the Carroll limit is in fact more straightforward than the Galilean one, for which the limit $c \to \infty$ of $mc^2\sqrt{1 + \frac{p^2}{m^2c^2}} dt$ requires regularization, achieved in [122] by adding a central extension i.e., working with the Bargmann model.

Now consider a massive relativistic model with spin. Generalizing the Jackiw-Nair ansatz
<figure><img src="image_8.png" /><figcaption>FIG. 8: Combining a constant external electric field $\mathbf{E} = \mathbf{E}_0$ with $\nabla \mathbf{B}$ induced by the spin-field term (III.17) yields effective electric field $\mathbf{E}^*$ (VI.43a). Velocity and force are perpendicular, as required by the Hall law. The number of equilibrium points has been considerably reduced.</figcaption></figure>

$s = \kappa c^2$ [121], we set $s = \chi + \kappa c^2$, whose Cartan form is (VII.2) plus a term describing the spin state of the particle, $\varpi = \varpi_0 + \varpi_s$. The rather complicated term $\varpi_s$ can be read off from # (4.1) in [122]. The result is seen not depend on the time coordinate. The Carrollian limit of $\varpi_s$ is therefore the same as in the Galilean case, and we end up with the Cartan form,

$$ \varpi_{any} \equiv \varpi = p_i dx^i - mds + \kappa c^2 d\varphi + \chi d\varphi + \frac{\theta}{2} \epsilon^{ij} p_i dp_j. \qquad (VII.3) $$

The term $\kappa c^2 d\varphi$ here diverges when $c \to \infty$, however its exterior derivative drops out, yielding the exotic part of the symplectic form (VI.11) with $\kappa_{exo} = \kappa$,

$$ \Omega_{exo} = dp_i \wedge dx^i + \frac{\kappa}{2m^2} \epsilon^{ij} dp_i \wedge dp_j. \qquad (VII.4) $$

Thus the exotic Carroll charge $\kappa_{exo}$ is deduced from the relativistic anyon model, just as in the Galilean case. However we do not recover (in this model) the second charge, $\kappa_{mag}$ by contraction.

Some more insight can be gained by regularizing $\varpi$. We follow [122], whose idea is to augment the Poincaré group by adding a trivial parameter, which naturally splits into two
contributions when computing the limit $c \to \infty$: one proportional to $c^2$ and thus diverging, which will exactly cancel out the diverging term of $\varpi$, and the next order term, in $c^0$, which will not diverge, and becomes a new physical parameter in the resulting model. Consider a trivial $\mathbb{R}$-central extension to the Poincaré group. Its Cartan 1-form will be $\tilde{\varpi} = \varpi + \alpha d\theta$. Now, we choose $\alpha = -\kappa c^2$$^{49}$ and $\theta = \phi - w/c^2$ for some $w \in \mathbb{R}$. This implies that $\alpha d\theta = -\kappa c^2 d\phi + \kappa dw$, and thus,

$$ \tilde{\varpi} = p_i dx^i - mds + \chi d\phi + \frac{\kappa}{2m^2} \epsilon^{ij} p_i dp_j + \kappa dw. \qquad (\text{VII.5}) $$

This expression is to be compared to $\varpi_{\text{exo}}$ in (VI.9). The presymplectic structure has gained one dimension, spanned by the parameter $w$. Such a situation arises in the “non-exotic” Galilean case, where the infinite energy is regularized similarly [78, 122].

In conclusion, the exotic term is recovered (up to a total derivative), but the magnetic term is missing, as it does also from the presymplectic 2-form $d\varpi$, cf. (VI.9).

## VIII. MOTION IN A PLANE GRAVITATIONAL WAVE

Our first application to general relativity is to motion in an exact plane gravitational wave [23–25]. In particular, we want to clarify the superficial contradiction between the claim that “particles with Carroll symmetry do not move” and those complicated motions found in a plane gravitational wave [24–27], — which happens to carry a “sort of” Carroll symmetry. The clue is precisely: “sort of”, as we now explain. We start with an exact gravitational wave written in Baldwin-Jeffery-Rosen (BJR) [23, 169] coordinates $(x, t, s)$, in terms of which the metric is

$$ ds^2 = a_{ij}(t)dx^i dx^j + 2dtds, \qquad (\text{VIII.1}) $$

where $a(t) = (a_{ij}(t))$ is a symmetric positive $2 \times 2$ matrix, which depends only on the light-cone coordinate $t$ [23, 25–27]. Particular interest will be devoted to the $2+1$ dimensional null submanifolds obtained by fixing the light-cone coordinate $t$,

$$ C_{t_0} = \{ (x, s), t = t_0 = \text{const.} \}. \qquad (\text{VIII.2}) $$

<footnote>$^{49}$ $\alpha d\theta$ is exact when $\alpha$ is either a constant or depends only on $\theta$.</footnote>
Restriction to $C_{t_0}$ yields the metric $ds^2 = a_{ij}(t_0)dx^i dx^j$ such that $\xi = \partial_s$ is in its kernel. Then the triple $(C_{t_0}, a_{ij}(t_0), \xi)$ provides us, for each fixed value of $t_0$, with a 2+1 dimensional Carroll manifold we will call a *Carroll slice* embedded into 4D gravitational wave spacetime [25, 92].

The isometry group of a gravitational wave has long been known to have (generically) 5-parameters [170], implemented by shifting $x$ and $s$ but leaving $t$ (and thus $C_{t_0}$) invariant [23, 25],

$$x \rightarrow x + H(t) \mathbf{b}_1 + \mathbf{c}, \qquad (\text{VIII.3a})$$

$$s \rightarrow s - \mathbf{b}_1 \cdot x - \frac{1}{2} \mathbf{b}_1 \cdot H(t) \mathbf{b}_1 + b_0, \qquad (\text{VIII.3b})$$

where the planar vectors $\mathbf{b}_1$ and $\mathbf{c}$, and the real number $b_0$ are constants [23, 25]. The key ingredient here is the $2 \times 2$ Souriau matrix $H(t)$, obtained by integrating the inverse of the BJR-profile-matrix $a = (a_{ij})^{50}$,

$$H(t) = \int_{0}^{t} a(u)^{-1} du. \qquad (\text{VIII.4})$$

For $t_0 = 0$ the Souriau matrix is zero, and (for $\mathbf{c} = b_0 = 0$) the action (VIII.3) reduces, for arbitrary BJR profile $(a_{ij})$, to the Carroll boost (I.1): the transverse coordinate $x$ is left invariant and it is only Carrollian time, $s$, which is boosted. For $t_0 \neq 0$ the implementation is determined by the Souriau matrix $H(t_0)$ (VIII.4), cf. (VIII.3).

The restriction of the non-transitive 5 parameter isometry group of the gravitational wave to each fixed Carroll manifold $C_{t_0}$ acts upon the latter transitively in a $t_0$-dependent way, as shown in fig.9$^{51}$. Conversely, the plane gravitational waves carries a Carroll-type symmetry obtained by “gluing together” all those $t = \text{const.}$ implementations (but dropping rotations).

Now we turn to the geodesics. If $\lambda$ is any affine parameter then,

$$\epsilon = \frac{1}{2} g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu}, \qquad (\text{VIII.5})$$

where $\{\cdot\} = d/d\lambda$, is a constant of the motion along any geodesic, is interpreted as the *relativistic mass square*. It is negative/zero/positive for timelike/null/spacelike geodesics.

<footnote>$^{50}$ (VIII.4) obviously assumes that $a$ is a regular matrix. However this remains true only in a finite time interval [23]: $\det(a)$ necessarily vanishes after a finite $t$-lapse [23, 25]. Thus the BJR description is only local; our investigations are limited to a finite time interval.</footnote>

<footnote>$^{51}$ Our illustration uses the the BJR profile $a = \cos^2 t \operatorname{diag}(e^{2t}, e^{-2t})$ studied in [26].</footnote>
<figure><img src="image_9.png" /><figcaption>FIG. 9: The (broken) Carroll group (VIII.3) leaves each “Carroll slice” $C_t$ invariant. The implementation is obtained from the broken Carroll one on $C_0$ through the Souriau matrix $H(t)$ (VIII.4) as indicated in **blue**. The 5 parameter isometry group of the gravitational wave acts (indicated in **orange**) transitively on each slice $C_t$, but not on the full spacetime.</figcaption></figure>

The rather complicated-looking general geodesic equations are,

$$ \frac{dx^2}{d\lambda^2} + \left( \frac{a'_{11}a_{22} - a_{12}a'_{12}}{\det(a)} \right) \frac{dx}{d\lambda} \frac{dt}{d\lambda} + \left( \frac{a'_{12}a_{22} - a_{12}a'_{22}}{\det(a)} \right) \frac{dy}{d\lambda} \frac{dt}{d\lambda} = 0, \qquad (\text{VIII.6a}) $$

$$ \frac{dy^2}{d\lambda^2} - \left( \frac{a'_{11}a_{12} - a_{11}a'_{12}}{\det(a)} \right) \frac{dx}{d\lambda} \frac{dt}{d\lambda} - \left( \frac{a'_{12}a_{12} - a_{12}a'_{22}}{\det(a)} \right) \frac{dy}{d\lambda} \frac{dt}{d\lambda} = 0, \qquad (\text{VIII.6b}) $$

$$ \frac{d^2 t}{d\lambda^2} = 0, \qquad (\text{VIII.6c}) $$

$$ \frac{d^2 s}{d\lambda^2} - \frac{1}{2} a'_{11} \left( \frac{dx}{d\lambda} \right)^2 - \frac{1}{2} a'_{22} \left( \frac{dy}{d\lambda} \right)^2 - a'_{12} \frac{dx}{d\lambda} \frac{dy}{d\lambda} = 0. \qquad (\text{VIII.6d}) $$

where $\det(a) = a_{11}a_{22} - a_{12}^2$ and the prime means $d/dt$. For a general profile $a_{ij}(t)$ these equations can only be solved numerically. The solution (for the same BJR profile $a = \cos^2 t \operatorname{diag}(e^{2t}, e^{-2t})$ as in fig.9) is depicted in fig. 10a. Analytic solutions arise in particular cases. When $t \neq \text{const.}$ then it can be chosen itself as affine parameter, $\lambda = t$. For arbitrary profile we get, trivially [26]:

**Theorem VIII.1.** For vanishing initial transverse velocity,

$$ \boldsymbol{x}'(t_0) = 0 \Rightarrow \boldsymbol{x}(t) \equiv \boldsymbol{x}_0 = \text{const.} \qquad (\text{VIII.7}) $$
<figure><img src="image_10.png" /><figcaption>FIG. 10: (a) For non-vanishing initial transverse momentum, $p \neq 0$, a massive particle moves along a timelike geodesic. (b) For $p = 0$ instead, the geodesics move, for any profile, only in time, $t$, and have fixed position, $x = x_0$.</figcaption></figure>

the transverse position $x$ does not move (while $s$ is a linear function of $t$), shown in fig. 10b.

1. Timelike [$t \neq \text{const.}$] solutions could be found also by using the symmetries [26]: the Noether theorem provides us with 5 conserved quantities,

$$ \pi = a x', \quad D = x - H(t) \pi, \quad m = 1, \qquad (\text{VIII.8}) $$

interpreted as linear momentum, boost-momentum; $m = 1$ in our parametrization$^{52}$. Conversely, our timelike geodesic is determined by these conserved quantities,

$$ x(t) = H(t) \pi + D, \quad s(t) = -\frac{1}{2} \pi \cdot H(t) \pi + et + d, \qquad (\text{VIII.9}) $$

where $d$ is an integration constant, as is confirmed by a straightforward calculation. This reduces the problem to calculating the Souriau matrix $H(t)$ in (VIII.4), which can be achieved by numerical integration, instead of solving the coupled equations (VIII.6).

<footnote>$^{52}$ The 4d gravitational wave as the Bargmann manifold of a 2 + 1 dimensional non-relativistic system with $m$ the “non-relativistic mass”, which is a priori independent of the relativistic mass $\epsilon$ in (VIII.5).</footnote>
The trivial “no-motion” solution (VIII.7) is recovered from (VIII.9) since the chosen initial condition $x' = 0$ in (VIII.7) implies (Theorem (VIII.1)), by (VIII.8), that the conserved transverse momentum vanishes$^{53}$,

$$ \pi = 0. \qquad (VIII.10) $$

Thus the boost momentum (VIII.8) becomes, for any profile $a_{ij}$, simply

$$ D = x \qquad (VIII.11) $$

cf. (I.5) with $m = 1$, whose conservation implies, once again, *immobility*, as explained in Corollary IV.1.

2. Let us now turn to a $t = t_0 = \text{const.}$ Carroll slice. $C_{t_0}$ is endowed with the singular metric $\tilde{g} = a_{ij}(t_0)dx^i dx^j$ whose kernel is generated by $\partial_s$, conforming that it is indeed a Carroll manifold. Carroll boosts, (I.1), act on it as (trivial) isometries. The choice $t_0 = 0$ can and will henceforth be assumed with no loss of generality, and the shorthand $C \equiv C_{t_0}$ will be used.

“Vertical” geodesics i.e. s.t. are constrained to the Carroll slice $C$ can also be considered. Our first and important observation is that (discarding tachyons for which $\epsilon > 0$) they are necessarily *lightlike*. The clue is that fixing $t$ the relativistic mass (VIII.5) reduces to

$$ \epsilon = a_{ij}(0) \dot{x}^i \dot{x}^j \ge 0 \qquad (VIII.12) $$

which, $(a_{ij}(0))$ being a positive matrix, *rules out a mass* for which $\epsilon < 0$. Thus we have

$$ \epsilon = \frac{1}{2} g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} = a_{ij} \dot{x}^{i} \dot{x}^{j} = 0 \qquad (VIII.13) $$

for any geodesic on $C$. Moreover, the transverse velocity vanishes,

$$ \dot{x}^i = 0 \quad \Rightarrow \quad x = x_0 = \text{const.} \qquad (VIII.14) $$

cf. (VIII.7). Then putting $dt/d\lambda = 0$ into (VIII.6) reduces the latter to

$$ \frac{d^2 x}{d\lambda^2} = 0, \qquad (VIII.15a) $$

$$ \frac{d^2 s}{d\lambda^2} - \frac{1}{2} a'_{11} \left(\frac{dx}{d\lambda}\right)^2 - \frac{1}{2} a'_{22} \left(\frac{dy}{d\lambda}\right)^2 - a'_{12} \frac{dx}{d\lambda} \frac{dy}{d\lambda} = 0. \qquad (VIII.15b) $$

<footnote>$^{53}$ All those complicated-looking trajectories plotted in [26, 27] come from the simple “no-motion” solution (VIII.14) above by switching from BJR to Brinkmann coordinates [171, 172].</footnote>
Eqn (VIII.15a) is automatically satisfied by (VIII.14) leaving us with,

$$s = s_0 + c\lambda + d, \qquad (VIII.16)$$

where $s_0$, $c$ and $d$ are real constants.

One can wonder what happens to a massive particle put on $C$? The answer comes from (VIII.12): having non-zero mass is *inconsistent* with staying on $C$: the particle will be expelled, as shown in fig.10a. In conclusion,

**Theorem VIII.2.** A particle can stay on a Carroll slice $C$ only if it is massless. Then its transverse position $x$ is fixed; its trajectory is a “vertical” straight line directed along the “Carroll time axis”, $s$.

In the massive case, immobility could also be derived from the conservation of the Carroll moment $D = mx$ (IV.8c), eqn. (VIII.11) for gravitational waves. However, massless Carroll particles have *vanishing* C-boost momentum, $D = 0$ in (IV.38), from which *no conclusion can be drawn*. As we have just seen, the “no motion” paradigm still holds on Carroll slices for a massless particle. Another example is provided by motion on the horizon of a black hole, to be presented in sec. IX.

# IX. HALL ON THE HOLE: MOTION ON THE HORIZON

Another physical application of (possibly doubly-extended) Carroll dynamics is to motion on the *horizon of a black hole* [18–22]. We first recall the following general properties:

* A black hole is characterized by its mass $M$, angular momentum $J$ and charge $Q$:

* The horizon $H$ is a Carroll manifold [18–20];

* The horizon is a null hypersurface, therefore the geodesics constrained to it are necessarily null [108],

$$\epsilon = \frac{1}{2}g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} = 0, \qquad (IX.1)$$

see chapters 34.3 and 34.4 in [173].

* The geodesic dynamics can be extended by adding both exotic & spin-field (commonly called spin-orbit) terms discussed in sec.VI.
This section is devoted to the study of an exotic photon (VI.48) whose key dynamical ingredients are, by (VI.45),

$$ \varpi_{spec} = \frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} v^i dv^j + \frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} x^i dx^j + \mu \chi B dx^i ds, \qquad (\text{IX.2a}) $$

$$ \sigma_{spec} = \frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} dv^i \wedge dv^j + \frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} dx^i \wedge dx^j + \mu \chi \partial_i B dx^i \wedge ds, \qquad (\text{IX.2b}) $$

$$ \mathcal{L}_{spec} = \frac{1}{2} \kappa_{\text{exo}} \epsilon_{ij} v^i \frac{dv^j}{ds} + \frac{1}{2} \kappa_{\text{mag}} \epsilon_{ij} x^i \frac{dx^j}{ds} + \mu \chi B. \qquad (\text{IX.2c}) $$

## A. Schwarzschild Horizons

We first study the simplest black hole, namely the static and spherically symmetric Schwarzschild metric, described by using (ingoing) Eddington-Finkelstein coordinates {$v, r, \vartheta, \varphi$} and natural units with the line element

$$ g \equiv g_{\mu\nu} dx^\mu dx^\nu = -\left(1 - \frac{2M}{r}\right) dv^2 - 2dvd\tau + r^2(d\vartheta^2 + \sin^2\vartheta d\varphi^2), \qquad (\text{IX.3}) $$

where $M$ is the mass of the Schwarzschild black hole. The Schwarzschild horizon, $\mathcal{H}$, is the hypersurface $r = 2M$, i.e.,

$$ \mathcal{H} = \{r = 2M, \vartheta, \varphi, v\} \cong \mathbb{S}^2 \times \mathbb{R}. \qquad (\text{IX.4}) $$

The $dv^2$ component of the metric (IX.3) induced on $\mathcal{H}$ vanishes, whereas the radial component disappears on the fixed-$r$ hypersurface $\mathcal{H}$, leaving us with the clearly degenerate “metric”,

$$ \tilde{g} = g|_{r=2M} = 0 \cdot dv^2 + 4M^2(d\vartheta^2 + \sin^2\vartheta d\varphi^2). \qquad (\text{IX.5}) $$

On the other hand,

$$ \tilde{\xi} = \partial_v \equiv \partial_s \qquad (\text{IX.6}) $$

is a nowhere vanishing vector which belongs to the kernel of $\tilde{g}$ where, to anticipate the role of Carrollian time that it will play, we renamed $v$ to $s$. We conclude that the Schwarzschild horizon ($\mathbb{S}^2 \times \mathbb{R}, \tilde{g}, \tilde{\xi}$) has a Carroll structure [19]$^{54}$, as defined in sec. IV.

The vector $\tilde{\xi}$ (called the generator of the horizon) is also a Killing vector field for the Carroll metric in that $L_{\tilde{\xi}} \tilde{g} = 0$. It is in fact a Killing vector field of the original Schwarzschild

<footnote>$^{54}$ Any null hypersurface in a Lorentzian spacetime is Carroll [16, 17, 92, 93].</footnote>
spacetime, $L_{\tilde{\xi}}g = 0$, which becomes a null vector on the horizon,

$$g(\tilde{\xi}, \tilde{\xi})|_{r=2M} = 0. \qquad (\text{IX.7})$$

The Schwarzschild horizon provides us with an insight into the peculiarities of Carrollian dynamics. One of them is that the horizon of (stationary) black holes can “trap” photons i.e., a photon may remain on the horizon forever. As a stationary horizon is always a null hypersurface, and since photons travel in null directions, a photon could move along the null direction defined by the (R component of the) horizon, thus staying on it forever. In the Schwarzschild case for instance, this amounts to emit a photon right on the horizon and radially outward, see Fig. 11. For details, see e.g. [173, §33.6].

<figure><img src="image_11.png" /><figcaption>FIG. 11: Eddington-Finkelstein diagram [173] of a Schwarzschild spacetime, with outgoing light rays $u = \text{const.}$ in green, highlighting key features such as the singularity, the horizon, and different trajectories of outgoing photons. Infinitely far away from the black hole, space is Minkowskian. However, the closer we get, the more the gravitational field bends the light cones inward, resulting in outgoing photons taking an ever longer time to escape from the black hole region. Past the horizon, the light cone is completely bent inward such that the only future-pointing directions are toward the singularity. An outgoing photon emitted right on the horizon would have a vertical trajectory, thus staying on the horizon.</figcaption></figure>

“Photon trapping” is a nice thought experiment to understand Carroll geometries. Consider a photon on a Schwarzschild horizon. We can look at its motion (or lack of motion)
intrinsically, on the horizon itself. On this surface, the photon is created at some point $(\boldsymbol{x}_0, s_0)$ in (Carroll) time (represented by the $\mathbb{R}$-axis generated by the null vector $\boldsymbol{\xi} = \partial_s$, and $\boldsymbol{x}_0 \in \mathbb{S}^2$). In order to stay on the horizon, the photon must have its momentum directed along the generator $\boldsymbol{\xi}$ of the horizon. Its momentum is then zero in the angular directions, and therefore the photon “stays in place” on $\mathbb{S}^2$. Its “trajectory” is precisely the Carrollian “no-motion”, $\boldsymbol{x}(s) = \boldsymbol{x}_0$.

The situation is illustrated by the Eddington-Finkelstein diagram on Fig. 11, cf. [173].

While this intuitive picture on the Schwarzschild horizon seems to imply no motion on this instance of a Carroll structure, the horizon is 2 + 1 dimensional and thus might accomodate potentially non-trivial dynamics associated with the double central extension, discussed in sec. VI. However, it has been shown in [34] and recalled in section VIC that the two charges $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$ only couple to the electromagnetic field but not the gravitational field. But the Schwarzschild spacetime carries no electromagnetic field. Hence,

**Theorem IX.1.** An uncharged massless (exotic or not) particle with purely outgoing momentum on the horizon of a Schwarzschild black hole stays fixed, but does not move.

This conclusion is valid also for spin-field interaction (VI.37) since $B = 0$.

## B. Motion on the Kerr-Newman horizon

Eqn. (VI.44) suggests that non-trivial motion may be possible on the horizon of a black hole for a massless particle with magnetic moment $\mu \neq 0$, non-zero anyonic spin $\chi \neq 0$ and non-zero magnetic charge $\kappa_{\text{mag}} \neq 0$, when the black hole has a non-uniform magnetic field $B \neq 0$ on the horizon.

Below we show that these conditions are met for a *Kerr-Newman black hole*, described by its mass $M$, angular momentum $J$, and charge $Q$. In terms of Eddington-like coordinates
$(v, r, \vartheta, \varphi)$ [174], a Kerr-Newman metric can be written in the form$^{55}$,

$$g = -\frac{\Delta}{\Sigma} \left( -dv + \frac{\Sigma}{\Delta} dr + a \sin^2 \vartheta d\varphi \right)^2 + \frac{\sin^2 \vartheta}{\Sigma} \left( adv - (r^2 + a^2)d\varphi \right)^2 + \Sigma d\vartheta^2 + \frac{\Sigma}{\Delta} dr^2 \qquad (\text{IX.8a})$$

$$\Sigma = r^2 + a^2 \cos^2 \vartheta, \quad \Delta = r^2 + a^2 + Q^2 - 2Mr, \qquad (\text{IX.8b})$$

where $a = J/M$. The (outer) horizon of a Kerr-Newman black hole is the $a = r_+ = \text{const.}$ hypersurface $\mathcal{H}$ defined by requiring$^{56}$

$$\Delta = 0 \quad \text{i.e.} \quad r = M + \sqrt{M^2 - (a^2 + Q^2)} = r_+. \qquad (\text{IX.9})$$

Then we consider again the 2 + 1 dimensional structure [19, 20] whose key ingredients are the induced metric and a vector,

$$\tilde{g} = g|_{\Delta=0} = \frac{\sin^2 \vartheta}{\Sigma} \left( a dv - (r_+^2 + a^2) d\varphi \right)^2 + \Sigma d\vartheta^2, \qquad (\text{IX.10a})$$

$$\xi = \partial_v + \Omega_H \partial_\varphi \quad \text{where} \quad \Omega_H = \frac{a}{r_+^2 + a^2}. \qquad (\text{IX.10b})$$

Here $\Omega_H$ is the angular velocity of the horizon.

The metric (IX.10a) is again singular, as seen from shifting the coordinates as $(\vartheta, \varphi, v) \mapsto (\vartheta, \tilde{\varphi} = \varphi - \Omega_H v, v)$,

$$\tilde{g} = \frac{(r_+^2 + a^2)^2 \sin^2 \vartheta}{\Sigma} d\tilde{\varphi}^2 + \Sigma d\vartheta^2 \quad \& \quad \xi = \partial_v. \qquad (\text{IX.11})$$

The kernel is generated by the vector $\xi$ which takes now the familiar form, $\tilde{g}(\xi) = 0$.

Thus we have a degenerate metric and a vector field in its kernel allowing us to conclude that the horizon of a Kerr-Newman black hole carries indeed a Carroll structure $(\mathbb{S}^2 \times \mathbb{R}, \tilde{g}, \tilde{\xi})$. Viewed in the original spacetime, $\xi$ is again null on the horizon and generates the $(\mathbb{R}$ component of the) horizon.

Now if we were to study the trapping of an “ordinary” photon (meaning one without extended charge) on the horizon $\mathcal{H}$ of a Kerr-Newman black hole as we did for Schwarzschild, we would arrive again at the same negative conclusion: we would find no motion. However the Carroll manifold $\mathcal{H}$ has 2+1 dimensions and therefore can accommodate the 2-parameter

<footnote>$^{55}$ The metric and its inverse are regular at $\Delta = 0$, as the $dr^2$ terms cancel each other out.</footnote>

<footnote>$^{56}$ In fact, there are two hypersurfaces which satisfying $\Delta = 0$. We are only interested in the behavior at the outer radius $r_+ .</footnote>
extension yielding an “exotic photon” (VI.48), which may again be coupled to the electro-magnetic field and which can move, as we will see it below.

In Eddington-like coordinates $(v, r, \vartheta, \varphi)$ [174]$^{57}$ the electromagnetic tensor in the Kerr-Newman spacetime is given by,

$$F = \frac{2aQr \sin\vartheta \cos\vartheta}{(r^2 + a^2 \cos^2\vartheta)^2} (adv - (r^2 + a^2)d\varphi) \wedge d\vartheta \\ + \frac{Q(r^2 - a^2 \cos^2\vartheta)}{(r_+^2 + a^2 \cos^2\vartheta)^2} (a \sin^2\vartheta d\varphi - dv) \wedge dr. \qquad (\text{IX.12})$$

The magnetic field is conveniently derived by using the Hodge operator which interchanges the electric and magnetic fields, $\star F = B_i dx^i \wedge dv + \dots$. We then identify the magnetic field induced on the horizon ($r = r_+$) as $B = B^r|_{\Delta=0} = (\star F)^{rv}$, which yields, in our case,

$$B = \frac{F_{\theta\varphi}}{\sqrt{|\det g|}} \Big|_{\Delta=0} = (2aQr_+(r_+^2 + a^2)) \frac{\cos\vartheta}{(r_+^2 + a^2 \cos^2\vartheta)^3}. \qquad (\text{IX.13})$$

We note for later reference that the magnetic field is a function of $\vartheta$ only, $B = B(\vartheta)$, as it follows from the axial symmetry of the Kerr-Newman spacetime. Non-vanishing $B$ is obtained for non-zero angular momentum $J = aM$ and charge $Q$. The electromagnetic field induced on the horizon,

$$\tilde{F} = F|_{\Delta=0} = \frac{aQr_+ \sin 2\vartheta}{(r_+^2 + a^2 \cos^2 \vartheta)^2} (adv \wedge d\vartheta + (r_+^2 + a^2) d\vartheta \wedge d\varphi), \qquad (\text{IX.14})$$

seems to have a non-vanishing electric field component,

$$E_{\vartheta} = F_{u\vartheta} = (a^2 Q r_+) \frac{\sin 2\vartheta}{(r_+^2 + a^2 \cos^2 \vartheta)^2}. \qquad (\text{IX.15})$$

However switching to co-moving coordinates, (IX.11), the coordinate transformation $\varphi \to \tilde{\varphi} = \varphi - \Omega_H v$ with $\Omega_H$ given in (IX.10b),

$$\tilde{F} \to \hat{F} = (aQr_+(r_+^2 + a^2)) \frac{\sin 2\vartheta}{(r_+^2 + a^2 \cos^2 \vartheta)^2} d\vartheta \wedge d\tilde{\varphi}, \qquad (\text{IX.16})$$

eliminates the electric field (IX.15): $E = 0$ on the horizon and we are left with the pure magnetic field (IX.13), which is indeed left invariant under this coordinate transformation.

<footnote>$^{57}$ $0 \le \vartheta \le \pi$ is oriented with $\vartheta = 0$ at the north pole.</footnote>
Dynamics on the Kerr-Newman Black Hole horizon

Turning to the dynamics of an “exotic photon” (VI.48) we recall that it has no electric charge, and therefore the terms $eE$ and $eB$ are switched off, leaving us with the effective fields,

$$B^* = \kappa_{\text{mag}}, \qquad (\text{IX}.17\text{a})$$

$$E_{\vartheta}^* = \mu\chi \partial_{\vartheta} B = -\mu\chi (2aQr_+(r_+^2 + a^2)) \frac{(r_+^2 - 5a^2 \cos^2 \vartheta)}{(r_+^2 + a^2 \cos^2 \vartheta)^4} \sin \vartheta, \qquad (\text{IX}.17\text{b})$$

respectively. The fields $B$ and $E^* = \nabla B$ are shown on Fig. 12.

<figure><img src="image_12.png" /><figcaption>FIG. 12: (a) The magnetic and (b) the effective electric field, $B = B(\vartheta)$ and $E^* = \nabla B$, respectively, on the horizon of a Kerr-Newman black hole.</figcaption></figure>

Thus the general (Hall) equations of motion (VI.44) reduce to$^{58}$

$$(x^{\vartheta})' = 0, \quad (x^{\varphi})' = \underbrace{(\mu\chi(2aQr_+)(r_+^2 + a^2)) \frac{(r_+^2 - 5a^2 \cos^2 \vartheta)}{(r_+^2 + a^2 \cos^2 \vartheta)^4}}_{E_{\vartheta}^*} \sin \vartheta \cdot \underbrace{\frac{1}{\kappa_{\text{mag}}}}_{(B^*)^{-1}} \qquad (\text{IX}.18)$$

which is the general Hall law with the cast (IX.17). The forces and velocities are depicted in Fig. 13. The trajectories are horizontal ($\vartheta = \text{const.}$) circles; the velocity is determined by $E_{\vartheta}^*$, shown in Fig. 12b. The angular velocity is maximal on the equator and goes smoothly to zero as we approach the poles. The angular velocity depends on the radius of the horizon roughly as $r_+^{-3}$, which implies higher speed for a smaller black hole.

<footnote>$^{58}$ Here the coefficient $\mu\chi$ is included into $E^*$ for convenience, cf. (VI.43a).</footnote>
<figure><img src="image_13.png" /><figcaption>FIG. 13: The horizon of a Kerr-Newman black hole (topologically a two-sphere) rotates around its vertical axis. For an uncharged particle the usual EM fields are switched off but the spin-field term induces an **effective electric field E*** $\propto \nabla B$ indicated by vertical **blue arrows**. The horizontal **red arrows** depict the **velocity**, (IX.18), of a massless, uncharged doubly-extended anyon on the horizon. The **force** and the **velocity** follow the Hall law (I.2) which, in the case of the Kerr-Newman black hole, has the form of (IX.18).</figcaption></figure>

In conclusion, we get:

**Theorem IX.2.** A massless uncharged anyon with non-vanishing exotic charges $\kappa_{\text{exo}}$ and $\kappa_{\text{mag}}$ moves on the horizon of a Kerr-Newman black hole by following the Hall law (IX.18).

The rotation we found is reminiscent of the known frame-dragging by a rotating black hole. It is however *unrelated* to it: frame-dragging is hidden in the coordinates we used in (IX.18). The “Carroll time” coordinate $s$ is defined indeed through $\partial_s = \partial_v + \Omega_H \partial_\varphi$, see (IX.10b), with $\Omega_H$ the angular velocity of the horizon: the coordinates are comoving with
the horizon. Equations (IX.13)-(IX.18) and Figs. 12-13 should be compared with those in flat space, discussed in sec. VI.

One could, theoretically, consider massless particles with electric charge, extending our flat-case study in sect. VI to the curved spaces. Combining the effective fields would yield a distorted version. Massless charged (quasi)particles are considered in condensed matter physics [36].

The general Carrollian equations of motion used to derive (IX.18) depend on the spin-orbit Hamiltonian (III.17), inspired from its flat spacetime form. However we cannot be sure of its precise expression in curved spacetime until 2+1 dimensional dynamics are derived from an established theory in the ambient spacetime — which does not exists yet, to our knowledge.

One could consider a general Hamiltonian of the form $\mathcal{H} = -\mu\chi\mathcal{B}(\vartheta)$, for some yet unknown $\mathcal{B}(\vartheta)$. It is straightforward to see that the conclusion of this section would not change: there would still be motion on the horizon of the black hole, of the same qualitative form: particles would still display an azimuthal motion.

### C. Partially broken Carroll and BMS symmetry on the horizon

Carroll manifolds were discussed in connection with BMS symmetry [14] – for the good reason that the BMS group is the conformal extension of the Carroll group [15–17]. Let us briefly summarize how the relation goes. For more details, the reader is advised to consult [16, 17]. The conformal symmetries of a (general) Carroll structure ($\mathcal{C} = \Sigma \times \mathbb{R}, g, \xi$) are vectorfields on $\mathcal{C}$ s.t.,

$$L_X g = \lambda g, \quad L_X \xi = \mu \xi \quad \text{with} \quad \lambda + 2\mu = 0. \qquad (\text{IX.19})$$

Such an $X$ has the form,

$$X = Y + \left( \frac{\lambda}{2}s + \mathcal{T}(x) \right) \partial_s \qquad (\text{IX.20})$$

where $(x, s)$ are coordinates on $\mathcal{C}$ and $Y$ is a conformal vector field of $(\Sigma, g_\Sigma)$. The $\mathcal{T}(x)$ (called “the super translations”) are arbitrary functions of the coordinate $x$. So the symmetry group of $(\mathcal{C} = \Sigma \times \mathbb{R}, g, \xi)$ is the semi-direct product $\text{Conf}(\Sigma, g_\Sigma) \times \mathcal{T}$ with $\mathcal{T}$ the group of supertranslations $\mathcal{T}$. When $\lambda$ and thus also $\mu$ vanish, such a transformation is a
Carrollian isometry, whose form is

$$X = Y + \mathcal{T}(x)\partial_s \qquad (\text{IX.21})$$

where Y is a Killing vector of the singular Carroll “metric” g on Σ.

For the round sphere $Σ = S^2$ with its usual metric, for example, one has that $\text{Conf}(S^2, d\Omega^2) = \text{SL}(2, \mathbb{C})$, and thus we get the BMS group: $\text{SL}(2, \mathbb{C}) \times \mathcal{T}$.

For the Kerr-Newman horizon ($S^2$, $\tilde{g}$) with $\tilde{g}$ as in (IX.11), the conformal isometries are found by solving the system of PDE,

$$2a^2Y^\vartheta \sin\vartheta \cos\vartheta + (r^2 + a^2 \cos^2\vartheta)(\lambda - 2\partial_\vartheta Y^\vartheta) = 0, \qquad (\text{IX.22})$$

$$2(r^2 + a^2)Y^\vartheta \cos\vartheta + (r^2 + a^2 \cos^2\vartheta) \sin\vartheta(2\partial_{\tilde{\varphi}}Y^{\tilde{\varphi}} - \lambda) = 0, \qquad (\text{IX.23})$$

$$(r^2 + a^2)^2 \sin^2\vartheta \partial_\vartheta Y^{\tilde{\varphi}} + (r^2 + a^2 \cos^2\vartheta)^2 \partial_{\tilde{\varphi}} Y^\vartheta = 0 \qquad (\text{IX.24})$$

with $Y^\vartheta$, $Y^{\tilde{\varphi}}$ and $\lambda$ functions of $(\vartheta, \tilde{\varphi})$. For isometries $\lambda$ (and thus $\mu$) vanish. The Killing vector of the (spatial part of the) horizon is a rotation around the axis of the black hole [173],

$$Y = \partial_{\tilde{\varphi}}. \qquad (\text{IX.25})$$

The spatial isometries of the Kerr-Newman horizon $\mathcal{H}$ are thus the same as that of the full Kerr-Newmann spacetime, $\text{SO}(2) \times \mathcal{T}$, generated by the

$$X = \partial_{\tilde{\varphi}} + \mathcal{T}(\vartheta, \tilde{\varphi})\partial_s. \qquad (\text{IX.26})$$

The symmetries of the equations of motion (VI.44), $(x^i)' = \epsilon^{ij}(\frac{E_j^*}{B^*})$, and $p_i' = 0$, spelled out on the horizon are conveniently encoded in the Lagrangian (IX.2c) [or equivalently, in the Cartan form (IX.2a)]:

$$\mathcal{L}_{\text{spec}} = \frac{1}{2}\kappa_{\text{exo}} \epsilon_{ij} v^i \frac{dv^j}{ds} + \frac{1}{2}\kappa_{\text{mag}} \epsilon_{ij} x^i \frac{dx^j}{ds} + \mu \chi B \, dx^i, \qquad (\text{IX.27})$$

where $x = (\vartheta, \varphi)$. The magnetic field is a function of $\vartheta$ only, $B = B(\vartheta)$ as seen from (IX.13).

Recall now the mechanical version of the Noether theorem: a vector field X is a symmetry if it changes the Cartan form / Lagrangian by a surface term, $\delta\mathcal{L} = df_X$. Then

$$Q_X = \frac{\partial \mathcal{L}}{\partial(x^i)'} X^i - f_X \qquad (\text{IX.28})$$

is a conserved quantity (see the Appendix).
* The only isometries of the black hole spacetime are “horizontal” rotations generated by $X = \partial_{\varphi}$ for which we find $\delta\mathcal{L} = -d(\kappa_{\text{mag}}\vartheta/2)$, which combines to yield

$$ \pi_{\varphi} = \kappa_{\text{mag}}\vartheta. \qquad (\text{IX.29}) $$

Its conservation is manifest: trajectories of (IX.18) are “horizontal” ($\vartheta = \text{const.}$). This unusual form agrees with the (3.18c) in [34].

* The zeroth order term of super translations, i.e. a (Carrollian) time-translation $X = \partial_s$ leaves the Lagrangian / Cartan form invariant, providing us with the conserved quantity

$$ Q_0 = \mu\chi B = -\mathcal{H}, \qquad (\text{IX.30}) $$

which is indeed the Hamiltonian (VI.47) of a Carroll anyon.

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

$$ L_X \varpi = \mu\chi B \partial_i \mathcal{T} dx^i, \qquad (\text{IX.31}) $$

which is not closed in general, $d(B d\mathcal{T}) = dB \wedge d\mathcal{T} \neq 0$. Thus the magnetic field breaks the supertranslation symmetry in general. However remembering that $B = B(\vartheta)$ on the horizon, cf. (IX.13), the obstruction vanishes when the supertranslation is “vertical”,

$$ \mathcal{T} = \mathcal{T}(\vartheta). \qquad (\text{IX.32}) $$

For a C-boost for example, the $\varphi$-component must vanish, $\mathcal{T}(\vartheta) = -\beta_1^\vartheta \vartheta$. The associated Noether quantity

$$ Q_{\vartheta} = \mu\chi \left( -B\vartheta + \int Bd\vartheta \right) \qquad (\text{IX.33}) $$

cf. # (V.8) in [35], is a combination of the Carroll Hamiltonian $\mathcal{H}$ (IX.30) and the integral of $B$. Complicated as it is, $Q_{\vartheta}$ depends on $\vartheta$ only, therefore its conservation implies, once again, that the trajectories must have $\vartheta = \text{const.}$ (as we had found it). On the contrary, $\varphi$-boost are broken and thus do not obstruct motion in the $\varphi$ direction.

The “allowed” and “forbidden” directions are correlated with the C-boost symmetry. In the example above; the allowed lines are plainly the “horizontal” motions depicted
in fig.13. The allowed directions follow the broken C-boost directions and are perpendicular to the residual conserved component of D. The “partial immobility”, is analogous to the one found for fractons – e.g. for “lineons” [41, 42], or for “X-cube” models [175].

* The last class of notable super translation symmetry is when the supertranslation is proportional to a power of the magnetic field itself, e.g. $\mathcal{T} \equiv \mathcal{T}_n \propto B^n$ for some $n = 0, \dots$, then, we get a conserved quantity for each $n$, $Q_n \propto (\mu\chi B)^{n+1} = \mathcal{H}^{n+1}$.

While these symmetries are interesting to notice, they bring no new physical information.

Related investigations of magnetized black hole horizons are reported in [176].

# X. CONCLUSION

Our investigations are devoted to particles with Carroll symmetry and to the rôle of the latter in Hall-type effects in various physical instances.

A fundamental question raised from the very beginning [2, 3] concerns the mobility of such particles [2, 7, 35, 41–43]. In the massive case, the “no-motion” statement follows directly from the eqns of motion, (IV.6) and (IV.11) and is due to the absence of a kinetic term in the Hamiltonian. Alternatively, it follows from the conservation of the quantity (IV.8c) associated with Carroll boost symmetry.

C-boost symmetry can also be partial: it may be broken in one, but not in another direction. In the first case, the particle is “liberated” and can move, while in the other direction motion remains forbidden. “Pinning” to a fixed point is then replaced by being constrained to one-dimensional submanifolds (“lineons” [42]) — leading to a Hall-type behavior (I.2). Typical examples arise in an external electromagnetic field, or on the horizon of a Kerr-Newman black hole.

After having long limited interest in the subject, the “no-motion property” of Carroll particles [2–4, 6] is now, conversely, attracting new attention: for quasiparticles called fractons (sect. V) studied in condensed matter physics, immobility is attributed to conserved dipole momentum [41, 42]. The relation of the latter to Carroll symmetry has been noticed — up
to terminology: in ref. [41] only the time-based one is called “Carroll”, but it is considered different from a rather mysterious dipole symmetry. The relation is clarified in sec.V: lifting the problem to one higher dimension we get indeed, that

1. In the free case we have *two*, *differently implemented* Carroll symmetries: one time (*t*)-based, the other “internal” i.e. *s*-based. The associated conserved quantities curiously coincide. They both extend, in the free case, to an infinite BMS-type symmetry.

2. Interactions break the *t*-based “external”, but *not* the *s*-based “internal” Carroll implementation. The higher order symmetries are generically broken, though.

Which implementation is the “true” one? It depends on the viewpoint. In *d* + 1 dimensions the second one is indeed “internal” while in the Kaluza-Klein-type *d* + 2 - dimensional framework they are both on the same footing up to being built upon different “time” coordinates : *t* or *s*, respectively. But in a gravitational wave, all 4 dimensions are ... just space-time dimensions of the Universe where natural phenomena happen, and it is the projection to 2 + 1 dimensional Newtonian spacetime which becomes a mere technical trick.

The “no-motion” property changes for *massless particles* (sec.IV C). The C-boost momentum vanishes, **D** = 0, and immobility can not be deduced from its conservation. For a spinless (“Fermat”) photon (sec.IV C) there is (instantaneous) motion; for a gravitational wave, or on the black hole horizon (sec.IX), it does prevent motion in the unbroken C-boost direction, as it does for massive particles.

The Carroll group has, in *d* = 2 dimensions, a double central extension [8, 34, 81] with additional parameters we denote by κ<sub>exo</sub> and κ<sub>mag</sub>, respectively. They are related to *non-commuting positions* (for κ<sub>exo</sub>), and to *non-commuting momenta* (for κ<sub>mag</sub>), respectively. The first one was familiar before from the Galilean theory recalled in sec.III. The second one is in turn explained by the particular role played by the euclidean subgroup. Taking them into account, and also that in *d* = 2 we can have anyons with any real spin χ and magnetic moment μ, yields an extended dynamics, which is however still consistent with the general formula (VI.44) in Theorem VI.7. The motion of an extended Carroll particle follows a generalized (anomalous) Hall law which includes a Zeeman force acting on the particle in inhomogeneous fields.

After presenting intuitively appealing toy models in sec.VI F and explaining the relation
to the Poincaré invariant relativistic theory (sec.VII), we illustrated this theory in the grav-
itational context. Firstly, for the “memory effect” of gravitational waves [24–26] (sec. VIII),
and then for motion of an hypothetical massless particle with anyonic magnetic moment
on the black hole horizon (sec. IX). See also recent investigations of magnetized black hole
horizons [176].

Concerning the physical relevance of these investigations, recent progress with meta-
materials makes it conceptually possible to create *condensed matter analogs* — including
Kerr-Newman black holes [177]. This hints at the possibility to *test experimentally* what
happens there. Other laboratory tests might exist in condensed matter physics.

Acknowledgments

The authors are grateful to P. Chruściel, G. Gibbons, J. Hartong, F. Peña-Benítez, M. Stone, P. Surowka and F. Ziegler for discussions and correspondence. LM is grateful to Prof. T. Harko for hospitality and discussions at Babeş-Bolyai University. PMZ was partially supported by the National Natural Science Foundation of China (Grant No. 11975320).

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