Carrollian Magnetic Limit

• Carrollian Magnetic Limit is the systematic ultra‐relativistic contraction (c → 0) that produces degenerate spacetimes with collapsing light cones and dominant magnetic interactions.
• It emerges in various contexts, including field theory contractions, gravitational dynamics near null hypersurfaces, and holographic models, by retaining spatial derivatives while freezing time evolution.
• This limit reveals unique physical regimes where traditional causal propagation vanishes, yet nontrivial spatial magnetic structures dictate the behavior of gauge and gravitational systems.

The Carrollian Magnetic Limit is a systematic ultra-relativistic contraction of field theories and gravitational models, defined by sending the speed of light $c \rightarrow 0$ while preserving specific dynamical structures. In this regime, causal propagation vanishes, the light cone collapses into the time axis, and the resulting Carrollian spacetimes exhibit degenerate metrics with unique transformation properties and symmetries. The "magnetic" label distinguishes this limit from the "electric" Carrollian limit, with the magnetic sector encoding non-trivial spatial, inductive, or geometric structures—often supported by sources such as magnetic fields, curvature invariants, or the magnetic part of the Weyl tensor—while time dynamics are ultra-local or entirely "frozen". The Carrollian Magnetic Limit arises in a broad range of contexts: field theory contractions, gravitational dynamics near null hypersurfaces, holography, and analog models in fluids. Its formal construction, physical content, and mathematical properties are central to the understanding of signal-free, ultra-relativistic regimes of modern theoretical physics.

1. Foundational Definition and Physical Framework

The Carrollian Magnetic Limit is obtained by taking $c \rightarrow 0$ in the covariant line element

$ds^2 = -c^2 dt^2 + d\vec{x}^2,$

leading to a degenerate metric structure where the contravariant form remains nondegenerate but the light cone “collapses” so that all events at distinct spatial points become causally disconnected. The associated symmetry group is the Carroll group, with boosts acting exclusively in the time direction. In this framework, the “magnetic” sector arises by selecting a limit (or dimensional reduction) in which spatial—rather than time—derivative terms dominate or survive, and magnetic-type interactions are retained.

This mechanism is universal across the contraction of field theories (via light-cone reductions (Majumdar, 3 Jul 2025), ultra-relativistic rescalings (1901.10147, Islam, 2023)), as well as in the formulation of gravity near null hypersurfaces (horizons, null infinity, and extreme limits) (Pérez, 2021, Campoleoni et al., 2022, Patil et al., 9 Sep 2025). The magnetic Carroll sector serves as a consistent truncation of the dynamics of the relativistic parent theory: spatial correlations and magnetic-type interactions survive, but all local time evolution is constrained or absent.

2. Magnetic Limit in Gravitational Theories

2.1 Linearized and Full General Relativity

The Carrollian Magnetic Limit bifurcates the gravitational dynamics into electric and magnetic sectors (Patil et al., 9 Sep 2025). Under a 1+3 decomposition of the linearized Einstein equations on an FLRW background, the limit $c \rightarrow 0$ does not preserve Carrollian invariance for the full system, necessitating a consistent split:

• Electric sector: Tidal (Weyl electric) tensor $E_{ab}$ survives, but all dynamics freeze.
• Magnetic sector: Gravito-magnetic tensor $H_{ab}$ remains, governing inductive/radiative modes. The field equations reduce to

$$\nabla_b H^{ab} = 0, \quad \dot{H}^{ab} = 0, \quad (\text{curl } H)^{ab} = \frac{1}{2} (\mu + p) \sigma^{ab}$$

with $E_{ab} = 0$. Only spatial curl and energy-momentum fluxes (via shear) source non-trivial $H_{ab}$, and all time evolution is ultra-local or strictly static (Patil et al., 9 Sep 2025).

2.2 Magnetic Contractions: Hamiltonian vs. Gauged Carroll Gravity

There is an exact correspondence between the magnetic Carrollian limit of Einstein gravity (from the Hamiltonian formalism) and a geometric gauging of the Carroll algebra (Campoleoni et al., 2022). The contraction singles out the spatial Ricci scalar as the surviving part of the Hamiltonian constraint,

$\mathcal{H}^{\rm M} = -\sqrt{g} R,$

and imposes a vanishing extrinsic curvature ($K_{ij} = 0$). The resulting theory admits the full contracted Carroll symmetry. The same structure emerges directly from a gauged Carroll connection, with the identification of auxiliary symmetric tensors with momenta, leading to physically indistinguishable "Carrollian gravity" actions at the magnetic point.

2.3 Asymptotic Symmetries, Boundary Structure, and BMS Carollian Algebra

Under appropriate asymptotic conditions (e.g. Regge–Teitelboim parity), the magnetic Carrollian limit yields a boundary symmetry algebra matching the full Carroll group; with alternative parity assignments (Henneaux–Troessaert), an infinite-dimensional BMS-like extension appears (Pérez, 2021, Fuentealba et al., 2022). In contrast, the electric Carrollian contraction leads to a reduced set, lacking an energy generator.

For coupled systems, as in Einstein–Yang–Mills theory, the magnetic Carrollian limit admits only those color charges or gauge transformations that survive the contraction and parity conditions, often leading to a "frozen" set of boundary charges and precluding the infinite enhancements observed in the electric Carrollian/Yang–Mills case.

2.4 Quadratic and Bimetric Gravity

Expanding higher-derivative or bimetric models involves precise scaling of coupling constants with $c$ (Tadros et al., 2023, Kluson, 30 Aug 2024). The magnetic limit, after truncating NLO fields, generically yields an action dominated by spatial curvature (Ricci scalar for each metric) plus interaction potentials, with temporal dynamics enforced by Lagrange multipliers ensuring the vanishing of the (now auxiliary) extrinsic curvature or analogous variables.

3. Field Theory Constructions and Magnetic Carroll Sectors

3.1 General Carrollian Field Theories

The ultra-relativistic ($c \rightarrow 0$) limit of Poincaré-invariant field theories produces Carrollian theories with degenerate kinetic operators. For scalars, the intrinsic Carroll action takes the form (Ciambelli, 2023): $$S[\phi] = g_2 \int \mathrm{vol}\; [ g_0 (v^\mu \partial_\mu \phi)^2 - h^{\mu\nu} \partial_\mu \phi \partial_\nu \phi ].$$ Here $g_0$ and $g_2$ are couplings, $v^\mu$ is a preferred time direction, and $h^{\mu\nu}$ is the spatial inverse degenerate metric. The "magnetic" term, $h^{\mu\nu} \partial_\mu \phi \partial_\nu \phi$, encodes the spatial action and is the only surviving kinetic piece if $g_0 \to 0$ (strict magnetic limit). Propagation is still possible due to spatial gradients, allowing nontrivial dynamics even for a single particle with nonzero energy.

For Carroll–Dirac or Yang–Mills theories, magnetic sectors are constructed by scaling the time components of fields to subleading order and retaining spatial derivatives or field strengths only (1901.10147, Islam, 2023). In $d=4$, all sectors (electric and magnetic) are invariant under an infinite-dimensional conformal Carrollian/BMS symmetry algebra.

3.2 Light-Cone Reduction and Magnetic Carrollian Actions

A direct algebraic derivation of magnetic Carrollian field theories uses null reduction from the Lorentzian light-cone action (Majumdar, 3 Jul 2025). Fields are projected onto a null hypersurface via a delta-function smearing in $x^-$, and only zero modes in the $x^-$ direction survive, with canonical analysis showing no second-class constraints in the magnetic sector. For example, for a scalar,

$$\mathcal{H}^{\rm mag} = \frac{1}{2} (\partial_i \phi)^2,$$

and for gauge theories in light-cone gauge, the Hamiltonian involves only the spatial field-strength components $F_{ij}$. The resulting theory is a reduction of the parent dynamics, with the magnetic solutions forming a consistent truncation of the $(d+1)$-dimensional theory.

3.3 Field Theory in Lower Dimensions and BMS3 Algebras

Ultrarelativistic Carrollian limits of 2d CFTs (e.g., via bounded current–current deformations) generate magnetic-like Carrollian sectors with energy–momentum densities obeying a BMS$_3$ algebra. The “magnetic” branch describes self-interacting null particles (“inner Carrollian structure”), realized in tensionless string models and encoding an infinite-dimensional symmetry (Parekh et al., 2023).

4. Carrollian Magnetic Limit in Electrodynamics and Nonlinear Theories

4.1 ModMax Theory and Nonlinear Carrollian Magnetism

The Carrollian limit of ModMax electrodynamics—the unique duality-invariant, conformal, nonlinear extension of Maxwell theory—produces two distinct Carrollian sectors: electric and magnetic (Correa et al., 26 Sep 2024). In the magnetic limit, field rescalings preserve the nonlinear parameter $\gamma$, which directly modulates the field equations: $$\partial_s B = 0, \qquad e^{-\gamma}(\nabla \times B_m - \partial_s E_m) = 0.$$ This sector possesses the full Carroll group, local time supertranslations, 2 internal symmetries (conformal, duality), and a diagonal conformal Carrollian algebra ($\mathfrak{ccarr}_2$). Notably, $\gamma$ cannot be absorbed in the magnetic case, ensuring physically distinct dynamics from the linear Maxwell limit.

4.2 Hydrodynamic and Gauge Theory Analogues

A striking correspondence arises between the gauge-theoretic description of shallow water hydrodynamics and the two Carrollian electromagnetic sectors (Bagchi et al., 6 Nov 2024). The flat band (“electric”) and Poincaré (“magnetic”) solutions map directly onto the respective Carrollian electrodynamics actions, with Carrollian invariance (boosts, vanishing stress-energy flux) emerging in both.

5. Magnetic Carrollian Dynamics on Null Hypersurfaces

5.1 Black Hole Horizons and Magnetized Environments

On black hole horizons (Kerr–Newman geometry), the induced $(2+1)$-dimensional Carrollian structure allows the realization of exotic “spin-Hall effect” dynamics for massless, charged, or spinning Carrollian particles coupled to a magnetic field (Gray et al., 2022, Bicak et al., 2023). The central extension of the Carroll algebra enables nontrivial horizon motions, with dynamics governed by equations: $$\frac{dx^A}{dv} = (\mu\chi/\kappa_{\rm mag})\,\epsilon^{AB} \partial_B B.$$ Tilted or time-dependent magnetic fields produce migratory flows (latitudinal oscillations), revealing that external electromagnetic structure can reactivate effective dynamics even in the ultra-relativistic, signal-free Carrollian regime.

5.2 Carrollian Geodesics and Extremal Surfaces

The Carrollian limit of geodesic actions naturally separates into electric and magnetic components (Ciambelli et al., 2023). In the magnetic sector (non-zero $g_1$ in the action), spatial dynamics are possible and interact nontrivially with the “frozen” time direction. For instance, in the Carroll–Schwarzschild and Carroll–Reissner–Nordström black hole backgrounds, magnetic–electric limits yield:

• Unique, often unstable circular orbits localized at “Carroll extremal surfaces” (CES), acting as mirror-like boundaries (Ciambelli et al., 2023, Chen et al., 27 Oct 2024).
• Rich phase space with energy-dependent effective potentials and novel global structures: ultra-relativistic squeezing of causal domains, finite numbers of asymptotic patches, and geodesic completeness partitioned by CESs.

For charged particles, the electromagnetic field can enforce strictly circular orbits at constant radii, strongly differing from standard relativistic geodesics (Chen et al., 27 Oct 2024).

6. Holography, Carrollian Magnetic Limit, and Boundary Dynamics

The holographic flat space limit (Bondi gauge, AdS radius $\ell \to \infty$) naturally implements the Carrollian limit at the boundary (Alday et al., 27 Jun 2024). The resulting Carrollian amplitudes, especially in the magnetic sector, are distributional correlators of Carrollian CFT primaries, structurally interpreting the flat space $S$-matrix in Carrollian terms. Tree-level correlators inherit bulk locality as “bulk point singularities” and delta-function support, directly tracing to spatial magnetic dynamics retained in the Carrollian limit and amplifying the characteristic ultra-local/non-propagating features of the magnetic Carroll sector.

7. Quantum and Effective Dynamics in the Carrollian Magnetic Limit

Quantum corrections (e.g., one-loop effective actions for scalars) behave nontrivially in the Carroll magnetic limit: performing the $c \rightarrow 0$ limit at the quantum level yields divergent terms removable by local counterterms, but the remaining finite part corresponds to an effective lower-dimensional action distinct from that obtained by naive classical reduction (Vassilevich, 31 Oct 2024). Importantly, the quantum Carroll limit “remembers” its higher-dimensional origins, introducing nonlocal finite contributions originating from the full parent theory’s spectra.

Summary Table: Magnetic vs. Electric Carrollian Limits

Property/Sector	Electric Carroll Limit	Magnetic Carroll Limit
Surviving dynamics	Time derivatives (ultra-local, "frozen")	Spatial derivatives / “magnetic” interactions (ultra-local spatial)
Gauge theory ex.	Carrollian electric equations	Magnetic Carrollian field strengths, e.g., $F_{ij}$ only
Gravity ex.	$E_{ab}$ (tidal), static	$H_{ab}$ (gravito-magnetic), static, $\nabla \times H = \text{source}$
Asymptotic symmetries	Reduced (time translations lost)	Full Carroll or extended BMS-type algebra
Local time evolution	Absent (“frozen”)	Absent (“static”); spatial/constraint evolution
Manifestation	Null reduction, deformation needed	Direct reduction/truncation
Holography	Electric Carrollian CFT	Magnetic Carrollian amplitudes, distributional correlators

Conclusion

The Carrollian Magnetic Limit systematically isolates the spatial, magnetic, or curvature-induced degenerate dynamics that survive in the strict $c \rightarrow 0$ limit of both classical and quantum field theories, gravitational models, and probe particle actions. Its mathematical structure, symmetry content, and physical implications (constraint-dominated, nontrivial spatial evolution, boundary or horizon dynamics) distinguish it sharply from both Galilean and electric Carrollian limits. The magnetic sector is fundamental to ultra-relativistic description of null hypersurfaces, fluid analogues, field theory reductions, and flat space holography, and it defines a robust, physically meaningful regime in the broader Carrollian landscape.