Magnetic Carrollian Regime in Field Theories

• Magnetic Carrollian Regime is defined by the c→0 ultra‐relativistic contraction that retains spatial-curvature and magnetic field strengths while suppressing kinetic time dynamics.
• In ghost-free bimetric gravity, the magnetic sector maintains a nontrivial interaction potential while reducing the Einstein–Hilbert kinetic terms to constraint forms.
• The regime spans diverse theories—including Maxwell, Yang–Mills, and string theory—revealing consistent Carrollian and BMS-type symmetry structures and ultralocal dynamics.

Searching arXiv for papers on the magnetic Carrollian regime and closely related limits. The magnetic Carrollian regime is the ultra-relativistic $c\to 0$ sector of a relativistic theory in which the surviving leading structures are typically spatial-curvature terms, magnetic field strengths, or non-derivative potentials, while the ordinary Lorentzian kinetic terms degenerate into constraints, first-order systems, or commuting local Hamiltonians. In ghost-free bimetric gravity this regime appears at next-to-leading order in a pre-ultralocal expansion, with the interaction potential between the two metrics surviving nontrivially while the Einstein–Hilbert pieces reduce to Lagrange-multiplier constraints (Kluson, 2024). Closely related magnetic Carrollian sectors occur in Einstein gravity, electrodynamics, nonlinear electrodynamics, scalar theory, Yang–Mills theory, light-front field theory, and string theory, where they are consistently associated with degenerate Carrollian geometry, reduced boost structure, and ultralocal or frozen time evolution (Campoleoni et al., 2022).

1. Definition through ultra-relativistic contraction

The magnetic Carrollian regime is not defined by a single universal prescription, but by a family of $c\to 0$ contractions that preserve the “magnetic” or spatial part of a relativistic theory. In bimetric gravity the starting point is the pre-ultralocal parametrization

$$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$

together with expansions

$$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$

and similarly for $f_{\mu\nu}$. The action then expands schematically as

$S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$

where $S^{(-2)}$ is the electric Carrollian sector and $S^{(0)}$ is the magnetic sector (Kluson, 2024).

In Maxwell theory the contraction is implemented directly on coordinates and fields. With $x^0=ct$, the magnetic limit rescales the potential as

$A_0=c\,a_0,\qquad A_i=a_i,$

so that the Carrollian action becomes

$c\to 0$0

and the Carrollian fields are

$c\to 0$1

(Banerjee et al., 2024).

For a free scalar field, the pure magnetic Carrollian sector retains only the spatial derivative term. On a Carroll manifold the action is

$c\to 0$2

which in adapted flat coordinates reduces to

$c\to 0$3

No Carrollian time derivative appears in this limit (Ciambelli, 2023).

A distinct but equivalent structural realization arises on the light front. For a free massless scalar written in light-cone coordinates and using $c\to 0$4 as Carrollian time, the reduced Hamiltonian density is

$c\to 0$5

with no further $c\to 0$6 rescaling required (Majumdar, 2024). This suggests that magnetic Carrollian dynamics is naturally adapted to null foliations as well as to explicit ultra-relativistic contractions.

2. Magnetic Carrollian bimetric gravity

In ghost-free bimetric gravity the magnetic regime is the next-to-leading Carrollian sector after factoring out an overall $c\to 0$7. Its defining feature is that the interaction potential survives, whereas the kinetic Einstein–Hilbert pieces collapse into constraint terms. Using Carroll-compatible extrinsic curvatures

$c\to 0$8

and transverse Lagrange multipliers $c\to 0$9 and $$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$0, the magnetic action is

$$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$1

with $$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$2 the Carroll limit of $$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$3 (Kluson, 2024).

The expansion mechanism is sharply asymmetric between electric and magnetic sectors. At leading order, the two Einstein–Hilbert terms yield two decoupled Carroll–GR actions, while the interaction potential is $$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$4 exactly and therefore does not participate in the electric limit. At next-to-leading order, after removing the overall $$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$5, the potential remains unchanged but the kinetic pieces appear only in Lagrange-multiplier form. The magnetic sector therefore keeps the bimetric coupling but loses genuine Carrollian graviton dynamics (Kluson, 2024).

The equations of motion reflect this structure. Variation with respect to the multipliers gives

$$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$6

Variation of the spatial metrics yields equations schematically of the form

$$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$7

and similarly for $$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$8, while variation of the fields entering $$g_{\mu\nu}=-c^2 T_\mu T_\nu+h_{\mu\nu},\qquad f_{\mu\nu}=-c^2 U_\mu U_\nu+\kappa_{\mu\nu},$$9 gives the algebraic constraint

$$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$0

(Kluson, 2024).

The symmetry content is reduced relative to the electric sector. The parent bimetric theory has only a single diagonal diffeomorphism group because the potential breaks the two independent diffeomorphism groups to the diagonal. After the Carroll limit one retains this diagonal $$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$1, acting on $$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$2, and only a single diagonal Carroll boost survives. By contrast, the electric sector has two decoupled Carroll boost symmetries and two separate Carroll–GR gauge algebras (Kluson, 2024).

Because $$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$3 on shell, there is no genuine Carrollian time evolution of either spatial metric, and the theory is ultralocal. The only nontrivial coupling is the algebraic potential at each spatial point, so there are no propagating degrees of freedom in the usual sense (Kluson, 2024).

3. Gravitational realizations beyond bimetric theory

For ordinary Einstein gravity, the magnetic Carrollian regime is obtained most directly in the ADM Hamiltonian formalism. Writing

$$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$4

with

$$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$5

the limit $$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$6 drops $$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$7 and yields

$$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$8

The surviving theory retains the spatial-curvature term, while the canonical momentum cannot be algebraically eliminated (Campoleoni et al., 2022).

The same theory admits a Cartan-like reformulation obtained by gauging the Carroll algebra. The first-order action

$$T_\mu=\tau_\mu+O(c^2),\quad h_{\mu\nu}=h^{(0)}_{\mu\nu}+c^2 H_{\mu\nu}+\cdots,$$9

is equivalent to the magnetic Hamiltonian action after solving torsion constraints, fixing time gauge, and defining

$f_{\mu\nu}$0

Variation enforces $f_{\mu\nu}$1, equivalently $f_{\mu\nu}$2, so the spatial metric is statically embedded up to spatial diffeomorphism (Campoleoni et al., 2022).

With negative cosmological constant, magnetic Carroll gravity in $f_{\mu\nu}$3 dimensions admits asymptotic conditions whose symmetry algebra is the conformal Carroll algebra in three dimensions, isomorphic to BMS$f_{\mu\nu}$4. The canonical constraints satisfy

$f_{\mu\nu}$5

so normal deformations form an abelian subalgebra. The Regge–Teitelboim surface charges are

$f_{\mu\nu}$6

and close without central terms under the BMS$f_{\mu\nu}$7 composition law (Pérez, 2022).

A later covariant linearized treatment on an FLRW background isolates a magnetic Weyl regime in which the electric Weyl tensor vanishes,

$f_{\mu\nu}$8

while the magnetic part obeys

$f_{\mu\nu}$9

Here the only nontrivial equation is a magnetostatic curl constraint sourced by the shear $S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$0 (Patil et al., 9 Sep 2025).

Quadratic gravity provides further magnetic sectors. For the two ghost-free $S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$1 Carrollian limits in four dimensions, the magnetic actions contain $S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$2 together with quartic extrinsic-curvature terms built from $S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$3 and $S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$4. In one of the two cases the quartic contribution takes the form

$S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$5

showing that magnetic Carrollian gravity can accommodate higher-curvature modifications while remaining a deformation of Carrollian GR (Tadros et al., 2023).

4. Gauge fields, nonlinear electrodynamics, and Yang–Mills theory

In action-based magnetic Carrollian Maxwell theory, varying the contracted action yields

$S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$6

In terms of Carrollian fields, the vacuum equations become

$S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$7

The magnetic action is invariant under gauge transformations, under the exchange $S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$8, $S=c^2 S^{(-2)}+c^4 S^{(0)}+O(c^6),$9, and under Carroll boosts for which

$S^{(-2)}$0

(Banerjee et al., 2024).

Off-shell realizations of magnetic Carrollian electrodynamics are not unique. One construction writes a minimal magnetic-sector action for $S^{(-2)}$1 and $S^{(-2)}$2 alone and shows invariance under the conformal Carroll algebra with supertranslations (1901.10147). Another concludes that the magnetic equations do not follow from any local action built out of $S^{(-2)}$3 alone and therefore introduces an extra Carroll scalar $S^{(-2)}$4 and vector $S^{(-2)}$5, reconstructing the action via Helmholtz integrability (Banerjee et al., 2020). Null-reduction constructions provide further first-order formulations: a Bargmann reduction produces an action involving $S^{(-2)}$6, $S^{(-2)}$7, and shifted momenta $S^{(-2)}$8 (Chen et al., 2023), while a deformed light-cone Kaluza–Klein-like null reduction preserves a first-class Gauss constraint

$S^{(-2)}$9

throughout the Carrollian limit and yields the canonical magnetic action

$S^{(0)}$0

(Zeng, 20 Jun 2026). This suggests that the magnetic Carrollian regime admits several inequivalent but structurally related off-shell completions.

Nonlinear electrodynamics sharpens the distinction between electric and magnetic Carrollian sectors. In the magnetic Carrollian limit of ModMax electrodynamics, the parameter $S^{(0)}$1 remains nontrivial, unlike in the electric contraction. The field equations retain

$S^{(0)}$2

while the inhomogeneous equations become nonlinear in $S^{(0)}$3 and $S^{(0)}$4; the resulting symmetry algebra contains the standard Carroll generators, infinite-dimensional supertranslations $S^{(0)}$5, a diagonal part forming $S^{(0)}$6, and two internal symmetries corresponding to conformal invariance and duality (Correa et al., 2024).

The Carrollian generalization of the Gaillard–Zumino condition gives a self-duality equation

$S^{(0)}$7

with $S^{(0)}$8 and $S^{(0)}$9. Its characteristic system admits closed-form solutions, and distinct seed data can flow to the same descendant self-dual Lagrangian; the data block identifies this as an attractor behavior. In the pure magnetic limit $x^0=ct$0, the Carrollian ModMax example reduces to

$x^0=ct$1

namely Carrollian Maxwell theory (Chen et al., 3 Jun 2025).

For non-Abelian gauge theory, Carrollian SU($x^0=ct$2) Yang–Mills has two magnetic sectors. Sector I is abelian and consists of $x^0=ct$3 decoupled copies with

$x^0=ct$4

Sector II is genuinely non-Abelian,

$x^0=ct$5

with

$x^0=ct$6

In $x^0=ct$7, both magnetic sectors are invariant under infinite Carrollian conformal symmetry, and the propagators exhibit ultra-local behavior (Islam, 2023).

5. Strings, light fronts, and null or near-horizon geometries

A magnetic-like Carrollian sector also appears in current-current deformations of toroidal CFT$x^0=ct$8. With deformation coupling $x^0=ct$9 or $A_0=c\,a_0,\qquad A_i=a_i,$0, the Hamiltonian becomes

$A_0=c\,a_0,\qquad A_i=a_i,$1

and the worldsheet action is

$A_0=c\,a_0,\qquad A_i=a_i,$2

Its first-class constraints satisfy the centrally trivial BMS$A_0=c\,a_0,\qquad A_i=a_i,$3 brackets

$A_0=c\,a_0,\qquad A_i=a_i,$4

The worldsheet metric is degenerate,

$A_0=c\,a_0,\qquad A_i=a_i,$5

and the theory is tensionless but remains relativistic in the target space (Parekh et al., 2023).

A different magnetic Carroll string arises from the small-$A_0=c\,a_0,\qquad A_i=a_i,$6 expansion of the relativistic closed bosonic string in phase space. After rescaling $A_0=c\,a_0,\qquad A_i=a_i,$7 and $A_0=c\,a_0,\qquad A_i=a_i,$8, the leading constraint

$A_0=c\,a_0,\qquad A_i=a_i,$9

defines the magnetic Carroll string sector. At next order, the phase-space action reduces to a Polyakov form with a Lorentzian worldsheet metric $c\to 0$00 and degenerate transverse target metric $c\to 0$01. Near the horizon of a four-dimensional Schwarzschild black hole, the geometry becomes a string-Carroll background of the form $c\to 0$02D Rindler $c\to 0$03, and the magnetic sector reproduces two classical branches: a null-geodesic branch with $c\to 0$04, and a folded-string or “yo-yo” branch with $c\to 0$05 (Bagchi et al., 2024).

The light front provides a particularly transparent realization. For a massless scalar on the null plane, the reduced Hamiltonian density

$c\to 0$06

satisfies

$c\to 0$07

and the finite generators $c\to 0$08, $c\to 0$09, and $c\to 0$10 obey the Carroll algebra. No singular contraction is needed: the first-order light-cone structure produces the magnetic Carroll Hamiltonian directly (Majumdar, 2024). This suggests a close structural relation between the magnetic Carrollian regime and theories formulated on null hypersurfaces.

Magnetic Carrollian geometry also reorganizes black-hole kinematics. In the magnetic–electric Carrollian Reissner–Nordström background, the Carroll extremal surfaces

$c\to 0$11

divide the spacetime into geodesically complete regions, and neutral geodesics cannot cross from one wedge to another in finite proper time. Charged particles, by contrast, are confined to circular motion at fixed radius by the residual electric field (Chen et al., 2024).

6. Structural features, symmetries, and interpretive issues

Across the examples above, the magnetic Carrollian regime is characterized by frozen or ultralocal Carrollian time evolution, but not by complete triviality. In scalar theory, the absence of time derivatives gives the primary constraint

$c\to 0$12

and the secondary constraint

$c\to 0$13

so the field is frozen on each Carrollian leaf (Ciambelli, 2023). In light-front scalar theory there is still a first-order action and nontrivial momentum density $c\to 0$14, but the local Hamiltonians commute (Majumdar, 2024). In magnetic bimetric gravity, the dynamics is frozen because $c\to 0$15; in magnetic Maxwell theory, by contrast, the magnetic field survives as a spatial field strength and the equations still organize into a nontrivial gauge system (Kluson, 2024).

The symmetry algebras are likewise theory-dependent but follow a recurring pattern. Magnetic sectors preserve Carrollian time translations, spatial rotations, and Carroll boosts in reduced or diagonal form. In four-dimensional gauge systems, these symmetries often enhance to an infinite-dimensional conformal Carroll algebra or BMS-type algebra (1901.10147). In AdS-contracted magnetic gravity the asymptotic symmetry algebra is the three-dimensional conformal Carroll algebra, isomorphic to BMS$c\to 0$16 (Pérez, 2022). In two-dimensional magnetic-like deformations of CFT$c\to 0$17, the Virasoro pair is replaced by BMS$c\to 0$18 at the marginality bound $c\to 0$19 (Parekh et al., 2023).

A common misconception is that “magnetic” always means “pure magnetic field theory.” The data do not support such a uniform reading. In bimetric gravity the surviving magnetic Carrollian term is the non-derivative interaction potential between two metrics (Kluson, 2024). In Einstein gravity it is the spatial-curvature Hamiltonian (Campoleoni et al., 2022). In scalar theory it is the spatial Laplacian sector (Ciambelli, 2023). In light-front formulations it is a purely spatial Hamiltonian density (Majumdar, 2024). The adjective therefore labels the contraction pattern rather than a single universal dynamical content.

Another recurrent issue is whether magnetic Carrollian theories admit conventional action principles. The electrodynamic literature contains distinct answers: one line derives explicit off-shell actions directly for the magnetic sector (Banerjee et al., 2024), another uses Bargmann or light-cone null reduction to preserve gauge constraints (Chen et al., 2023), and a third introduces additional fields to satisfy Helmholtz integrability (Banerjee et al., 2020). This indicates that the magnetic regime is best understood as a family of closely related non-Lorentzian formulations rather than as a uniquely normalized theory.

At the quantum level, magnetic Carrollian structure also affects state spaces. For the magnetic Carrollian scalar, the first-order action

$c\to 0$20

implies

$c\to 0$21

In the massless case, the natural vacuum state is pure but nonregular, and the GNS Hilbert space factorizes into a Fock sector and a nonseparable soft sector,

$c\to 0$22

a structure the data block associates with infrared degrees of freedom and flat-space holography (Fredenhagen et al., 24 Apr 2026).

Taken together, these results identify the magnetic Carrollian regime as a broad non-Lorentzian universality class. Its defining hallmarks are a degenerate Carrollian geometry, suppression or constraint of ordinary time-kinetic terms, survival of spatial or algebraic “magnetic” structures, and enlarged Carroll/BMS-type symmetries. The precise realization varies sharply by theory, but the regime consistently provides a controlled language for null boundaries, near-horizon dynamics, ultra-relativistic field theory, and interacting non-Lorentzian gauge and gravitational systems.