Carrollian Regime of Gravity

• Carrollian Regime of Gravity is the ultra-relativistic limit where c→0 yields a degenerate Carrollian geometry that suppresses spatial propagation.
• It unifies null boundaries, black-hole horizons, and asymptotic symmetries by distinguishing between electric and magnetic sectors.
• Gauge-theoretic formulations and three-dimensional Chern–Simons models enrich its applications in holography and gravitational dynamics.

The Carrollian regime of gravity is the ultra-relativistic limit $c\to 0$, in which light cones collapse onto the time direction, spatial propagation is suppressed, and the relevant geometry becomes Carrollian rather than Lorentzian. On null hypersurfaces this structure is encoded by a degenerate metric $q_{\alpha\beta}$ together with a preferred vector $n^\alpha$ satisfying $n^\alpha q_{\alpha\beta}=0$; in gauge-theoretic and first-order formulations it is often described by a clock form, spatial vielbein, and Carroll boost and rotation connections (Nguyen, 13 Nov 2025, Concha et al., 16 Dec 2025). Recent work has developed the Carrollian regime as a common language for null infinity, black-hole horizons, asymptotic symmetries, ultra-relativistic limits of General Relativity and its extensions, three-dimensional Chern–Simons gravity, and flat-space holography (Ciambelli et al., 2018, Donnay et al., 2019, Tadros et al., 2023).

1. Geometric definition and ultra-relativistic limit

In the Carrollian limit, the speed of light is sent to zero, causal propagation across space is lost, and the surviving kinematics is adapted to a degenerate temporal structure. One formulation starts from the relativistic metric in Randers–Papapetrou form and shows that the $c\to 0$ limit leaves three independent fields: a lapse-like scalar $\Omega(t,\mathbf x)$, a spatial one-form $b_i(t,\mathbf x)$, and a spatial metric $a_{ij}(t,\mathbf x)$. The corresponding diffeomorphisms reduce to Carrollian diffeomorphisms,

$$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$

and the basic Carroll-covariant derivatives are $\Omega^{-1}\partial_t$ and $q_{\alpha\beta}$0 (Ciambelli et al., 2018).

A second, widely used formulation is the pre-ultralocal decomposition

$q_{\alpha\beta}$1

with analytic expansion in $q_{\alpha\beta}$2. The leading fields $q_{\alpha\beta}$3 define the Carrollian geometry, and the extrinsic curvature of spatial leaves is

$q_{\alpha\beta}$4

This decomposition underlies recent analyses of Carrollian limits of General Relativity, quadratic gravity, and bimetric gravity (Tadros et al., 2023, Kluson, 2024).

At null infinity, Penrose conformal compactification yields a conformal Carrollian structure $q_{\alpha\beta}$5, with

$q_{\alpha\beta}$6

and in adapted coordinates $q_{\alpha\beta}$7,

$q_{\alpha\beta}$8

Under Weyl rescaling, $q_{\alpha\beta}$9 and $n^\alpha$0, so the intrinsic null-boundary data is a conformal Carrollian class rather than an ordinary boundary metric (Nguyen, 13 Nov 2025).

2. Null boundaries, horizons, and Carrollian conservation laws

The event horizon of a generic non-extremal black hole provides a concrete realization of the Carrollian regime. In null Gaussian coordinates,

$n^\alpha$1

the horizon sits at $n^\alpha$2, and the induced metric becomes

$n^\alpha$3

Comparing the stretched-horizon metric with Randers–Papapetrou form identifies

$n^\alpha$4

so approaching the horizon is an ultra-relativistic limit. In this sense, the horizon is the ultra-relativistic endpoint of the family of timelike stretched horizons (Donnay et al., 2019).

In that setting, the null Raychaudhuri and Damour equations become Carrollian conservation laws. The Brown–York tensor on the stretched horizon diverges as $n^\alpha$5, but its divergent behavior reorganizes into finite Carrollian momenta $n^\alpha$6, $n^\alpha$7, $n^\alpha$8, and $n^\alpha$9, and the conservation equations take the form

$n^\alpha q_{\alpha\beta}=0$0

$n^\alpha q_{\alpha\beta}=0$1

Substituting the horizon/Carroll dictionary reproduces exactly the null Raychaudhuri and Damour equations (Donnay et al., 2019).

More generally, Carrollian covariance yields intrinsic conservation laws for the Carrollian momenta $n^\alpha q_{\alpha\beta}=0$2: $n^\alpha q_{\alpha\beta}=0$3

$n^\alpha q_{\alpha\beta}=0$4

In asymptotically flat gravity, these equations arise as the ultra-relativistic limit of relativistic stress-tensor conservation and reinterpret the boundary equations of motion at null infinity as Carrollian conservation laws (Ciambelli et al., 2018).

At null infinity itself, the BMS generators are conformal Carrollian vector fields. In Bondi frame,

$n^\alpha q_{\alpha\beta}=0$5

and the general conformal Carrollian generator is

$n^\alpha q_{\alpha\beta}=0$6

which is precisely the boundary restriction of BMS symmetry (Donnay et al., 2022).

3. Electric and magnetic sectors of Carrollian gravity

A recurring result is that the Carrollian limit of gravity is not unique. In linearized General Relativity on an FLRW background, the $n^\alpha q_{\alpha\beta}=0$7 covariant decomposition shows that the electric and magnetic Weyl tensors,

$n^\alpha q_{\alpha\beta}=0$8

do not admit a single Carrollian contraction preserving the full Einstein system. Instead, the theory bifurcates into an electric sector dominated by $n^\alpha q_{\alpha\beta}=0$9 and a magnetic sector dominated by $c\to 0$0 (Patil et al., 9 Sep 2025).

The Carrollian electric limit is defined by

$c\to 0$1

and yields the reduced system

$c\to 0$2

Assuming $c\to 0$3, the remaining equation forces $c\to 0$4, so the electric Carrollian regime is a frozen tidal theory (Patil et al., 9 Sep 2025).

The Carrollian magnetic limit is defined by

$c\to 0$5

and yields

$c\to 0$6

The paper repeatedly interprets this as the natural gravito-magnetic sector for horizons and null boundaries, but it also emphasizes that the derived equations are static; a plausible implication is that the magnetic sector retains nontrivial shear-coupled structure without recovering propagating wave dynamics at the level of the reduced linearized system (Patil et al., 9 Sep 2025).

In the Hamiltonian formulation of ultrarelativistic gravity, this split appears as inequivalent magnetic and electric contractions of General Relativity. For the magnetic contraction, one keeps the spatial-curvature term,

$c\to 0$7

while for the electric contraction one keeps the kinetic term,

$c\to 0$8

Under Regge–Teitelboim parity conditions, the magnetic theory has the Carroll algebra as asymptotic symmetry, whereas the electric theory is truncated to the semidirect sum of spatial rotations and spatial translations. With Henneaux–Troessaert parity conditions, the magnetic asymptotic symmetry algebra becomes a BMS-like extension of the Carroll algebra, while the electric theory yields only rotations plus parity-odd supertranslations (Pérez, 2021).

Higher-curvature and multi-metric theories preserve the same basic electric–magnetic logic. In quadratic gravity, the admissible Carrollian modifications of Carrollian GR up to leading and next-to-leading order occur only for the scalings

$c\to 0$9

and all modify the Carrollian limit of General Relativity by quartic extrinsic-curvature terms (Tadros et al., 2023). In ghost-free bimetric gravity, the electric Carrollian limit is the sum of two decoupled Carroll gravities, whereas in the magnetic limit the bimetric interaction survives but the dynamics becomes constrained, with $\Omega(t,\mathbf x)$0 and $\Omega(t,\mathbf x)$1 acting as Lagrange multipliers (Kluson, 2024).

4. Gauge-theoretic and topological realizations

In three dimensions, the Carrollian regime admits a fully gauge-theoretic and topological formulation. The most general relativistic parent in this setting is Mielke–Baekler gravity,

$\Omega(t,\mathbf x)$2

whose ultra-relativistic contraction produces the Carroll Mielke–Baekler algebra $\Omega(t,\mathbf x)$3 and a Carrollian Chern–Simons theory (Concha et al., 16 Dec 2025).

The Carroll connection is

$\Omega(t,\mathbf x)$4

with temporal vielbein $\Omega(t,\mathbf x)$5, spatial vielbein $\Omega(t,\mathbf x)$6, rotation connection $\Omega(t,\mathbf x)$7, and Carroll boost connection $\Omega(t,\mathbf x)$8. The deformed Carroll algebra has nonvanishing commutators

$\Omega(t,\mathbf x)$9

$b_i(t,\mathbf x)$0

$b_i(t,\mathbf x)$1

so the parameters $b_i(t,\mathbf x)$2 and $b_i(t,\mathbf x)$3 deform the ordinary Carroll algebra to support curvature and torsion (Concha et al., 16 Dec 2025).

A central technical result is that the contraction preserves a non-degenerate invariant bilinear form, which makes a Chern–Simons action possible without introducing extra generators. The field equations then reduce to vanishing of the deformed curvatures, but ordinary torsion and curvature need not vanish. In particular,

$b_i(t,\mathbf x)$4

whenever $b_i(t,\mathbf x)$5, so the theory has non-zero temporal torsion on shell. Likewise, for $b_i(t,\mathbf x)$6,

$b_i(t,\mathbf x)$7

This realizes the first fully general torsional three-dimensional Carrollian gravity with non-zero temporal torsion and non-zero curvature (Concha et al., 16 Dec 2025).

Several known theories arise as special limits:

• Carroll gravity: $b_i(t,\mathbf x)$8,
• AdS-Carroll gravity: $b_i(t,\mathbf x)$9,
• ultra-relativistic torsional gravity: $a_{ij}(t,\mathbf x)$0.

This unifying role suggests that the three-dimensional Chern–Simons setting captures, in a particularly explicit way, how the Carrollian regime can retain intrinsic torsion and curved sectors rather than collapsing to a naive flat limit (Concha et al., 16 Dec 2025).

5. Scaling symmetry, dynamical Carroll gravity, and related phases

A broader gauge-theoretic construction starts from the anisotropic scaling extension of the Carroll algebra $a_{ij}(t,\mathbf x)$1, generated by $a_{ij}(t,\mathbf x)$2, $a_{ij}(t,\mathbf x)$3, $a_{ij}(t,\mathbf x)$4, $a_{ij}(t,\mathbf x)$5, and $a_{ij}(t,\mathbf x)$6, with commutators

$a_{ij}(t,\mathbf x)$7

The gauge field is

$a_{ij}(t,\mathbf x)$8

and conventional curvature constraints solve part of the spin connection in terms of the Carrollian geometric variables (Afshar et al., 23 Dec 2025).

After introducing a compensator scalar and fixing the scaling symmetry by

$a_{ij}(t,\mathbf x)$9

the remaining independent gravity multiplet is

$$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$0

The trace of the extrinsic curvature $$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$1 fixes the temporal component of the dilatation gauge field,

$$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$2

while the spatial part $$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$3 survives and transforms under Carroll boosts as

$$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$4

This is the mechanism by which the post-gauge-fixing theory retains a genuinely dynamical Carrollian sector (Afshar et al., 23 Dec 2025).

The resulting framework contains distinct regimes. In dynamical Carroll gravity, local Carroll boosts remain unfixed and the extrinsic curvature is allowed to be dynamical. Varying the higher-derivative Carrollian action with respect to $$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$5 imposes

$$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$6

which admits the nontrivial solution

$$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$7

This is an explicit realization of a Carrollian regime with evolving spatial geometry (Afshar et al., 23 Dec 2025).

For $$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$8, one may instead use boosts to fix

$$t' = t'(t,\mathbf x),\qquad \mathbf x'=\mathbf x'(\mathbf x),$$9

which removes the boost redundancy and produces Aristotelian gravity. A different choice is to keep boosts, write $\Omega^{-1}\partial_t$0, and impose

$\Omega^{-1}\partial_t$1

Then the boost parameter is reinterpreted as a vector-charge gauge symmetry, yielding a fracton gauge theory coupled to Aristotelian geometry (Afshar et al., 23 Dec 2025). This suggests that the Carrollian regime is one phase of a larger non-Lorentzian gauge-theoretic structure, rather than a single isolated model.

6. Holography, amplitudes, quantization, and solution-space geometry

Carrollian holography treats the null conformal boundary $\Omega^{-1}\partial_t$2 as the natural arena for asymptotically flat quantum gravity. A sourced $\Omega^{-1}\partial_t$3 conformal Carrollian field theory has been proposed as the boundary theory whose Ward identities reproduce the Bondi evolution equations, with the external source identified with Bondi news,

$\Omega^{-1}\partial_t$4

In this formulation, Carrollian currents match Bondi mass and angular-momentum aspects, and the Ward identities of the boundary theory reproduce those of celestial CFT after an integral transform along the null generators (Donnay et al., 2022).

A complementary formulation uses Carrollian amplitudes, namely massless scattering amplitudes written in position space at null infinity. Tree-level graviton amplitudes become correlators of Carrollian primary fields depending on $\Omega^{-1}\partial_t$5, and the collinear limit defines a Carrollian OPE. Smearing this OPE along the generators of null infinity yields the action of celestial symmetry algebras; for gravity, the paper identifies $\Omega^{-1}\partial_t$6 as the relevant enhanced celestial symmetry acting on Carrollian operators (Mason et al., 2023). In string theory, the Fourier transform to null infinity converts the $\Omega^{-1}\partial_t$7-expansion into a tower of $\Omega^{-1}\partial_t$8-descendants of the field-theoretic Carrollian amplitude, including four-graviton amplitudes in heterotic string theory (Stieberger et al., 2024). At loop level, one-loop four-point Carrollian amplitudes in $\Omega^{-1}\partial_t$9 supergravity retain a differential-operator relation to tree-level data, while eikonal gravitational amplitudes develop logarithmic dependence on Carroll time $q_{\alpha\beta}$00 and exhibit discontinuities that are descendants of Carrollian Born amplitudes (Nenmeli et al., 9 Apr 2026).

The quantization of simple Carrollian field theories reveals an important obstruction. Two-derivative Carrollian theories are strongly sensitive to the ultraviolet, can be regulated on a spatial lattice at finite inverse temperature, and yield continuum limits that are generalized free rather than genuinely interacting. In those limits, non-Gaussian correlations are suppressed by positive powers of the lattice spacing, and supertranslation symmetry remains unbroken (Cotler et al., 2024). This suggests that any Carrollian boundary dual of flat-space gravity must go beyond the simplest two-derivative models if it is to capture the expected nontrivial infrared and symmetry-breaking structure.

A further refinement comes from the symplectic geometry of Einstein gravity with a generic null boundary. The on-shell symplectic form splits into boundary and bulk parts, but neither is separately closed because of symplectic flux through the null boundary. Within the bulk sector, a $q_{\alpha\beta}$01-dimensional Lagrangian submanifold carries a Carrollian structure whose nondegenerate part is the Wheeler–DeWitt metric and whose kernel direction is the outgoing Robinson–Trautman mode (Adami et al., 2023). This suggests that the Carrollian regime is not only a geometry of spacetime null boundaries, but also a geometry of gravitational solution space itself.

Taken together, these developments define the Carrollian regime of gravity as an intrinsic regime of null-boundary and ultra-relativistic gravitational physics. It includes exact conservation laws on null hypersurfaces, inequivalent electric and magnetic contractions, fully torsional three-dimensional Chern–Simons realizations, dynamical gauge-theoretic Carroll phases, and a holographic program in which graviton scattering data is organized by Carrollian symmetry at $q_{\alpha\beta}$02. Open directions explicitly identified in the literature include supersymmetric extensions, matter couplings, higher-dimensional generalizations, a fuller treatment of radiative and nonlinear sectors, and the construction of Carrollian quantum field theories with the symmetry-breaking and non-Gaussian structure expected of flat-space quantum gravity (Concha et al., 16 Dec 2025, Nguyen, 13 Nov 2025).