Carrollian Gravity: Ultra-Relativistic Dynamics

• Carrollian gravity is a geometric framework obtained by taking the c → 0 limit of General Relativity, leading to a degenerate light-cone structure and ultra-local dynamics.
• It underpins the dynamics of null hypersurfaces, where the decoupling of time and space facilitates a novel holographic correspondence with boundary field theories.
• Distinct electric and magnetic Carrollian limits reveal bifurcated dynamics, with static tidal forces versus active radiative modes, impacting black hole horizon and flat space holography studies.

Carrollian gravity is a geometric and dynamical framework arising as the ultra-relativistic ($c \rightarrow 0$) limit of General Relativity. In this limit, the light-cone degenerates so that causal propagation is “frozen,” leading to structures where time decouples from space and null hypersurfaces—such as event horizons or null infinity—carry a degenerate geometry governed by Carrollian symmetry. Carrollian gravity is relevant both for understanding the ultra-local regime of gravitational dynamics and as the boundary geometry for holography in asymptotically flat spacetime, with deep connections to the BMS symmetry group.

1. Carrollian Limit: From Lorentzian Geometry to Carrollian Structure

The Carrollian limit is defined by taking the speed of light $c \to 0$. At the algebraic level, this contraction of the Poincaré algebra yields the Carroll algebra, where boosts and time translations decouple from spatial translations. In geometric terms, the Lorentzian metric splits as: $g_{\mu\nu} = -c^2 T_\mu T_\nu + \Pi_{\mu\nu}$ with $T_\mu$ a “clock” one-form and $\Pi_{\mu\nu}$ a degenerate spatial metric satisfying $T^\mu T_\mu = -1$ and $\Pi_{\mu\nu}T^\nu = 0$ (Tadros et al., 1 Jan 2024). As $c \to 0$, the metric degenerates: causal signals cannot propagate in space, and only time evolution remains.

All methods for obtaining this limit—for instance, the pre-ultralocal parametrization or the ADM decomposition with vanishing signature—are equivalent and produce the same Carrollian geometry (Tadros et al., 1 Jan 2024). On null hypersurfaces of Lorentzian spacetimes, the induced geometry is naturally Carrollian, characterized by a degenerate metric and a distinguished vector field (the “Carrollian time”).

2. Carrollian Dynamics and Conservation Laws

In standard GR, conservation of the energy–momentum tensor arises from diffeomorphism invariance. In the Carrollian context, since the metric degenerates, the energy–momentum tensor splits into a triplet of “Carrollian momenta”: $$\mathcal{O} = \frac{1}{\Omega a} \frac{\delta S}{\delta \Omega}, \quad \mathcal{B}^i = \frac{1}{\Omega a} \frac{\delta S}{\delta b_i}, \quad \mathcal{A}^{ij} = \frac{1}{\Omega a} \frac{\delta S}{\delta a_{ij}}$$ where $\Omega$ is the time lapse, $a_{ij}$ is the spatial metric, and $b_i$ is a connection field (Ciambelli et al., 2018). The key conservation equations, derived from Carrollian covariance, are: $$\left(\frac{1}{\Omega} \partial_t + \theta\right)\mathcal{E} - (\widehat{\nabla}_i + 2\varphi_i)\mathcal{B}^i - \mathcal{A}^{ij} \frac{1}{\Omega} \partial_t a_{ij}=0$$

$$2(\widehat{\nabla}_i + \varphi_i)\mathcal{A}^i_j + 2\mathcal{B}^i\omega_{ij} - \mathcal{E}\varphi_j = 0$$

where $\theta$ and $\varphi_i$ encode expansion and acceleration, respectively. These equations generalize the relativistic energy–momentum conservation laws to the ultra-relativistic regime.

On physical null boundaries (such as event horizons or future null infinity), bulk Einstein equations induce Carrollian conservation laws for Carrollian momenta, including in the presence of radiation, where extra fields (e.g., a heat current and a stress tensor) emerge (Ciambelli et al., 2018, Donnay et al., 2019).

3. Carrollian Symmetry, Killing Vectors, and Charges

The symmetry algebra of Carrollian gravity is the Carroll group or its extensions, depending on the contraction chosen. For the “magnetic” contraction, the full Carroll (or conformal Carroll) algebra appears; in the “electric” contraction, only a truncated subalgebra (spatial rotations, spatial translations, or supertranslations) is present, as determined by parity conditions on the fields (Pérez, 2021).

Carrollian Killing vectors $\xi$ are defined by their invariance properties under degenerate geometry: $$\delta_\xi \Omega = 0, \qquad \delta_\xi a_{ij} = 0$$ Their explicit form depends on a set of equations involving time and spatial derivatives (e.g., (Donnay et al., 2019, Ciambelli et al., 2018)). Conserved charges associated to these symmetries are constructed by contracting the Carrollian momenta with the Killing vectors, yielding: $$\mathcal{C}_\xi = \int_\Sigma a\, [X\mathcal{E} - \xi^i \pi_i + 2b_i\xi^j \mathcal{A}^{ij}]$$ where $X = \Omega \xi^t - b_i \xi^i$ (Ciambelli et al., 2018). In the context of black hole horizons, these charges generalize energy and (for rotational Killing vectors) angular momentum to dynamical backgrounds (Donnay et al., 2019).

Imposing suitable parity conditions is crucial for realizing extended symmetry algebras (e.g., BMS-like extensions in the magnetic limit, as compared to more truncated algebras in the electric limit) (Pérez, 2021). The presence or absence of time translations and boosts as asymptotic symmetries is a precise diagnostic of the contraction considered (Pérez, 2022).

4. Ultra-Relativistic Holography: Carrollian and Celestial Dualities

Carrollian gravity underpins frameworks for flat space holography. In this paradigm, 4D asymptotically flat gravity is dual to a $D-1$ dimensional conformal Carrollian field theory (CCarrFT) living on the null boundary. Scattering data and gravitational radiation (“news”) are encoded by correlators and external sources in the CCarrFT, leading to modified, sourced Ward identities that match the BMS flux-balance laws (Donnay et al., 2022, Donnay et al., 2022). For example, the Noether current in the CCarrFT obeys a source-coupled non-conservation law,

$\partial_a j^\xi = F^\xi[\sigma]$

with $F^\xi[\sigma]$ corresponding to gravitational radiation through the boundary.

There is a direct connection between Carrollian and celestial holography: Carrollian operators, after a Fourier–Mellin transform in the retarded time (or energy), become the operators of a celestial CFT—the latter encoding the action of the Lorentz group and its infinite-dimensional extensions on the celestial sphere (Donnay et al., 2022, Donnay et al., 2022).

Carrollian correlators for massless fields in general dimensions take explicit universal forms (see (Kulkarni et al., 8 Aug 2025)): $$\mathcal{C}_2^{\Delta_1,\Delta_2}(u_{12},\mathbf{x}_{12}) \propto \frac{\delta^{D-2}(\mathbf{x}_{12})\, \Gamma(\Delta_1 + \Delta_2 - (D-2))}{(u_{12} - i\varepsilon)^{\Delta_1 + \Delta_2 - (D-2)}} \delta_{\epsilon_1,-\epsilon_2}$$ These Carrollian amplitudes are directly related to gravitational S-matrix elements in flat spacetime, establishing the precise holographic dictionary.

5. Bifurcated Dynamics: Electric and Magnetic Carrollian Theories

Carrollian gravity admits multiple consistent dynamical sectors. Following the analogy to Galilean electromagnetism, there are two robust Carrollian limits of linearized gravity (Patil et al., 9 Sep 2025):

• Electric Carrollian Limit: Dynamics are frozen, yielding a static theory where only the “electric” (tidal force) part of the Weyl tensor survives. The time evolution of tidal fields vanishes ($\dot{E}_{ab}=0$) and their spatial curl is zero.
• Magnetic Carrollian Limit: The “magnetic” Weyl tensor $H_{ab}$ (governing gravito-magnetic and radiative effects) remains dynamically nontrivial and is sourced by the spacetime shear, while the electric part is suppressed or algebraically fixed. Unlike the electric case, a boundary theory with dynamic (though Carrollian) propagation survives.

This bifurcation resolves ambiguities and provides schemes for defining Carrollian gravitational theories with distinct properties, crucial for describing black hole horizons and dynamical null infinity.

6. Carrollian Gravity in Applied Contexts: Null Boundaries, Horizons, and Phase Space

Carrollian geometry naturally describes the intrinsic structure of null hypersurfaces, such as black hole event horizons and future null infinity (Adami et al., 2023, Freidel et al., 10 Jun 2024). For a generic null boundary, the solution phase space splits into boundary and bulk sectors; the bulk sector possesses a Carrollian metric (of Wheeler–DeWitt type) with a degenerate kernel aligned with physically relevant gravitational wave solutions (e.g., Robinson–Trautman modes).

On dynamical horizons, the Raychaudhuri and Damour equations—governing evolution of intrinsic geometric data—are recast as Carrollian conservation laws (Donnay et al., 2019). The associated symplectic structure is nontrivial: the total symplectic form is closed, but neither the boundary nor the bulk part is individually closed, due to the flux of bulk modes across the null surface (Adami et al., 2023).

Stretched Carrollian structures (sCarrollian structures) on causal surfaces generalize the classical Carrollian structures of null hypersurfaces by introducing a parameter encoding the “stretching” from null to timelike geometry. This allows a unified treatment of the induced geometry, dynamics, symplectic structure, and Noether charges for both null and timelike boundaries (Freidel et al., 10 Jun 2024), with direct applications to the membrane paradigm and black hole hydrodynamics.

7. Action Principles, Extensions, and Quantization

The dynamical content and gauge structure of Carrollian gravity are subtle. For magnetic theories, gauging the Carroll algebra (or its relatives, such as AdS–Carroll) yields first-order Cartan geometries that consistently produce nontrivial Chern–Simons actions in 3D and analogues of higher-dimensional structures (Figueroa-O'Farrill et al., 2022, Concha et al., 31 Dec 2024). For electric Carrollian gravity, a first-order Cartan-like action can be derived directly from the Carrollian limit of the Einstein–Cartan action, but cannot be obtained by the standard gauging due to a no-go constraint from the vanishing torsion imposed by gauge invariance (Pekar et al., 3 Jun 2024, Figueroa-O'Farrill et al., 2022).

The quantization of Carrollian field theories reveals distinctive features. Naive continuum quantization leads to singular UV behavior—every spatial point behaves as an independent quantum-mechanical system, making the lattice regulator indispensable (Cotler et al., 16 Jul 2024). Correlators are ultra-local in space, and perturbative interactions can be systematically included by power counting lattice spacing. Supertranslation symmetry, acting as a space-dependent time translation, remains unbroken in the quantum theory and enforces the ultralocality of correlators, providing a candidate framework for the dual theory to flat space quantum gravity.

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Table: Carrollian Limits of Gravity—Key Features

Limit	Dynamical Degrees	Symmetry Algebra
Magnetic	Gravito-magnetic (radiation) $H_{ab}$	Full conformal Carroll / BMS
Electric	Static tidal (electric) $E_{ab}$	Truncated Carroll
Stretched	Unified via sCarrollian structures	Interpolates

This comprehensive structure situates Carrollian gravity as the natural geometric and dynamical framework for describing gravitational physics on null and ultra-relativistic hypersurfaces, both in the classical gravitational sector and as the boundary arena for holographic dualities in asymptotically flat spacetimes. The bifurcated dynamics, extended symmetry algebras, and connections to celestial holography constitute current frontiers in the subject.