Action principle for the equation-level Carrollian electric and magnetic limits of general relativity

The Carrollian ($c\to 0$) limit of general relativity can be taken at the level of the field equations or at the level of the action, and the two routes need not agree. Patil and Shankaranarayanan (2025) contracted the linearized Einstein equations on a Friedmann-Lemaitre-Robertson-Walker background using the 1+3 covariant formalism and found that the system bifurcates into a static, ultralocal electric limit and a magnetostatic magnetic limit governed by the magnetic part of the Weyl tensor. They showed that the electric sector follows, in the ultralocal regime, from the known electric Carroll action, but their magnetic equations are not the equations of motion of any existing action-level magnetic Carrollian gravity, and they leave the construction of a consistent action principle for these limits explicitly open, recalling the well-established fact that for non-relativistic and Carrollian limits taking the limit and varying the action do not in general commute. This problem asks for a Carroll-invariant action whose Euler-Lagrange equations reproduce the equation-level Carrollian magnetic (and electric) limits of general relativity, or a proof that no such action exists, together with a precise account of how the equation-level contraction relates to the known action-level magnetic Carrollian theories.

Background

The Carrollian limit of a relativistic theory is the ultrarelativistic contraction $c\to 0$ in which light cones close up onto lines: the resulting Carrollian geometry carries a degenerate metric with a preferred vector field along which signals propagate, and it is the intrinsic geometry of any null hypersurface, such as a black-hole event horizon or null infinity. Carroll boosts are the transformations that preserve this degenerate structure; unlike Lorentz boosts, they act trivially on the preferred direction and shift spatial coordinates. Because questions about horizon dynamics, gravitational memory, and flat-space holography are naturally posed on null surfaces, deriving the Carrollian limit of general relativity (GR) itself has become a central problem of non-Lorentzian gravity. There are two inequivalent routes: contract the field equations, or contract the action. That these need not agree is classical knowledge — in the Galilean ($c\to\infty$) limit, Newton-Cartan gravity does not follow from naively contracting the Einstein-Hilbert action. On the action side, Henneaux and Salgado-Rebolledo (2021) implemented the Carrollian contraction on the Hamiltonian form of Lorentz-invariant theories and showed that, exactly as in Galilean electromagnetism (Levy-Leblond; Le Bellac and Levy-Leblond), there are two inequivalent contractions — an electric one and a magnetic one — obtained by different consistent rescalings of the canonical variables; for $p$-form gauge fields the manifestly covariant action of the magnetic contraction requires an additional pure gauge field whose elimination modifies the Carroll transformations, and for gravity the two contractions were given in Hamiltonian form. Campoleoni, Henneaux, Pekar, Perez and Salgado-Rebolledo (2022) proved that this magnetic Hamiltonian contraction of Einstein gravity is equivalent to a theory built by gauging the Carroll algebra (a construction pioneered by Hartong in 2015). Separately, Hansen, Obers, Oling and Sogaard (2022) expanded the Einstein-Hilbert action in powers of the speed of light: the leading-order action is the electric (ultralocal) Carrollian theory, and a truncation of the next-to-leading-order action corresponds to a magnetic Carroll limit, in which they constructed a Carrollian analogue of the Schwarzschild solution; Musaeus, Obers and Oling (2024) gave a Palatini-type formulation of these expansions. First-order (Cartan-like) actions for electric Carrollian gravity were derived from a smooth limit of the Einstein-Cartan action by Pekar, Perez and Salgado-Rebolledo (2024), and Barbero G., Margalef-Bentabol, Varo and Villasenor (2026) performed a full Hamiltonian constraint analysis of the most general Carroll-invariant Lagrangian descending from the Holst action — the first-order form of the GR action plus a duality-antisymmetric term weighted by the Immirzi parameter, a free coupling that drops out of the classical equations of motion — introducing Ashtekar-like variables (self-dual connection variables of the kind used in loop quantum gravity) for the magnetic Carrollian regime. On the equation side, Patil and Shankaranarayanan (2025) used the 1+3 covariant formalism — the decomposition of spacetime relative to a timelike observer congruence $u^a$, developed by Ellis and collaborators, in which the Weyl curvature tensor splits into its electric part $E_{ab}=C_{acbd}u^c u^d$ (the tidal-force field) and its magnetic part $H_{ab}$ (built by contracting the Weyl tensor with $u^a$ and the spatial Levi-Civita tensor; it encodes frame-dragging and the complementary radiative degrees of freedom of gravitational waves). Studying how linear tensor perturbations on a Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological background transform under Carroll boosts, they found that the full linearized Einstein system is not Carrollian-invariant: it bifurcates into a Carrollian electric limit (a static, ultralocal theory of $E_{ab}$, of Belinski-Khalatnikov-Lifshitz or silent-universe type, where spatial gradients are negligible) and a Carrollian magnetic limit whose invariant form is the magnetostatic system $\tilde\nabla_b H^{ab}=0$, $\dot H^{ab}=0$, $(\mathrm{curl}\,H)^{ab}=\frac12(\mu+p)\sigma^{ab}$, $E^{ab}=0$, where $\tilde\nabla$ is the spatial covariant derivative, an overdot is the derivative along the observer congruence, $\sigma_{ab}$ is its shear, and $\mu,p$ are matter density and pressure. In their Appendix C they verified that the electric equations do follow, in the ultralocal/silent regime, from the electric Carroll action $S_E=\int d^Dx\,\Omega\,(K_{ab}K^{ab}-K^2)$ of Henneaux and Salgado-Rebolledo, where $K_{ab}$ is the extrinsic curvature of the spatial slices and $\Omega$ the measure. However, their magnetic equations do not coincide with the equations of motion of the magnetic Hamiltonian theory of that same reference, and they state that deriving a consistent action principle for these limits is under investigation, recalling that for non-relativistic and Carrollian limits taking the limit and varying the action do not in general commute. The open problem is therefore the variational completion of the equation-level Carrollian limits of GR and the reconciliation of the two contraction routes.

Problem Statement

Construct a Carroll-invariant action principle whose Euler-Lagrange equations reproduce the equation-level Carrollian limits of general relativity obtained by covariant 1+3 contraction — in particular the Carrollian magnetic system of Patil and Shankaranarayanan (arXiv:2509.07601, Eq. (109)): $\tilde\nabla_b H^{ab}=0$, $\dot H^{ab}=0$, $(\mathrm{curl}\,H)^{ab}=\frac12(\mu+p)\sigma^{ab}$, $E^{ab}=0$ — or prove that no such action exists within a precisely stated class of local Carroll-invariant Lagrangians; the class invoked in a no-go must be the maximal class of local Carroll-invariant Lagrangians built from the 1+3 variables and their derivatives to all finite orders, modulo total derivatives and field redefinitions — so that a no-go cannot rest on an artificially narrow choice — and must contain, up to the field redefinitions, truncations, or additional pure-gauge fields that cast them in the same variables, both the known action-level magnetic Carrollian gravities named below and the spatial-derivative field content from which $H_{ab}$ is built in the 1+3 formalism. In either case, establish the precise relation to the known action-level magnetic Carrollian gravities (the Hamiltonian contraction of Henneaux-Salgado-Rebolledo, the Carroll-algebra gauging of Campoleoni et al., and the magnetic truncation of the small-speed expansion of Hansen et al.): field redefinitions, truncations, additional pure-gauge fields, or a proof of inequivalence. The electric Carrollian limit is already reproduced, in the ultralocal/silent regime, by the known electric Carroll action $S_E$ of Henneaux and Salgado-Rebolledo, and is required only as a consistency check; the action principle sought here is the one for the magnetic system.

References

1. Tanmay Patil and S. Shankaranarayanan, Covariant Carrollian Electric and Magnetic Limits of General Relativity, JHEP 01 (2026) 096, arXiv:2509.07601, DOI 10.1007/JHEP01(2026)096, https://arxiv.org/abs/2509.07601
2. Marc Henneaux and Patricio Salgado-Rebolledo, Carroll contractions of Lorentz-invariant theories, JHEP 11 (2021) 180, arXiv:2109.06708, DOI 10.1007/JHEP11(2021)180, https://arxiv.org/abs/2109.06708
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