Carrollian Fluid Helicity

• Carrollian fluid helicity is a geometric invariant that reinterprets traditional helicity in the ultrarelativistic (c→0) regime using inverse velocity and vorticity from Carrollian structures.
• It is characterized by a degenerate metric, frame fields, and Carroll-covariant derivatives that underpin its role in flat holography and black hole horizon dynamics.
• The invariant preserves hydrodynamic-frame symmetry, offering a novel framework to study twist and topological aspects in non-conventional fluid flows.

Carrollian fluid helicity refers to the geometric and topological characterization of "twist," or handedness, in Carrollian hydrodynamics—the limit of relativistic hydrodynamics as the speed of light approaches zero ($c \rightarrow 0$). In this ultrarelativistic regime, conventional notions of motion and flow cease to exist: macroscopic velocities vanish, and the conventional velocity field plays no role. Instead, the residual kinematic and dynamical information that survives this contraction limit is encoded in geometric fields, emergent "inverse velocity" variables, and the underlying Carrollian structure of the fluid's spacetime substrate. The concept of helicity in this context must therefore be reinterpreted in terms of the degenerate geometry, emergent connections, and the frame fields that characterize the Carrollian fluid, often in direct contrast to the Galilean regime ($c \rightarrow \infty$), where the standard velocity field and its vorticity control the dynamics and topology.

1. Carrollian Hydrodynamics: Framework and Key Geometric Structures

The Carrollian limit is implemented by starting from a relativistic fluid in a Randers–Papapetrou background, with the spacetime geometry given as

$ds^2 = -Q^2(dt - b_i dx^i)^2 + a_{ij} dx^i dx^j,$

where $Q$ is a lapse-like function, $b_i$ is a frame field, and $a_{ij}$ is the spatial metric. In the $c \to 0$ limit, the proper velocity (or local velocity field) vanishes rapidly ($v_i = c^2 B_i + \mathcal{O}(c^4)$), so the only surviving dynamical variable is the "inverse velocity field" $B_i$.

The Carrollian geometry is characterized by

• A degenerate metric, projecting onto the purely spatial sector,
• A Carrollian connection (the Levi–Civita–Carroll connection) enabling well-defined Carroll-covariant spatial and time derivatives,
• Frame data (e.g., $b_i$) encoding inertial structure,
• Geometric two-forms measuring vorticity and inertial rotation.

The equations of motion, derived from the $c\to 0$ limit of the relativistic divergence of the energy–momentum tensor, split into constraints manifestly covariant under Carrollian diffeomorphisms (arbitrary time reparametrizations, local spatial diffeomorphisms).

2. Helicity in Carrollian Fluids: Definition and Geometric Interpretation

Helicity in Galilean and relativistic hydrodynamics is

$$\mathcal{H} = \int u \cdot (\nabla \times u)\, d^3x,$$

measuring the topological linking of vortex lines. In Carrollian fluids, since $u_i \to 0$, this expression becomes trivial, but the physical content re-emerges through the remaining geometric and kinematical variables:

• The inverse velocity field $B_i$,
• The Carrollian vorticity two-form $W_{ij} = \partial_i b_j - \partial_j b_i + (...)$.

A natural Carrollian fluid helicity density is then

$\mathcal{H}_C = \epsilon^{ijk} B_i W_{jk}.$

This object captures the "twisting" of the background geometry and the $B$-field, which encode information about residual rotation and topology even when the ordinary velocity is zero.

For 2D Carrollian fluids (such as conformal fluids on time-dependent surfaces), helicity per se does not generally exist; instead, the dynamical structure can exhibit geometric flow properties governed by quantities like the curvature scalar $K$ and conformally invariant vorticities, leading to Calabi-type flows (e.g., Robinson–Trautman dynamics).

3. Carrollian Vorticity, Inverse Velocity, and Invariance Properties

The Carrollian vorticity arises both from the nontrivial structure of the one-form $b_i$ and from curvature in the background spatial geometry. Crucially, the Levi–Civita–Carroll connection ensures that all relevant derivatives and contractions (including those appearing in helicity) are covariant under local Carrollian boosts and diffeomorphisms.

The inverse velocity field $B_i$ is not a velocity but acts as a residual, inverse kinematical degree of freedom; together with $W_{ij}$ it captures the permitted "rotational structure" of the Carrollian fluid. The resulting helicity invariant is independent of hydrodynamic frame and local boost transformations (i.e., choices of local observers), a direct consequence of the Carrollian limit preserving full hydrodynamic-frame invariance, in contrast to the Galilean case (Petkou et al., 2022).

In 3D, a general expression takes the form: $\mathcal{H}_C = \epsilon^{ijk} B_i D_j B_k,$ where $D_j$ is the Carroll–covariant derivative.

4. Carrollian Helicity and Flat Holography

Carrollian fluids are central in flat holography, appearing naturally as duals to the geometry on null boundaries (future or past null infinity) of asymptotically flat spacetimes (Ciambelli et al., 2018). The Carrollian fluid's data (energy density, pressure, heat current, viscous stress) and the intrinsic Carrollian geometry (spatial metric $a_{ij}$, frame $b_i$, curvature) fully determine the dynamics and reconstruct the spacetime via a resummed derivative expansion.

In this correspondence, the "helicity content" is encoded via duality relations involving dissipative observables (heat currents, viscous stresses) and Cotton descendants (third-derivative geometric objects that encode gravito-magnetic information). Functions like $Q_+ \sim i X_z$ or $E_+ \sim X_{zz}$ (in complex coordinates) serve as complex combinations representing the twist or helicity, providing direct links to gravitational charges and asymptotic symmetries. In examples like conformal Carrollian fluids on a 2D time-dependent geometry, the Robinson–Trautman equation (a Calabi flow) becomes the "holographic image" of conformal Carrollian fluid evolution, with geometric invariants playing the role of topological twistedness when conventional helicity is not available.

5. Comparison with (Non-)Relativistic and Galilean Hydrodynamics

Framework	Velocity Field	Vorticity Definition	Helicity Invariant	Conservation
Galilean	$u_i$	$w_i = \epsilon_{ijk} \partial_j u_k$	$u \cdot w$	Ideal, non-dissipative flow
Relativistic	$u^\mu$	$\omega_{\mu\nu}$	$u \cdot \ast \omega$	Ideal, non-dissipative flow
Carrollian	$u_i \to 0$, $B_i$	$W_{ij}$ (geometry, $b_i$)	$B_i W_{jk} \epsilon^{ijk}$	Frame-invariant, geometric

In the Galilean and relativistic cases, helicity is associated with the transport and conservation of momentum and vorticity. In the Carrollian limit, helicity becomes a "geometric charge," entirely constructed from background Carrollian geometric data and inverse kinematical fields, and its conservation is dictated by geometric symmetries (Carrollian diffeomorphisms).

6. Carrollian Helicity in Horizons, Black Hole Membrane Paradigm, and Symplectic Structure

The hydrodynamics of black hole horizons can be cast as Carrollian fluid equations living on the null boundary. Projecting Einstein's equations onto the horizon yields conservation laws for Carrollian momenta, where the Hajíček one-form $\mathcal{H}_A$ plays the role of fluid momentum density, and its curl $$\omega^\mathcal{H} = \frac{1}{2} q^{AB} \nabla_{[A} \mathcal{H}_{B]}$$ captures the "twisting," i.e., fluid helicity (Donnay et al., 2019, Freidel et al., 2022, Redondo-Yuste et al., 2022, Hüsnügil et al., 27 Aug 2025).

The canonical phase space structure (expressed via a pre-symplectic potential intrinsic to the horizon) and the evolution of canonical charges along the transverse (radial) direction are naturally linked to the spin–2 (shear, helicity) sector. Equilibration of the horizon—when sourced (e.g., by scalar matter, as in (Hüsnügil et al., 27 Aug 2025))—is reflected in the decay and relaxation of the Carrollian fluid's energy, momentum, and helicity content.

7. Mathematical Formulation, Symmetry Structures, and Invariants

• Carrollian helicity density in three spatial dimensions:

$\mathcal{H}_C = \epsilon^{ijk} B_i W_{jk},$

with $W_{jk} = \partial_j b_k - \partial_k b_j +$ (inverse velocity corrections).

• Fluid/gravity dictionary: in horizon dynamics, the canonical momentum and its curl (the vorticity of $\mathcal{H}_A$) encode the dynamical helicity content.
• Covariance: all helicity-like constructs are invariant under Carrollian diffeomorphisms and local boosts, a reflection of the preserved hydrodynamic-frame invariance in Carrollian fluids (Petkou et al., 2022, Freidel et al., 2022).
• Conservation or evolution laws: the helicity (or its analog) may be conserved in non-dissipative sectors (e.g., in the absence of heat currents or geometric expansion) but generally evolves according to the intrinsic geometric and symmetry structure, especially in the presence of sources or horizon dynamics (Freidel et al., 2022, Redondo-Yuste et al., 2022, Hüsnügil et al., 27 Aug 2025).

8. Applications and Physical Relevance

Carrollian fluid helicity is relevant in:

• Flat holography, as boundary correlates of bulk gravitational charges (including soft hair and angular momentum multipoles).
• Hydrodynamics of null surfaces, stretched horizons, and black hole horizon dynamics.
• Theoretical models where "motion" is suppressed but geometric/topological information survives (e.g., ultrarelativistic limits, fractonic states in condensed matter).
• Understanding non-equilibrium relaxation processes in strong-gravity or ultrarelativistic regimes (in horizon relaxation, the decay of helicity marks the return to equilibrium).
• The design of symplectic and canonical phase spaces for quantum gravity on null boundaries, with helicity content serving as a key observable.

Conclusion

Carrollian fluid helicity is a geometric and topological construct encoding the residual “twisting” or rotational content in Carrollian hydrodynamics, intrinsically tied to the degenerate geometry, background frame fields, and “inverse” kinematics that survive in the $c \to 0$ limit. Its mathematical manifestation—typically as an invariant contraction between the inverse velocity and the Carrollian vorticity—serves as a central ingredient in holographic correspondences, characterizes the dynamics of null surfaces and black hole horizons, and provides a unifying topological observable in the absence of conventional motion. Carrollian helicity thus generalizes the familiar fluid helicity in a purely geometric, diffeomorphism-invariant setting and underpins many of the emergent phenomena found in flat-space holography, horizon dynamics, and non-Lorentzian field theories.