Carrollian Hydrodynamics

• Carrollian hydrodynamics is a framework that describes fluid dynamics in the ultra‐relativistic c→0 limit, where time and space decouple.
• It emerges from the c→0 contraction of the Poincaré group, employing specialized covariant derivatives and modified energy-momentum tensors to capture dissipative and geometric effects.
• The theory underpins diverse applications including holography, black hole horizon dynamics, heavy-ion collisions, and condensed matter systems, unifying them through Carroll symmetry.

Carrollian hydrodynamics is the framework for describing fluid dynamics in the ultra-relativistic regime where the speed of light $c$ tends to zero. This contraction limit of the Poincaré group yields the Carroll symmetry algebra, in which time and space decouple and causal propagation along spatial directions is suppressed. Carrollian and conformal Carrollian symmetries, initially regarded as mathematical curiosities, have now emerged as central organizing principles in a diverse array of physical systems, prominently including asymptotically flat holography, black-hole horizon dynamics, heavy-ion collisions, condensed matter systems (e.g., fracton phases, flat bands), and certain integrable models. The theoretical structure, applications, and dualities of Carrollian hydrodynamics are therefore of central interest in mathematical physics and high-energy theory.

1. Carrollian Symmetry and the $c \to 0$ Limit

Carrollian symmetry emerges from the singular limit $c \to 0$ of the relativistic symmetry group. The resulting algebra, sometimes with conformal enhancement (“conformal Carroll group”), is characterized by:

• Degenerate “Carrollian” spacetimes, admitting an absolute spatial foliation with vanishing lightcones.
• Carrollian diffeomorphisms: spatial coordinates transform among themselves, $x' = x'(x)$, while the time coordinate transforms as $t' = t'(t, x)$, reflecting a lack of boost mixing between space and time compared to the Galilean case ($c \to \infty$) (Ciambelli et al., 2018).
• Invariance under spatial isometries and “Carrollian boosts” $B_i = -x_i \partial_t$ (Bagchi et al., 2023).

Physically, this means that propagation along spatial directions is frozen; motion or transport must be carried either by collective fields (hydrodynamics), field-theoretic degrees of freedom, or via background geometry. Despite this “ultralocality,” Carrollian systems can exhibit rich dynamics, particularly when embedded on curved backgrounds, null boundaries, or in nontrivial phase spaces.

2. Fundamentals and Equations of Carrollian Hydrodynamics

Carrollian hydrodynamics can be constructed both as the $c \to 0$ limit of relativistic hydrodynamics and directly from symmetry principles. The core steps are:

• Start from the relativistic energy-momentum tensor

$$T^{\mu\nu} = (\epsilon + p) \frac{u^\mu u^\nu}{c^2} + p\,g^{\mu\nu} + \tau^{\mu\nu} + \ldots,$$

where $\epsilon$ is the energy density, $p$ the pressure, and $\tau^{\mu\nu}$ dissipative corrections.

• In a manifestly Carroll-covariant coordinate system (Papapetrou–Randers metric),

$$ds^2 = -c^2 \Omega dt^2 - 2c\,b_i dt\,dx^i + a_{ij} dx^i dx^j,$$

expand all fluid variables in powers of $c$ (Ciambelli et al., 2018, Boer et al., 2023).

• The velocity field must scale as $v^i \sim c\,\Omega\,\beta^i$ to avoid divergences as $c \to 0$, with $\beta^i$ the Carrollian “inverse velocity.”

The resulting hydrodynamics features:

• Scalar and vector conservation laws:

$$\begin{align*} &\text{(a) Energy:} \quad c\,\nabla_\mu T^{\mu 0} = F + E + \mathcal{O}(c^2),\ &\text{(b) Momentum:} \quad \nabla_\mu T^{\mu i} = 2H + G + \mathcal{O}(c). \end{align*}$$

• Dissipative structures are “doubled”: two scalars and two vectors appear, typically interpreted as two sectors of heat currents and viscous stresses.
• Carrollian hydrodynamic equations are covariant under Carrollian diffeomorphisms, often requiring the construction of specialized covariant derivatives (“Levi-Civita–Carroll” or Weyl–Carrollian connections) to preserve symmetry (Freidel et al., 2022).

Table 1: Key Ingredients of Carrollian Hydrodynamics

Variable	Origin/Interpretation	Transformation Property
$\epsilon$, $p$	Carrollian energy density, pressure	Weyl weight 3 (conformal)
$\beta^i$	Carroll inverse velocity	Spatial vector
$Q_C^i$, $T^i$	Heat currents	Weyl weight 2
$\Sigma_C^{ij}$, $E^{ij}$	Viscous stresses	Weyl weight 1, tracefree

3. Carrollian Hydrodynamics from Symmetry and Effective Field Theory

A comprehensive, symmetry-based construction incorporates both hydrodynamics and spontaneous symmetry breaking:

• The spontaneous breaking of boost symmetry (present in any thermal rest frame) in Carrollian fluids implies a true dynamical Goldstone mode (the “boost Goldstone”) which cannot be removed by local redefinitions. This Goldstone field couples directly to the fluid velocity, yielding $u^\mu = -\ell^\mu$ in equilibrium (Armas et al., 2023).
• The equilibrium partition function and associated currents must be constructed to be invariant under Carrollian boosts and Stueckelberg symmetries, leading to an extended set of hydrostatic and dissipative transport coefficients (up to 12 first-order dissipative terms).
• The temperature and entropy must be defined in a manner that incorporates the Goldstone field to maintain invariance—a key distinction from conventional relativistic fluids, where a globally defined time direction ensures natural invariants.
• Carrollian effective actions, even for a free scalar field, may combine “electric” (time-kinetic) and “magnetic” (spatial) sectors. This allows for nontrivial classical evolution:

$$S[\phi] = g_2 \int d\mu\, [ g_0\, (v^\mu D_\mu\phi)^2 - h^{\mu\nu} D_\mu\phi D_\nu\phi ],$$

with $g_0$ controlling the effective propagation “speed.” Such actions reveal that even single-particle Carrollian dynamics need not be frozen (Ciambelli, 2023).

4. Holography, Black Hole Horizons, and Carrollian Fluids

Carrollian hydrodynamics naturally underpins asymptotically flat holography, the membrane paradigm, and the dynamics of gravitational horizons:

• In the flat limit of AdS/CFT, the holographic boundary theory becomes a co-dimension one Carrollian Conformal Field Theory (CCFT), living on $\mathbb{R}\times S^2$ at null infinity (Bagchi et al., 19 Jun 2025).
• The “modified Mellin transform” approach restores full Poincaré invariance to celestial correlators by explicitly treating the null (Carrollian) time as a dynamical boundary coordinate.
• The gravity/fluid correspondence for null event horizons and stretched horizons is now framed in Carrollian variables. Projecting the bulk Einstein equations onto a null or stretched horizon, one obtains constraints and evolution equations matching those of Carrollian hydrodynamics. The energy density corresponds to the expansion, the Carrollian pressure to minus the surface gravity, and the Hájíček one-form to Carrollian fluid momentum (Freidel et al., 2022, Redondo-Yuste et al., 2022, Freidel et al., 10 Jun 2024, Hüsnügil et al., 27 Aug 2025).
• In the presence of external perturbations (e.g., scalar field infall), the relaxation of the horizon maps precisely to the equilibration of the dual Carrollian fluid, both being governed by background geometric evolution (e.g., expansion, shear).

5. Phenomenological Applications: Heavy-Ion Collisions and Condensed Matter

Carrollian hydrodynamics provides a universal framework for several classes of real-world systems:

• Heavy-Ion Collisions: Carrollian symmetry is manifest in the dynamics near light speed, central to the analysis of Bjorken and Gubser flows. These flows are precisely recovered as solutions to Carrollian hydrodynamics on appropriate Cartan–Carrollian backgrounds (Bagchi et al., 2023, Bagchi et al., 2023, Kolekar et al., 27 Sep 2024). Boost invariance, rapidity-independent evolution, and dissipative corrections all have Carrollian geometric interpretations. Including subleading $c^2$ corrections enables modeling of violations of boost invariance and other observed phenomena.
• Black Hole Horizons: The membrane paradigm of horizon dynamics is properly described using Carrollian hydrodynamic equations, rather than the usual Navier-Stokes equations, with full Noether charge structure and symplectic geometry mapped to Carrollian fluid phase space (Freidel et al., 2022, Redondo-Yuste et al., 2022).
• Condensed Matter Systems: Carrollian symmetry (and its variants) arises in the theory of fracton phases, flat-band systems, and in hydrodynamics of shallow water waves. The duality between Carrollian and Galilean fluids in one dimension, enabled by the isomorphism of Carroll and Galilei algebras, exposes new correspondences and allows mapping equilibrium variables to dissipative observables (Athanasiou et al., 8 Jul 2024, Bagchi et al., 6 Nov 2024).
• Flat-Band and Shallow Water Hydrodynamics: Recent results demonstrate a direct mapping between flat-band wave solutions and sectors (“electric” and “magnetic”) of Carrollian electrodynamics, revealing Carrollian symmetry as an organizing principle in shallow water dynamics (Bagchi et al., 6 Nov 2024).

6. Mathematical Structure and Dualities

The mathematical architecture of Carrollian hydrodynamics is enriched by dualities and structural features:

• Carrollian–Galilean Duality: In $1+1$ dimensions, Carrollian and Galilean algebras are isomorphic, leading to a precise duality between one-dimensional Carrollian and Galilean fluids (Athanasiou et al., 8 Jul 2024). Hydrodynamic equations for stress and velocity in Carrollian fluids can be mapped to density and velocity equations in their Galilean counterparts under exchange of “longitudinal” and “transverse” variables, with equilibrium and out-of-equilibrium roles reversed.
• Frame Invariance: Carrollian hydrodynamics retains hydrodynamic-frame invariance (the fluid velocity can be redefined locally with no change to observables), in contrast to the Galilean case where imposing standard constraints (e.g., vanishing loss/gain current) breaks this invariance (Petkou et al., 2022).
• Shock Formation and Well-Posedness: Systematic mathematical analysis in recent work has established rigorous global existence results for entropy ($L^\infty$) solutions to the isentropic Carrollian fluid equations (e.g., with polytropic constitutive law $\varepsilon = (1/\gamma)\sigma^\gamma$), together with criteria for $C^1$ blow-up and entropy-based compensated compactness techniques (Athanasiou et al., 8 Jul 2024, Petropoulos et al., 8 Jul 2024).
• Limitations of Strict Carroll Limit: The strict $c \to 0$ limit can lead to pathological partition functions and degeneracies, but a “Carrollian regime”—retaining leading corrections in $c$—permits well-defined thermodynamics and sensible hydrodynamic theories, e.g., with an equation of state $w = -1$ for Carrollian massless gases (Boer et al., 2023).

7. Outlook and Future Directions

Carrollian hydrodynamics continues to expand its reach:

• New geometric frameworks (“ruled stretched Carrollian structures”) systematically unify null and timelike horizon treatments, providing the tools for direct quantization of Carrollian fluid phase space and gravitational charges (Freidel et al., 10 Jun 2024).
• Holographic duality suggests Carrollian CFTs as natural duals for flat space physics, fully realizing Poincaré (BMS) invariance at null infinity via modified operators dependent on Carrollian time (Bagchi et al., 19 Jun 2025).
• Dualities and mappings between Carrollian hydrodynamics, Galilean hydrodynamics, fracton phases, and even shallow water waves indicate that Carrollian symmetry is a robust and widely-applicable concept, not a mere mathematical curiosity but a “kaleidoscope” (sensu (Bagchi et al., 19 Jun 2025)) for unifying seemingly disparate physical systems.

Carrollian hydrodynamics thus underlies a broad range of physical phenomena, connecting flat-space holography, gravitational physics, ultra-relativistic flows, and condensed matter models through its unique structural and symmetry properties.