Carroll Hydrodynamics with Spin

• Carroll hydrodynamics with spin is a theoretical framework describing fluid flow in the ultrarelativistic (c→0) limit with persistent spin degrees of freedom.
• It utilizes the pre-ultralocal parametrization to transition from Lorentzian to Carrollian geometry, introducing degenerate spatial metrics and nontrivial spin dynamics.
• The framework maps to boost-invariant flows like Bjorken and Gubser flows, offering analytic insights into spin polarization in quark–gluon plasma.

Carroll hydrodynamics with spin is a framework describing fluid dynamics in the Carrollian regime—the ultrarelativistic limit where the speed of light $c$ approaches zero—with explicit inclusion of a spin current. This approach emerges as the $c\to0$ limit of relativistic spin hydrodynamics, yielding a degenerate geometric and dynamical structure in which spatial motion is frozen yet spin degrees of freedom persist and evolve nontrivially. Recent developments (Shukla et al., 21 Jan 2026) provide a covariant formalism for constructing Carroll hydrodynamics with spin, clarify its geometric foundations via the pre-ultralocal (PUL) split, and demonstrate its mapping to boost-invariant flows relevant for quark–gluon plasma phenomenology.

1. Pre-Ultralocal Parametrization of Carrollian Geometry

The pre-ultralocal (PUL) parametrization is a manifestly covariant scheme for taking the $c\to0$ Carrollian limit of Lorentzian geometry. The spacetime vielbein $E_\mu{}^A$ is decomposed into a timelike component scaled by $c$ and spatial components: $$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$ with tangent-space indices $A=(0,a)$. The metric and its inverse become

$$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$

The spatial projector is $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$ and $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$. Orthogonality and completeness conditions enforce

$c\to0$0

The Carrollian limit $c\to0$1 yields:

• A degenerate spatial metric $c\to0$2 of signature $c\to0$3,
• Kernel generator $c\to0$4 with $c\to0$5,
• A clock form $c\to0$6 satisfying $c\to0$7.

Local Lorentz boosts reduce to local Carroll boosts: $c\to0$8 with $c\to0$9. Carroll-invariant constructs must be built from $c\to0$0, $c\to0$1, and their derivatives.

2. Relativistic Spin Hydrodynamics and Carroll Expansion

Relativistic ideal spin hydrodynamics prescribes conservation of stress tensor and spin current: $c\to0$2 with constitutive relations

$c\to0$3

where $c\to0$4 projects orthogonal to the four-velocity $c\to0$5 ($c\to0$6), and $c\to0$7 is the antisymmetric spin-polarization tensor.

The “electric/magnetic” (EM) split is

$c\to0$8

with $c\to0$9.

Expanding in PUL variables and powers of $E_\mu{}^A$0, the fluid velocity is

$E_\mu{}^A$1

with $E_\mu{}^A$2 as $E_\mu{}^A$3, subject to

$E_\mu{}^A$4

This leaves three Carrollian velocity components. Thermodynamic quantities and spin-tensors expand regularly: $E_\mu{}^A$5

$E_\mu{}^A$6

The Carrollian spin density is a spatial tensor

$E_\mu{}^A$7

The energy-momentum tensor and spin current decompose as

$E_\mu{}^A$8

with leading term

$E_\mu{}^A$9

In the Carrollian limit,

$c$0

For the spin current: $c$1 yielding the Carroll spin density

$c$2

where $c$3 and $c$4 are spatial vectors and $c$5 is the Carrollian vierbein measure.

3. Carrollian Equations of Motion with Spin and Constitutive Extension

The equations of motion project the energy-momentum conservation along $c$6 and $c$7: $c$8

$c$9

with Carroll-compatible connection $$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$0 ($$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$1, $$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$2), Carroll extrinsic curvature $$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$3, and acceleration $$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$4.

The spin-current dynamics in the Carroll limit is: $$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$5 or equivalently, for the ideal Carroll spin current $$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$6,

$$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$7

Spin non-conservation is sourced by torsion in the Carroll connection,

$$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$8

in contrast to the torsion-free Levi–Civita connection.

For a conformal parent theory ($$E_\mu{}^A = (c\,L_\mu ,\, E_\mu{}^a), \quad E^\mu{}_A = \left(-\tfrac{1}{c}\,K^\mu ,\, E^\mu{}_a \right),$$9, $A=(0,a)$0), in the Carroll limit the equation of state becomes $A=(0,a)$1, with the spin-trace constraint: $A=(0,a)$2

At first order in derivatives, Carroll-invariant terms may be constructed from $A=(0,a)$3, $A=(0,a)$4, $A=(0,a)$5, $A=(0,a)$6, $A=(0,a)$7, $A=(0,a)$8, $A=(0,a)$9, and their Carroll-covariant derivatives, weighted by transport coefficients. Notably:

• Spin-shear viscosity $$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$0 multiplies $$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$1,
• Spin-diffusion $$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$2 multiplies $$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$3.

Schematically,

$$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$4

$$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$5

4. Mapping Carrollian Hydrodynamics with Spin to Boost-Invariant Flows

Boost-invariant hydrodynamic flows for ultrarelativistic fluids—Bjorken and Gubser flows—are realized as special solutions of Carroll hydrodynamics in suitable Carrollian geometries, now extended to include spin.

Bjorken Flow with Spin

In Milne coordinates $$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$6, the metric is $$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$7. Symmetries impose $$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$8, so in PUL variables,

$$g_{\mu\nu} = -c^2 L_\mu L_\nu + H_{\mu\nu}, \quad g^{\mu\nu} = -\tfrac{1}{c^2} K^\mu K^\nu + H^{\mu\nu}.$$9

The Carroll fluid equations reduce to

$H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$0

which matches the Bjorken energy loss law.

For spin, the only compatible longitudinal unit vector is $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$1 and similarly for $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$2, with scalar functions $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$3, $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$4: $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$5 and the Carroll spin equation

$H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$6

has solutions $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$7, $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$8.

Gubser Flow with Spin

Coordinates $H_{\mu\nu} = E_\mu{}^a E_\nu{}^b\,\delta_{ab}$9 on global $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$0 enable $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$1 invariance and boosts in $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$2: $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$3 with $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$4 and Carroll data

$H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$5

The equations become

$H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$6

with solution $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$7. For the spin current, the relevant unit vector is $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$8, and the Carroll spin equation prescribes $H^{\mu\nu} = E^\mu{}_a E^\nu{}_b\,\delta^{ab}$9: $c\to0$00 yielding $c\to0$01, $c\to0$02.

5. Distinctive Properties and Phenomenological Implications

Several novel features arise in Carroll hydrodynamics with spin:

• Intrinsic torsion and spin non-conservation: The compatible Carroll connection possesses torsion proportional to $c\to0$03, directly sourcing spin non-conservation terms. This sharply contrasts with the symmetric Levi–Civita connection of relativistic hydrodynamics.
• Two classes of Carroll fluids: The generating-functional approach (Armas–Jain–Jensen) shows that inclusion of spin and Carroll Goldstone modes yields two inequivalent Carroll fluids: standard $c\to0$04 limit and a distinct class with $c\to0$05. Coupling of spin in the latter remains an open question.
• Ultralocality and fracton analogies: The $c\to0$06 collapse of lightcones enforces ultralocality—suppression of spatial dynamics. However, nontrivial extrinsic curvature $c\to0$07 and Carroll acceleration $c\to0$08 encode geometric memory, giving the spin sector features reminiscent of fractons: internal angular momentum evolution without net transport.
• Applications to polarized quark–gluon plasma (QGP): The Carroll mapping for Bjorken and Gubser flows with spin yields analytic templates for early-time spin polarization, relevant for off-central heavy-ion collisions. The characteristic $c\to0$09 (Bjorken) and $c\to0$10 (Gubser) decay laws for spin polarization provide benchmarks for simulations incorporating additional effects such as shear and vorticity.

In summary, Carroll hydrodynamics with spin is obtained by systematically expanding relativistic spin hydrodynamics in the $c\to0$11 regime, utilizing the PUL split, and projecting conservation laws onto Carrollian data $c\to0$12. The incorporation of spin enriches both the mathematical structure (via torsion-induced non-conservation) and phenomenological potential (notably in QGP and Carrollian condensed-matter analogs) (Shukla et al., 21 Jan 2026).