Carrollian Viscous Stress Tensor

• Carrollian viscous stress tensor is defined as the c → 0 limit of the relativistic viscous stress tensor, capturing essential dissipative and equilibrium properties on null boundaries.
• It is constructed via dualities with geometric data, such as Cotton tensor descendants, and is tightly constrained by Carrollian geometry and BMS symmetries.
• Its conservation laws and holographic implications enable the reconstruction of Ricci-flat spacetimes and offer insights into universal memory effects in gravitational scattering.

The Carrollian viscous stress tensor is a central object in the geometric and hydrodynamic description of field theories and fluids defined on null boundaries or in the ultra-relativistic (c → 0) limit, such as in asymptotically flat holography and Carrollian conformal field theories. Its structure emerges from contraction limits of relativistic hydrodynamics, inherits dualities with geometric data (such as descendants of the Cotton tensor), and encodes both equilibrium and dissipative (non-equilibrium) features of the theory. The mathematical and physical properties are tightly constrained by the underlying Carrollian geometry, symmetry group (notably the BMS algebra), and the requirements of consistent conservation laws and holographic correspondence.

1. Definition and Carrollian Limit Construction

The Carrollian viscous stress tensor originates as the c → 0 limit of the relativistic viscous stress tensor in hydrodynamics. For a relativistic fluid on a (d+1)-dimensional pseudo-Riemannian manifold, the energy-momentum tensor is decomposed as: $$T_{\mu\nu} = \epsilon\,u_\mu u_\nu + p\,h_{\mu\nu} + \tau_{\mu\nu}$$ where $u_\mu$ is the fluid velocity, $h_{\mu\nu}$ is the projector transverse to $u_\mu$, $p$ is the pressure, and $\tau_{\mu\nu}$ is the symmetric, traceless viscous stress tensor. In the “Carrollian limit,” where $c\to 0$, the structure transmutes:

• The fluid velocity decouples; dynamics reduce to a two-dimensional spatial surface $S$ with Carrollian time $t$.
• The cosmological constant $\Lambda\to 0$ leads to a degenerate metric on the boundary.

The limit produces two Carrollian viscous stress tensors associated with different orders in the expansion parameter $k$ (effectively proportional to $\sqrt{-\Lambda}$): $$\Sigma_{ij} = -\lim_{k \to 0} \tau_{ij}\,,\quad \Xi_{ij} = -\lim_{k \to 0} \frac{1}{k^2} \tau_{ij}$$ with $i,j$ labeling directions on the spatial surface $S$. In many models, only $\Sigma_{ij}$ contributes nontrivially; the higher-order $\Xi_{ij}$ can play a role in more general settings (Ciambelli et al., 2018).

The spatial part of the Carrollian energy-momentum tensor, as constructed from variational principles with respect to the Carrollian geometric data $(\Omega, b_i, a_{ij})$, is decomposed into pressure and viscous (traceless) stress components as: $$\mathcal{A}^{ij} = -\frac{1}{2} (\mathcal{P}\,a^{ij} - \Xi^{ij})$$ where $\Xi^{ij}$ is the traceless Carrollian viscous stress tensor (Ciambelli et al., 2018).

2. Geometric and Duality Structure

The geometric underpinning of the Carrollian viscous stress tensor is provided by the Carrollian structure:

• A degenerate spatial metric $a_{ij}$, Carrollian “clock” function $\Omega$, and frame connection $b_i$ (Ciambelli et al., 2018).
• An Ehresmann connection which splits time and space (Ciambelli et al., 2019).

The viscous tensor is determined (in fluid/gravity holography) by the duality with Cotton-like descendants—third-derivative objects constructed from the Carrollian geometry. Explicitly,

$\Sigma_{ij} \sim \mathrm{dual}\bigl[X_{ij}\bigr]$

where $X_{ij}$ is a symmetric, traceless tensor built from the Carrollian Cotton tensor and its descendants (e.g., via Eqs. (4.11)-(4.12) of (Ciambelli et al., 2018)). This duality is crucial to reconstructing Ricci-flat spacetimes holographically and ensures that dissipation in the boundary theory is entirely fixed by geometric data.

A typical phenomenological form for the Carrollian viscous stress tensor, constructed from geometric scalars and tensors, is: $$\pi_{ij} = -2\eta\,\zeta_{ij} - \zeta_\mathrm{bulk}\,a_{ij}\,\theta$$ where $\zeta_{ij}$ is the shear (trace-free part of the “time” derivative of $a_{ij}$), $\theta$ is the Carrollian expansion, and $\eta$, $\zeta_\mathrm{bulk}$ are shear and bulk viscosities (Ciambelli et al., 2019).

3. Conservation Laws and Hydrodynamic Role

Carrollian energy-momentum conservation is expressed via covariant derivative operators adapted to the Carrollian structure. For the energy density and dissipative (friction) tensors, the conservation equation takes the schematic form: $$-\Omega D_t\varepsilon + D_i Q^i + \Xi^{ij}\,\xi_{ij} = 0$$ where $Q^i$ is the Carrollian heat current and $\xi_{ij}$ denotes the Carrollian shear associated with the expansion of the degenerate metric (Ciambelli et al., 2018).

Covariant conservation follows from Carrollian symmetry: $$(1/\Omega)\,\partial_t \mathcal{E} + \theta \mathcal{E} - (\hat\nabla_i + 2\varphi_i) \mathcal{B}^i - \mathcal{A}^{ij} (1/\Omega) \partial_t a_{ij} = 0$$ and a spatial equation involving the viscous stress (Ciambelli et al., 2018).

These conservation laws, together with the structure of $\Sigma_{ij}$ and $\Xi_{ij}$, govern the possible dissipative dynamics in Carrollian fluids, as well as the evolution of energy and (in higher dimensions or nontrivial boundary geometries) angular momentum and other conserved quantities.

4. Symmetry Constraints and Holographic Implications

In conformal Carrollian field theories (CCFTs) in three dimensions, the viscous stress tensor is heavily constrained by symmetry:

• Local Carrollian and Weyl invariance enforce $T^i_u=0$ and tracelessness $T^i_j = -\frac{1}{2} T^u_u \delta^i_j$ on a flat background (Dutta, 2022).
• The stress-tensor OPE structure implements the infinite-dimensional (BMS-like) algebra of conserved charges, and the algebra closes on the stress-tensor multiplet and its components.

Operator product expansions (OPEs) for the Carrollian stress tensor reflect this structure. For instance: $$T_z(z, \bar{z})\,T_z(\omega, \bar{\omega}) ~ 2\partial_{\omega}\delta^2(z-\omega) T_z(\omega, \bar{\omega}) + \delta^2(z-\omega)\,\partial_\omega T_z(\omega, \bar{\omega})$$ This endows the stress-tensor multiplet with a role analogous to the Virasoro generators in 2D CFTs, but in a BMS or $w_{1+\infty}$ algebraic setting (Saha, 2023, Nguyen et al., 19 Mar 2025).

In the context of flat holography, the Carrollian viscous stress tensor allows the reconstruction of Ricci-flat (Einstein) spacetimes from boundary data, encoding both energy transport and dissipative corrections. The duality with Cotton-like tensors is essential for summing the derivative expansion in the bulk and ensuring the consistency of the boundary/bulk correspondence (Ciambelli et al., 2018).

5. Explicit Realizations: Scalar Fields, Stretched Horizons, and Lower-Dimensional Models

For Carrollian scalar fields, the energy-momentum tensor receives contributions from both “electric” (time-like) and “magnetic” (spatial) terms. Variation with respect to the degenerate metric gives spatial stress components directly related to viscous stress, even in single-particle dynamics (Ciambelli, 2023).

In the geometry of stretched horizons (sCarrollian structures), one defines a generalized news tensor $N_i{}^j$ and constructs the (s)Carrollian stress tensor as: $T_i{}^j = N_i{}^j - (N_k{}^k)\delta_i^j$ The viscous (shear) part is the symmetric, traceless component and directly encodes gravitational radiation and horizon dynamics (Freidel et al., 10 Jun 2024).

For one-dimensional Carrollian fluids (e.g., in the duality with Galilean fluids), the viscous stress tensor appears as a scalar $\sigma$, related via the Carrollian limit of the relativistic stress tensor as $\tau = (\sigma / c^2) + \tilde{\sigma} + ...$. The fluid equations are often formulated as coupled conservation laws for $\sigma$ and the Carrollian velocity $\beta$, with constitutive relations $\varepsilon = \gamma^{-1} \sigma^\gamma$ (Athanasiou et al., 8 Jul 2024, Athanasiou et al., 8 Jul 2024, Petropoulos et al., 8 Jul 2024).

6. Mathematical Properties and Well-Posedness

The evolution equations involving the Carrollian viscous stress tensor display rich mathematical structure:

• The constituent variables (e.g., $\sigma$ for 1D fluids) can exhibit finite-time blow-up (shock formation) for certain initial data, diagnosed via the evolution of Riemann invariants (Athanasiou et al., 8 Jul 2024).
• For particular constitutive laws (e.g., $\gamma=3$), the equations may be written in strict conservative form; global entropy solutions exist in $L^\infty$ under appropriate initial conditions, often established by vanishing viscosity approximations and compensated compactness (Petropoulos et al., 8 Jul 2024).
• The OPE structure in CCFT controls the short-distance expansion of correlators and amplitudes, with the stress-tensor multiplet organizing into indecomposable representations determined by the symmetry algebra (Nguyen et al., 19 Mar 2025).

7. Physical Applications and Holographic Correspondence

The Carrollian viscous stress tensor governs dissipative phenomena in a wide range of contexts:

• In asymptotically flat holography, it encodes the universal memory effects and soft graviton theorems; its Ward identities are equivalent to these low-energy limits in gravitational scattering (Saha, 2023, Ruzziconi et al., 7 Nov 2024).
• It is essential for describing regimes such as the large-rapidity expansion (Bjorken/Gubser flow) in heavy-ion collisions, where the Carrollian limit refines the mapping between QGP hydrodynamics and boundary theory, even when including subleading corrections in $c$ (Kolekar et al., 27 Sep 2024).
• The connection with celestial operators and the localization of the stress tensor enables a unified description of soft theorems, celestial holography, and asymptotic symmetries (Bagchi et al., 10 Aug 2024).
• In black hole horizon dynamics, the sCarrollian stress tensor provides a hydrodynamical description of gravitational radiation and horizon response (Freidel et al., 10 Jun 2024).

The unique feature of the Carrollian viscous stress tensor is that, unlike in standard relativistic fluids, dissipation is fully determined (for example, via duality with geometric data) by the Carrollian geometry on the boundary. This ensures both the geometric (fluid/gravity) and algebraic (symmetry/OPE) perspectives are consistent and robust for applications in both mathematical physics and holography.