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Carrollian hydrodynamics from symmetries

Laurent Freidel, Puttarak Jai-akson

TL;DR

This work advances Carrollian hydrodynamics by embedding it in a rich geometric framework and by deriving its conservation laws from enhanced near-Carrollian symmetries. By incorporating a fluid velocity field V^A and subleading sphere metric data λ_{AB} as essential phase-space variables, the authors obtain the complete Carrollian hydrodynamic equations via a near-Carrollian diffeomorphism and construct a finite Carrollian fluid action whose variations reproduce these laws. The approach unifies previous results (which lacked the extra phase-space data) and provides explicit Noether charges and their time evolution within this extended symmetry context. The work also clarifies the role of the Randers-Papapetrou metric and Ehresmann connection in organizing Carrollian geometry and sets the stage for applications to null boundaries, horizons, and holography. Future directions include higher-order corrections, thermodynamics of Carrollian fluids, and connections to stretched-horizon membranes."

Abstract

In this work, we revisit Carrollian hydrodynamics, a type of non-Lorentzian hydrodynamics which has recently gained increasing attentions due to its underlying connection with dynamics of spacetime near null boundaries, and we aim at exploring symmetries associated with conservation laws of Carrollian fluids. With an elaborate construction of Carroll geometries, we generalize the Randers-Papapetrou metric by incorporating the fluid velocity field and the sub-leading components of the metric into our considerations and we argue that these two additional fields are compulsory phase space variables in the derivation of Carrollian hydrodynamics from symmetries. We then present a new notion of symmetry, called the near-Carrollian diffeomorphism, and demonstrate that this symmetry consistently yields a complete set of Carrollian hydrodynamic equations. Furthermore, due to the presence of the new phase space fields, our results thus generalize those already presented in the previous literatures. Lastly, the Noether charges associated with the near-Carrollian diffeomorphism and their time evolutions are also discussed.

Carrollian hydrodynamics from symmetries

TL;DR

This work advances Carrollian hydrodynamics by embedding it in a rich geometric framework and by deriving its conservation laws from enhanced near-Carrollian symmetries. By incorporating a fluid velocity field V^A and subleading sphere metric data λ_{AB} as essential phase-space variables, the authors obtain the complete Carrollian hydrodynamic equations via a near-Carrollian diffeomorphism and construct a finite Carrollian fluid action whose variations reproduce these laws. The approach unifies previous results (which lacked the extra phase-space data) and provides explicit Noether charges and their time evolution within this extended symmetry context. The work also clarifies the role of the Randers-Papapetrou metric and Ehresmann connection in organizing Carrollian geometry and sets the stage for applications to null boundaries, horizons, and holography. Future directions include higher-order corrections, thermodynamics of Carrollian fluids, and connections to stretched-horizon membranes."

Abstract

In this work, we revisit Carrollian hydrodynamics, a type of non-Lorentzian hydrodynamics which has recently gained increasing attentions due to its underlying connection with dynamics of spacetime near null boundaries, and we aim at exploring symmetries associated with conservation laws of Carrollian fluids. With an elaborate construction of Carroll geometries, we generalize the Randers-Papapetrou metric by incorporating the fluid velocity field and the sub-leading components of the metric into our considerations and we argue that these two additional fields are compulsory phase space variables in the derivation of Carrollian hydrodynamics from symmetries. We then present a new notion of symmetry, called the near-Carrollian diffeomorphism, and demonstrate that this symmetry consistently yields a complete set of Carrollian hydrodynamic equations. Furthermore, due to the presence of the new phase space fields, our results thus generalize those already presented in the previous literatures. Lastly, the Noether charges associated with the near-Carrollian diffeomorphism and their time evolutions are also discussed.
Paper Structure (21 sections, 113 equations, 1 figure)

This paper contains 21 sections, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Carroll Structures
  3. Acceleration, Vorticity, and Expansion
  4. Horizontal Covariant Derivative
  5. Adapted coordinates for the Carroll structure
  6. Carrollian transformations
  7. Carrollian Hydrodynamics
  8. Metric on $H$
  9. Covariant derivative
  10. Carrollian Hydrodynamics
  11. Hydrodynamics from Symmetries
  12. The Action for Carrollian Fluid
  13. Near-Carrollian Diffeomorphism
  14. Hydrodynamics from Near-Carrollian Diffeomorphism
  15. Conclusion
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The space $H$ endowed with the Carroll structure. The general coordinates are $x^i = (u,y^A)$ where the surfaces at the cuts $u = \text{constant}$ are identified with the sphere $S$. The vertical vector $\ell$ and the horizontal vector $e_A$ span the tangent space $TH$