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Carrollian hydrodynamics and symplectic structure on stretched horizons

Laurent Freidel, Puttarak Jai-akson

TL;DR

The paper develops a unified, non-singular geometric framework for treating stretched horizons and null boundaries via the rigging technique, enabling a Carrollian fluid interpretation of horizon dynamics at finite distance from the true null boundary.By constructing a rigged metric and a Carrollian intrinsic structure on the stretched horizon, it shows that Einstein equations projected onto the horizon are precisely the Carrollian conservation laws for a horizon fluid with energy density $\mathscr{E}$, pressure $\mathscr{P}$, momentum density $\pi_A$, heat current $\mathscr{J}^A$, and viscous stress $\mathscr{T}_{AB}$.A gravitational dictionary connects the extrinsic geometry (Weingarten tensor and related scalars) to Carrollian fluid variables, and the pre-symplectic potential is expressed in terms of Carrollian conjugate pairs, revealing a Carrollian phase-space structure for gravity on the horizon and yielding Noether charges associated with tangential symmetries.The results provide a robust bridge between the membrane paradigm and Carrollian hydrodynamics, offering a platform for horizon thermodynamics, near-horizon expansions, and potential extensions to null infinity and holographic contexts.

Abstract

The membrane paradigm displays underlying connections between a timelike stretched horizon and a null boundary (such as a black hole horizon) and bridges the gravitational dynamics of the horizon with fluid dynamics. In this work, we revisit the membrane viewpoint of a finite distance null boundary and present a unified geometrical treatment to the stretched horizon and the null boundary based on the rigging technique of hypersurfaces. This allows us to provide a unified geometrical description of null and timelike hypersurfaces, which resolves the singularity of the null limit appearing in the conventional stretched horizon description. We also extend the Carrollian fluid picture and the geometrical Carrollian description of the null horizon, which have been recently argued to be the correct fluid picture of the null boundary, to the stretched horizon. To this end, we draw a dictionary between gravitational degrees of freedom on the stretched horizon and the Carrollian fluid quantities and show that Einstein's equations projected onto the horizon are the Carrollian hydrodynamic conservation laws. Lastly, we report that the gravitational pre-symplectic potential of the stretched horizon can be expressed in terms of conjugate variables of Carrollian fluids and also derive the Carrollian conservation laws and the corresponding Noether charges from symmetries.

Carrollian hydrodynamics and symplectic structure on stretched horizons

TL;DR

The paper develops a unified, non-singular geometric framework for treating stretched horizons and null boundaries via the rigging technique, enabling a Carrollian fluid interpretation of horizon dynamics at finite distance from the true null boundary.By constructing a rigged metric and a Carrollian intrinsic structure on the stretched horizon, it shows that Einstein equations projected onto the horizon are precisely the Carrollian conservation laws for a horizon fluid with energy density $\mathscr{E}$, pressure $\mathscr{P}$, momentum density $\pi_A$, heat current $\mathscr{J}^A$, and viscous stress $\mathscr{T}_{AB}$.A gravitational dictionary connects the extrinsic geometry (Weingarten tensor and related scalars) to Carrollian fluid variables, and the pre-symplectic potential is expressed in terms of Carrollian conjugate pairs, revealing a Carrollian phase-space structure for gravity on the horizon and yielding Noether charges associated with tangential symmetries.The results provide a robust bridge between the membrane paradigm and Carrollian hydrodynamics, offering a platform for horizon thermodynamics, near-horizon expansions, and potential extensions to null infinity and holographic contexts.

Abstract

The membrane paradigm displays underlying connections between a timelike stretched horizon and a null boundary (such as a black hole horizon) and bridges the gravitational dynamics of the horizon with fluid dynamics. In this work, we revisit the membrane viewpoint of a finite distance null boundary and present a unified geometrical treatment to the stretched horizon and the null boundary based on the rigging technique of hypersurfaces. This allows us to provide a unified geometrical description of null and timelike hypersurfaces, which resolves the singularity of the null limit appearing in the conventional stretched horizon description. We also extend the Carrollian fluid picture and the geometrical Carrollian description of the null horizon, which have been recently argued to be the correct fluid picture of the null boundary, to the stretched horizon. To this end, we draw a dictionary between gravitational degrees of freedom on the stretched horizon and the Carrollian fluid quantities and show that Einstein's equations projected onto the horizon are the Carrollian hydrodynamic conservation laws. Lastly, we report that the gravitational pre-symplectic potential of the stretched horizon can be expressed in terms of conjugate variables of Carrollian fluids and also derive the Carrollian conservation laws and the corresponding Noether charges from symmetries.
Paper Structure (28 sections, 136 equations, 1 figure)

This paper contains 28 sections, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometries of stretched horizons and null boundaries
  3. Rigged Structures
  4. Null rigged structures and Induced Carroll Structures
  5. Local Boost and rescaling symmetries
  6. Coordinates
  7. Rigged metric, rigged derivative and rigged connection
  8. Conservation Laws on Stretched Horizons
  9. Conservation laws
  10. Gravitational dictionary
  11. Viscous stress tensor and Energy density
  12. Momentum density
  13. Carrollian heat current
  14. Surface gravity and Pressure
  15. Rigged derivative summary
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Stretched horizons $H$ are chosen to be hypersurfaces at $r = \text{constant}$ and the null boundary $N$ is the limit $r \to 0$ of the sequence of stretched horizons. The surface $H$ is endowed with the rigging vector $k$ and its dual form $\bm{n}$. The Carroll structure with the vertical vector $\ell$ and the horizontal vector $e_A$ is induced from the rigged structure, and together with $k$, they form a complete basis for the tangent space $TM$.