Carrollian hydrodynamics and symplectic structure on stretched horizons

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.