Revisiting Spherically Symmetric Spacetime I: Geometro-Hydrodynamics

Abstract: This series of works revisits the geometry, dynamics, and covariant phase space of spherically symmetric spacetimes with the aim of exploring the thermodynamics of spacetime from their dynamical properties. In this first paper, we examine the geometry from the perspective of a foliation by spherical hypersurfaces. Using the rigging technique, we first define a local frame adapted to these slices and reconstruct the geometry and dynamics fully. We clarify the connection of the frame adapted to constant-radius slices, to the Kodama vector and Misner-Sharp energy. Through frame transformations, we then show that the gravitational dynamics in a general foliation-adapted frame can be interpreted as hydrodynamics, i.e., geometro-hydrodynamics: the Einstein equations exhibit the gravitational analogs of the Euler and Young-Laplace equations, and the spacetime can be viewed as the worldvolume of a concentric stack of "gravitational bubbles" -- spherical collective modes with the Misner-Sharp energy density and a geometric pressure. We apply this framework to apparent horizons and study the dynamics. Finally, we demonstrate that a similar geometro-hydrodynamic picture holds in Lovelock gravity. These results provide a fresh perspective on this class of spacetimes and lay the foundation for understanding their thermodynamic properties.