Revisiting Spherically Symmetric Spacetime I: Geometro-Hydrodynamics

TL;DR

This work introduces a geometric framework for spherically symmetric spacetimes by foliating spacetime into spherical slices and employing a rigging-based, foliation-adapted frame. It identifies a Kodama-driven temporal flow and Misner-Sharp energy to recast the Einstein equations in a geometro-hydrodynamic form, with the energy density $E = M/A$ and gravitational pressure $P = \frac{1}{8\pi G_N}\kappa - \frac{d-3}{d-2}E$, yielding Euler-like and Young-Laplace-like relations that describe a concentric stack of gravitational bubbles. The formalism is frame-covariant and extends to Lovelock gravity, where a Lovelock energy $E_L$, pressure $P_L$, and dual pressure $ar P_L$ preserve the hydrodynamic structure, thereby offering a universal dictionary between geometry and thermodynamics for spherical spacetimes. The results establish a robust foundation for covariant phase-space analyses and horizon thermodynamics, and pave the way for connecting spacetime dynamics to its microscopic, thermodynamic constituents.

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.

Figures (2)

• Figure 1: Spacetime as the worldvolume of a concentric stack of gravitational bubbles. (Left) Spherically symmetric spacetime $\mathcal{M}$ is viewed as a stack of spherical slices $\Sigma_{\sigma}$. The tangent space $T\mathcal{M}$ is spanned by $(\uline{u}, \uline{k}, \uline{\partial}_A)$, where $\uline{u}$ lies along $\Sigma_\sigma$, $\uline{\partial}_A$ are tangent to the gravitational bubble on $\mathcal{S}_r$, and $\uline{k}$ is transverse, connecting one slice $\Sigma_{\sigma}$ to another $\Sigma_{\sigma + \Delta \sigma}$. The radius of $\mathcal{S}_r$ can vary in both directions $\uline{u}$ and $\uline{k}$. Each gravitational bubble carries the Misner-Sharp energy density $\esstix{E}$. Its infinitesimal change along the temporal direction, $\Delta_{\uline{u}}\esstix{E} = (\Delta t)\mathcal{L}_{u}\esstix{E}$, over a time duration $\Delta t$, is governed by the Einstein equation and depends on the matter energy flux $T_{un}$, analogous to the Euler equation for energy. The change $\Delta_{\uline{k}}\esstix{E} = (\Delta\sigma)\mathcal{L}_{k}\esstix{E}$ describes how $\esstix{E}$ varies in the transverse direction and is likewise determined by the Einstein equation. (Right) On each gravitational bubble on $\mathcal{S}_r$, one can also define the notion of pressure (or surface tension), which equilibrates with the matter pressure $T_{kn}$ through the Einstein equation, interpreted as the Young-Laplace equation.
• Figure 2: (Left) Spherically symmetric spacetime foliated by a series of radial slices $\Sigma_r = \mathcal{T}_r \times \mathcal{S}_r$, which are non-expanding surfaces (i.e., $\theta_{(\bdx{u})} =0$). (Right) The 2-dimensional normal plane $\mathcal{N}$, whose points are attached to the round sphere $\mathcal{S}_r$. The radial slice $\Sigma_r$ is represented as a vertical line. The tangent space and cotangent space to $\mathcal{N}$ split into the vertical (temporal) part spanned by $(\uline{\bdx{u}}, \bdx{k})$ and the transverse (radial) part spanned by the null rigging structure $(\uline{\bdx{k}}, \bdx{n})$. Locally, the bases $(\uline{u}, {k})$ and $(\uline{k},{n})$, adapted to an arbitrary foliation $\sigma(y) = \text{constant}$ (the green line), are linear combinations of those adapted to the radial slices.