# Spherical Spacelike Geometries in Static Spherically Symmetric Spacetimes: Generalized Painlev\`e-Gullstrand Coordinates, Foliation, and Embedding

## TL;DR

This paper proves the uniqueness and existence conditions for foliations of static spherically symmetric spacetimes by spherical spacelike geometries, generalizing Painlevé-Gullstrand coordinates and exploring their embedding properties.

## Contribution

It provides a geometric proof of foliation uniqueness and existence, introduces a natural generalization of Painlevé-Gullstrand coordinates, and links foliation properties to embeddability in higher-dimensional spacetimes.

## Key findings

- Foliation by flat hypersurfaces implies foliation by non-positive Ricci curvature hypersurfaces.
- Derived algebraic conditions for the existence of spherical spacelike foliations.
- Connected foliation properties to embeddability in higher-dimensional static spherically symmetric spacetimes.

## Abstract

It is well known that static spherically symmetric spacetimes can admit foliations by flat spacelike hypersurfaces, which are best described in terms of the Painlev\`{e}--Gullstrand coordinates. The uniqueness and existence of such foliations were addressed earlier. In this paper, we prove, purely geometrically, that any possible foliation of a static spherically symmetric spacetime by an arbitrary codimension-one spherical spacelike geometry, up to time translation and rotation, is unique, and we find the algebraic condition under which it exists. This leads us to what can be considered as the most natural generalization of the Painlev\`{e}--Gullstrand coordinate system for static spherically symmetric metrics, which, in turn, makes it easy to derive generic conclusions on foliation and to study specific cases as well as to easily reproduce previously obtained generalizations as special cases. In particular, we note that the existence of foliation by flat hypersurfaces guarantees the existence of foliation by hypersurfaces whose Ricci curvature tensor is everywhere non-positive (constant negative curvature is a special case). The study of uniqueness and the existence concurrently solves the question of embeddability of a spherical spacelike geometry in one-dimensional higher static spherically symmetric spacetimes, and this produces known and new results geometrically, without having to go through the momentum and Hamiltonian constraints.