Rigid spheres in Riemannian spaces

TL;DR

The paper introduces a new method for constructing special topological two-spheres, called 'rigid spheres', to simplify spacetime foliations in General Relativity, potentially reducing supertranslation ambiguities.

Contribution

It defines a four-parameter family of rigid spheres in Riemannian three-manifolds and discusses their application in constructing advantageous spacetime foliations.

Findings

Existence of a four-parameter family of rigid spheres in generic Riemannian 3-manifolds.

Rigid spheres can serve as fundamental elements for various spacetime foliations.

Potential to avoid supertranslation ambiguities in spacetime analysis.

Abstract

Choice of an appropriate (3+1)-foliation of spacetime or a (2+1)-foliation of the Cauchy space, leads often to a substantial simplification of various mathematical problems in General Relativity Theory. We propose a new method to construct such foliations. For this purpose we define a special family of topological two-spheres, which we call "rigid spheres". We prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold (in case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere). The rigid spheres can be used as building blocks for various ("spherical", "bispherical" etc.) foliations of the Cauchy space. This way a supertranslation ambiguity may be avoided. Generalization to the full 4D case is discussed. Our results generalize both the Huang foliations (cf. \cite{LHH}) and the foliations used by us (cf. \cite{JKL}) in the analysis of the two-body problem.