On the existence of rigid spheres in four-dimensional spacetime manifolds

TL;DR

This paper generalizes the concept of rigid spheres from flat Minkowski spacetime to arbitrary four-dimensional spacetime manifolds, establishing local existence results using geometric analysis and the implicit function theorem.

Contribution

It introduces a new definition of rigid spheres in curved spacetimes and proves their local existence near Minkowski space, expanding understanding of geometric structures in general relativity.

Findings

Existence of rigid spheres in curved spacetimes near Minkowski space

Introduction of conditions on curvature and torsion for rigid spheres

Development of a solution family parameterized by eight parameters

Abstract

This paper deals with the generalization of usual round spheres in the flat Minkowski spacetime to the case of a generic four-dimensional spacetime manifold $M$. We consider geometric properties of sphere-like submanifolds in $M$ and introduce conditions on external curvature and torsion, which lead to a definition of a {\em rigid sphere}. The main result is a local existence theorem concernig such spheres. For this purpose we apply the surjective implicit function theorem. The proof is based on a detailed analysis of the linearized problem and leads to an eight-parameter family of solutions in case when the metric tensor $g$ of $M$ is from a certain neighbourhood of the flat Minkowski metric. This contribution continues the study of rigid spheres in (Class. Quantum Grav. \textbf{30} (2013), 175010, doi:10.1088/0264-9381/30/17/175010, 18 pp.).