Embeddings for Schwarzschild metric: classification and new results

TL;DR

This paper introduces a method for classifying embeddings of highly symmetric Riemannian spaces, applies it to Schwarzschild metric, and constructs all minimal-dimensional embeddings, including two novel solutions with potential physical applications.

Contribution

It presents a new systematic method for classifying and constructing embeddings of symmetric Riemannian spaces, specifically applied to Schwarzschild metric, revealing two previously unknown embeddings.

Findings

Classified all embeddings of Schwarzschild metric in six-dimensional space.

Constructed two new embeddings, one asymptotically flat.

Identified potential applications in gravity and many-body problems.

Abstract

We suggest a method to search the embeddings of Riemannian spaces with a high enough symmetry in a flat ambient space. It is based on a procedure of construction surfaces with a given symmetry. The method is used to classify the embeddings of the Schwarzschild metric which have the symmetry of this solution, and all such embeddings in a six-dimensional ambient space (i.e. a space with a minimal possible dimension) are constructed. Four of the six possible embeddings are already known, while the two others are new. One of the new embeddings is asymptotically flat, while the other embeddings in a six-dimensional ambient space do not have this property. The asymptotically flat embedding can be of use in the analysis of the many-body problem, as well as for the development of gravity description as a theory of a surface in a flat ambient space.