Classification of ten-dimensional embeddings of spherically symmetric static metrics

TL;DR

This paper classifies all possible four-dimensional static, spherically symmetric surfaces embedded in a ten-dimensional flat space, providing a comprehensive framework useful for embedding gravity theories and analyzing their equations of motion.

Contribution

It introduces a classification of 52 embedding classes based on elementary block dimensions, enhancing understanding of embeddings in Regge-Teitelboim gravity.

Findings

52 classes of embeddings summarized in a table

Analysis of unfolding properties of embeddings

Identification of smooth Minkowski embeddings

Abstract

The group-theoretic method for constructing symmetric isometric embeddings is used to describe all possible four-dimensional surfaces in flat $(1,9)$-dimensional space, whose induced metric is static and spherically symmetric. For such surfaces, we propose a classification related to the dimension of the elementary blocks forming the embedding function. All suitable 52 classes of embeddings are summarized in one table and analyzed for the unfolding property (wich means that the surface does not belong locally to some subspace of the ambient space), as well as for the presence of smooth embeddings of the Minkowski metric. The obtained results are useful for the analysis of the equations of motion in the Regge-Teitelboim embedding gravity, where the presence of unfolded embeddings of the Minkowski metric is essential.