Nontrivial isometric embeddings for flat spaces

TL;DR

This paper explores the construction of nontrivial, symmetric, and unfolded isometric embeddings of flat spaces, which are crucial for embedding gravity models and their relation to General Relativity.

Contribution

It introduces a method of sequential surface deformations to construct unfolded symmetric embeddings of flat Euclidean and Minkowski spaces.

Findings

Constructed explicit symmetric embeddings for flat Euclidean 3-space.

Developed a new method of sequential surface deformations for embedding construction.

Showed embeddings can be used to analyze embedding gravity equations.

Abstract

Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be required to have the same symmetry as the metric. On the other hand, it is possible to require the embedding to be unfolded so that the surface in the ambient space would occupy the subspace of the maximum possible dimension. In the weak gravitational field limit, such a requirement together with a large enough dimension of the ambient space makes embedding gravity equivalent to General Relativity, while at lower dimensions it guarantees the linearizability of the equations of motion. We discuss symmetric embeddings for the metrics of flat Euclidean three-dimensional space and Minkowski space. We propose the method of sequential surface deformations for the construction of unfolded embeddings. We use it to construct such embeddings of flat Euclidean three-dimensional space and Minkowski space, which can be used to analyze the equations of motion of embedding gravity.