Nontrivial Isometric Embeddings for Flat Spaces

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.

Weak Field Limit for Embedding Gravity

“…Therefore, bI ij can be represented as a 6 × 6 matrix. Since this matrix is nonsingular for "unfolded" embeddings (see details in [18]), you can introduce symmetric by l, m and transverse by I value ᾱlm I , which is inverse to bI ij in the matrix sense:…”

“…Let us present an embedding ȳI (x i ) of the flat Euclidean 3D metric into 9D Euclidean space that would be spherically symmetric in the sense discussed in [20] and would also be unfolded in terms of [18]. The latter means that, for such a surface, the second fundamental form (7) is nonsingular as a 6 × 6 matrix; see the text after (7).…”

“…The simplest choice in the form of ȳa (x µ ), which defines a 4D plane in the ambient space, is unsuitable since, in such a background, Regge-Teitelboim Equations (2) are not linearized [2]. To linearize the equations, it is necessary to choose "unfolded" embedding [18] as the background, which means that the second fundamental form of the surface is nondegenerate in some sense.…”