Embeddings for General Relativity

TL;DR

This paper develops a systematic method to embed n-dimensional vacuum general relativity into higher-dimensional spacetimes with various sources, exploring their properties, singularities, and physical parameter variations.

Contribution

It provides a comprehensive classification of all possible embeddings in Einstein spaces, including new classes free of singularities and with variable physical parameters.

Findings

Constructed all physically distinct embeddings in Einstein spaces.

Identified embeddings with no singularities and variable physical parameters.

Showed many embeddings have physical singularities at finite extra coordinate values.

Abstract

We present a systematic approach to embed $n$-dimensional vacuum general relativity in an $(n + 1)$-dimensional pseudo-Riemannian spacetime whose source is either a (non)zero cosmological constant or a scalar field minimally-coupled to Einstein gravity. Our approach allows us to generalize a number of results discussed in the literature. We construct {\it all} the possible (physically distinct) embeddings in Einstein spaces, including the Ricci-flat ones widely discussed in the literature. We examine in detail their generalization, which - in the framework under consideration - are higher-dimensional spacetimes sourced by a scalar field with flat (constant $\neq 0$) potential. We use the Kretschmann curvature scalar to show that many embedding spaces have a physical singularity at some finite value of the extra coordinate. We develop several classes of embeddings that are free of singularities, have distinct non-vanishing self-interacting potentials and are continuously connected (in various limits) to Einstein embeddings. We point out that the induced metric possesses scaling symmetry and, as a consequence, the effective physical parameters (e.g., mass, angular momentum, cosmological constant) can be interpreted as functions of the extra coordinate.