Embeddings for General Relativity

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.