Embeddings for general relativity

• Published in: Classical and quantum gravity 2015-10, Vol.32 (19), p.195018-195035
• Main Author: Ponce de Leon, J
• Format: Article
• Language: English
• Subjects: Angular momentum; Constants; Cosmological constant; embeddings for general relativity; Kaluza-Klein Gravity; Mathematical analysis; modified general relativity; Quantum gravity; Relativity; Scalars; Singularities
• ISSN: 0264-9381; 1361-6382
• DOI: 10.1088/0264-9381/32/19/195018

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 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 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.