An investigation of embeddings for spherically symmetric spacetimes into Einstein manifolds

Embeddings into higher dimensions are very important in the study of higher-dimensional theories of our Universe and in high-energy physics. Theorems which have been developed recently guarantee the existence of embeddings of pseudo-Riemannian manifolds into Einstein spaces and more general pseudo-Riemannian spaces. These results provide a technique that can be used to determine solutions for such embeddings. Here we consider local isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate embedded spaces that admit bulks of a specific type. We show that the general Schwarzschild–de Sitter spacetime and Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space of a particular form, and we discuss their five-dimensional solutions.