The approach to gravity as a theory of embedded surface

We study the approach to gravity in which our curved spacetime is considered as a surface in a flat ambient space of higher dimension (the embedding theory). The dynamical variable in this theory is not a metric but an embedding function. The Euler-Lagrange equations for this theory (Regge-Teitelboim equations) are more general than the Einstein equations, and admit "extra solutions" which do not correspond to any Einsteinian metric. The Regge-Teitelboim equations can be explicitly analyzed for the solutions with high symmetry. We show that symmetric embeddings of a static spherically symmetric asymptotically flat metrics in a 6-dimensional ambient space do not admit extra solutions of the vacuum Regge-Teitelboim equations. Therefore in the embedding theory the solutions with such properties correspond to the exterior Schwarzchild metric.