Deformations of the embedded Einstein spaces

Abstract

We discuss a method of studying the stability of solutions of Einstein's equations, which can be outlined as follows: Consider an embedding of a given Einstein space V₄ into a pseudo-Euclidean space E ^N_p,q (N>4, p+q=N),(p,q) describing the signature of the space E ^N_p,q . Then all the geometrical objects of V₄ can be expressed in terms of the embedding functions, Z^A(xⁱ), A=1,2,...,N,i=0, 1,2,3.

Then let us deform the embedding: Z^A → Z^A+εv^A, ε being an infinitesimal parameter. The Einstein equations can be developed then in the powers of ε; we study the equations arising by requirement of the vanishing of the first or second order terms. Some partial results concerning the de Sitter, Einstein and Minkowskian spaces are given.