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 spaceV ₄ into a pseudo-Euclidean spaceE ^N _p,q (N > 4,p + q =N) (p,q) describing the signature of the spaceE ^N _p,q . Then all the geometrical objects ofV ₄ can be expressed in terms of the embedding functions,Z ^A (x ⁱ),A = 1, 2,...,N, i = 0, 1, 2, 3. Then let us deform the embedding:Z ^A→Z ^A +ευ ^A, ε being an infinitesimal parameter. The Einstein equations can be developed then in the powers ofε; 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.