New approximation methods in General Relativity: geodesic deviations and deformations of embeddings

Abstract

We show how approximate solutions of the two-body problem in General Relativity, and the approximate solutions of Einstein's equations in vacuo can be constructed using small deformations of geodesics and of Einstein space-times embedded into a pseudo-Euclidean flat space of higher dimension. The method consists in using expansions of equations around a given simple solution (a circular orbit in the case of geodesics, and Minkowskian or Schwarzschild space in the case of Einstein's equations) in a series of powers of small deformation parameter, and then solving by iteration the corresponding linear systems of differential equations.5th International School on Field Theory and Gravitation,