General results on stability by linearization when the submanifold M of V is compact

© 2010, Birkhäuser, Springer Basel AG. We continue with the situation from the previous chapter. We have a 4-dimensional manifold V with an initial Lorentzian metric (Formula Presented.) and an initial stress-energy tensor T corresponding to a perfect fluid and which together fulfil Einstein’s equation (Formula Presented.). Let us consider a hypersurface M of V such that at each point x∈M the velocity vector u of the perfect fluid is perpendicular to M with respect to the initial metric v. As observed in the previous chapter, study of the linearization stability of Einstein’s equation at the initial metric leads us to the study of the linearization stability of the mapping (Formula Presented.) at the initial pair (g, k).