General Results on Stability by Linearization when the Submanifold M of V is Compact

Abstract

We continue with the situation from the previous chapter. We have a 4-dimensional manifold V with an initial Lorentzian metric $\tilde g$ and an initial stress-energy tensor T corresponding to a perfect fluid and which together fulfil Einstein’s equation $G(\tilde g) = \chi T,$. 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 $\tilde g$. 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

$$ \begin{gathered} \mathcal{S}^\mathcal{S} (g) \times \mathcal{S}^{\mathcal{S} - 1} (k) \hfill \\ \cup \hfill \\ \Phi : \mathcal{U}\, \to \mathcal{F}^{\mathcal{S} - 2} (F) \times \Omega ^{\mathcal{S} - 2} (X) \hfill \\ (g',k') \to (\mathcal{H}(g',k'), \gamma (g',k')) \hfill \\ \end{gathered} $$

at the initial pair (g, k).