Linearization stability of the Einstein equation for Robertson–Walker models. I

This is the first part of a series of two papers. In this article we study the linearization stability of the Einstein equation in the presence of matter. We have slightly changed the classic definition of this concept for the vacuum spacetime and a more general one adapted to our case is given. We consider a Robertson–Walker model $(V,g,T)$ where $V$ stands for the spacetime, g for a Robertson–Walker metric, and T for a stress-energy tensor of a perfect fluid. We write $V=S×I$ where S is a spacelike hypersurface of $V$ and I an ↛-interval. We show that in the case S has a constant curvature K equal to 0, the Einstein equation $G(g)=χT$ is linearization stable at g. In a subsequent paper we shall prove that in the case $K=1$ the opposite occurs. The case $K=−1$ remains as an open question.