Linear instability of nonvacuum spacetimes

We study the problem of linear instability in nonvacuum spacetimes. For vacuum spacetimes linear instability occurs when the spacetime has Killing vectors. In the nonvacuum case, one must prescribe how the sources are to vary. For one natural choice, we show that the signal for instability is the existence of integral constraint vector fields. These vector fields lead, as in the vacuum case, to nonlinear constraints on the first-order perturbations to the metric and momentum. For other choices for variations of the sources, we show how to modify the definition of integral constraint vectors appropriately. Since Robertson-Walker spacetimes have integral constraint vectors our results may have cosmological applications.