A fundamental question arising in the study of the Einstein equations in general relativity is that of global stability of solutions under perturbations of their initial state. In this thesis, we will establish two results pertinent to instability phenomena for the Einstein equations, associated to the presence of two distinct geometric structures on the underlying spacetimes.

In the first part of the thesis, we will focus on the class of stationary and asymptotically flat spacetimes; such spacetimes arise naturally as models of the final state of the evolution of isolated self-gravitating systems in the universe. Our first main result in this setting will be a general instability statement for solutions psi to the scalar wave equation squaregpsi=0 (a simple linear toy model for the non-linear Einstein equations) on spacetimes (M,g) possessing a non-empty ergoregion but no event horizon. This statement was conjectured and heuristically supported by Friedman in 1978. As a corollary of this result, we will also obtain an instability statement for the acoustical wave equation on the hydrodynamic vortex, a phenomenon first studied numerically by Oliveira--Cardoso--Crispino.

In the second part of the thesis, we will turn our attention to the class of asymptotically anti-de Sitter spacetimes. These spacetimes arise as solutions of the Einstein equations with a negative cosmological constant and have a prominent role in the high energy physics literature. In this setting, it was conjectured by Dafermos--Holzegel in 2006 that anti-de Sitter spacetime is unstable as a solution of the vacuum Einstein equations when reflecting boundary conditions are imposed on conformal infinity. The second main result of this thesis will be a proof of the instability of AdS for the Einstein--null dust system.