Horizons of some asymptotically stationary spacetimes

On a class of dynamical spacetimes which are asymptotic as $t\to\infty$ to a stationary spacetime containing a horizon $\mathcal{H}_0$, we show the existence of a unique null hypersurface $\mathcal{H}$ which is asymptotic to $\mathcal{H}_0$. This is a special case of a general unstable manifold theorem for perturbations of flows which translate in time and have a normal sink at an invariant manifold in space. Examples of horizons $\mathcal{H}_0$ to which our result applies include event horizons of subextremal Kerr and Kerr-Newman black holes as well as event and cosmological horizons of subextremal Kerr-Newman-de Sitter black holes. In the Kerr(-Newman) case, we show that $\mathcal{H}$ is equal to the boundary of the black hole region of the dynamical spacetime.