Past exam of the general relativity II course of the University of Oxford / 2015 / 2 / c

A regular hypersurface $h=0$, with $dh\ne 0$, is a null hypersurface if its normal is null:The raised normal is then also tangent to the hypersurface, and its induced metric is degenerate. This is a local geometric property.

In an asymptotically flat spacetime, choose its future null infinity ${\mathcal{I}}^{+}$. The black hole region and future event horizon areHere $J^{-}({\mathcal{I}}^{+})$ is the causal past of future null infinity: the region from which a future-directed causal curve can escape to arbitrarily distant observers. An event horizon therefore depends on the global future causal structure, and an arbitrary null hypersurface need not be an event horizon.