Event horizon

Remarks

Example: for the Schwarzschild spacetime of mass $m$ in ingoing Eddington–Finkelstein coordinates, $J^-(\mathcal{I}^+) \cap M = \set{r > 2m}$, so $\mathcal{H}$ is the null hypersurface $r = 2m$. Check that it is null: there $g_{vv} = -(1 - 2m/r) = 0$, so the tangent vector $\partial_v$ of the horizon generators is null, and the induced metric on $r = 2m$ is degenerate.

Non-example: the photon sphere $r = 3m$ of the same spacetime is not an event horizon. Points on it can send outward light rays to $\mathcal{I}^+$, so it lies inside $J^-(\mathcal{I}^+)$, not on its boundary.

Basic properties: $\mathcal{H}$ is an achronal, closed, Lipschitz hypersurface. It is generated by null geodesics that have no future endpoint in $M$, although they may enter it from the past. For stationary black holes in many settings, $\mathcal{H}$ is a Killing horizon. Replacing $\mathcal{I}^+$ by the world line of one observer gives the event horizon of that observer. An example is the Rindler horizon of a uniformly accelerated observer in Minkowski space.