Proposition: the event horizon and the apparent horizon of a black hole always coincide.
As a reminder: the event horizon is defined as the boundary of the closure of the causal past of future null infinity, i.e. the boundary of the closure of the set of points which can reach infinity by a timelike path (so the boundary of the black hole). The apparent horizon is defined as the boundary of the set of trapped surfaces, where a trapped surface is defined as a surface for which all outgoing null rays converge.
Question: The above proposition is known to be false. Can someone give a counterexample? As always for counterexamples, the simpler the better!
I know the following:
- For a counterexample we necessarily need to consider a non-stationary black hole, as Hawking proved that the proposition is true for stationary ones.
- The apparent horizon is always inside the black hole. But doesn't this imply that there are points inside the black hole from which light rays can diverge?
- Apparently, one counterexample is given by stellar collapse, though I fail to understand this counterexample, i.e. I don't recognize the points which are outside the apparent horizon and inside the black hole.
- Unlike the event horizon, the apparent horizon is observer dependent. Wikipedia states: "For example, it is possible to slice the Schwarzschild geometry in such a way that there is no apparent horizon, ever, despite the fact that there is certainly an event horizon." Clearly they don't coincide then, but doesn't this contradict the fact that they do coincide for stationary spacetimes?
So basically, I know some differences between the two concepts, but I cannot think of a concrete example for which we can identify the apparent horizon and the event horizon, and see that they are different.