# Black hole hair, gravitational wave emission, and information, from outside observer

I have a few conceptual puzzles that I cannot see my way through. Time and again, I've read that from the point of view of a static outside observer, infalling matter just reaches a black hole's event horizon but is never observed to pass. At the event horizon, outgoing light is infinitely redshifted, and so no signal that the matter has entered the black hole reaches the observer in a finite amount of time.

However, this does not seem to jive with the no-hair theorem: The infalling matter that falls into the horizon leads to a slight non-uniformity in density that is quickly radiated away as gravitational waves. These would seem to reach the static outside observer in a finite amount of time. Wouldn't that serve as a signal that the infalling matter has, indeed, gone past the event horizon? Moreover, if matter never appeared to pass the black hole horizon, from the outside frame, then how would the black hole ever gain mass? Wouldn't this lead to an asymmetric build-up of matter just outside of the event horizon, that would be detectable in the gravitational field? Or would this asymmetry appear to be radiated away before/as the matter reached the horizon?

On the other hand, maybe this is an illusion. I remember reading about alternatives to the Schwarszchild metric (maybe I am thinking of the Eddington-Finkeltstein coordinates?) that correspond to static observers asymptotically far from the black hole, but infalling matter near the black hole. Or perhaps when we are dealing with infalling matter, since the Schwarzschild metric is no longer exact, we must use perturbational methods that change the situation. Maybe in this more physical set of coordinates or perturbed solution, outside observers do see matter enter the black hole?

But if they do see matter enter the black hole, then from their frame of reference there is a finite (at least) amount of time before the matter reaches the inner singularity, and so the observer knows something about the internal structure of the black hole. In this case, if the observer wanted to make an accounting of the "information" inside the black hole, wouldn't some of the information in the black hole then correspond to the matter known to exist inside, in addition to whatever information is associated with the horizon itself? Is this just a temporary situation that only lasts until the the black hole radiates away waves and again has no hair?

Finally, a question about the firewall paradox. As I understand it, the firewall paradox holds that the event horizon has physical significance due the presence of high energy radiation trapped at it. This is said to be in contradiction with the equivalence principle, because the event horizon is not gravitational special according to an infalling observer. I don't understand why this is a contradiction of the equivalence principle. If I jump into the sun, I will be fried as I cross its surface. There is nothing gravitationally special about this surface, either; it just so happens that the laws of gravity and nuclear physics, applied to the mass, temperature, and contents of the Sun, determine an outer boundary. Why isn't a similar interpretation possible for a firewall -- that the black hole has a certain mass, and as an incidental consequence that a lot of high energy radiation tends to cluster at this particular surface?

• I don't know enough to answer the question specifically but the no hair theorem is probably not 100% true about real black holes as pretty much everyone at this point agrees that there is no information loss and to get around the information loss paradox, information would leave the black hole somehow and that violates the no hair theorem. Much of the discussion I see about the problems of black holes seem to point to a differentiation of theoretical black holes which we can only understand as eternal object and the real observed ones which obviously formed at some point. – A. C. A. C. Dec 21 '17 at 22:34
• My answer to the last question is that, the fa t that there is nothing special about the horizon means that all matter will always cross and fall through the horizon, so the horizon should be an empty space. And that's the contradiction with firewall. To my best knowledge. – Ismasou Dec 22 '17 at 2:44