Spinning Top Falling Into a Schwarzchild Black Hole

[amigocabal]

If a spinning top free falls into a Schwarzchild black hole, the law of conservation of angular momentum means that its angular momentum will be added to the black hole, transforming it to a Kerr black hole.

However, outside observers at a distance will never observe the spinning top crossing the event horizon. (Indeed, they will never actually observe the original collapsed object shrink below the Schwarzchild radius.) Does this mean that, from the point of view of distant observers at rest with respect to the black hole, the hole will always remain a Schwarzchild black hole? Or does the spinning top somehow transfer its angular momentum to the black hole before entering the event horizon?

[Borgholio]

The spinning of a black hole imparts some of it's momentum to the accretion disc around it, and it affects the gravity waves coming off of the singularity itself. All of which can be detected.

[Kuroneko]

No, it does not mean that. For a simpler scenario of a spherically symmetric shell falling into a Schwarzschild black hole, the event horizon will expand toward the shell before any matter crosses it. Outside that kind of toy problem, the exact behavior of the event horizon is hard to determine, but it's morally the same. The event horizon is defined in terms of future lightlike infinity, so one cannot be certain where it is without knowing almost the entire future history of the black hole, and more importantly here, the location of the event horizon depends on and responds to what will fall into the black hole in the future.

So I think the main conceptual problem here is the impression that the event horizon needs something to fall into it before it can respond appropriately. This is incorrect.

[Ziggy Stardust]

So I think the main conceptual problem here is the impression that the event horizon needs something to fall into it before it can respond appropriately. This is incorrect.

Would you mind elaborating on this point?

I thought the event horizon was defined roughly as the point beyond which the gravitational pull of the black hole is so powerful that even objects traveling at the speed of light (e.g. light itself) cannot escape. So while I see your point as to how it is difficult to precisely define, I don't understand what you mean when you say that it does not need something to "fall into it before it can respond appropriately." If we simply define it as a point with respect to the speed of light, by definition it isn't responding to anything going less than the speed of light. Unless you are referring to the way we perceive objects to slow indefinitely as they approach the event horizon, but I don't really see how that is an "interaction" per se. It seems to me that there is simply an asymptotic process on which we've defined a certain threshold.

[Kuroneko]

I thought the event horizon was defined roughly as the point beyond which the gravitational pull of the black hole is so powerful that even objects traveling at the speed of light (e.g. light itself) cannot escape. So while I see your point as to how it is difficult to precisely define, I don't understand what you mean when you say that it does not need something to "fall into it before it can respond appropriately."

That definition isn't really wrong, but perhaps it's better operationally as a thought experiment: to find out whether a given event is inside or outside the event horizon, one shines some idealized light rays from it and observes where they end up after an infinite amount of time. If they escape infinitely far away, the event is outside the event horizon. Through this procedure one can find the boundary of the region from which they don't escape, which is the event horizon itself.

If the spacetime geometry is evolving, the light rays you sent out interact now with some snapshot of the spacetime as it is "now", but rather what it will be. Since that can affect whether or not the light rays will ultimately escape, the location of the event horizon now depends on the future history. The event horizon responds in advance of what will happen.

This shouldn't be too surprising--we've defined the event horizon in terms of what happens even if one waits an infinite amount of time (which has more formal definition), so of course it's a highly non-local concept that depends on the future. Although it does mean that the event horizon is not always a useful concept to analyze black holes; actually, there's about a dozen of inequivalent types of "horizons" used to characterize black holes, depending on the theoretical needs one has. For example, one could try to formalize the notion of "locus of outwardly-directed light rays fail to actually expand outward", and thus define a notion of a "non-expanding horizon". This is a local notion (in particular, one don't need to wait an infinite amount of time), and for a Schwarzschild (or Kerr-Newman) black hole that will match the event horizon... but in a general dynamical situation, the two horizons won't coincide.

If we simply define it as a point with respect to the speed of light, by definition it isn't responding to anything going less than the speed of light.

No. But I hope the above made it more clear.

[Ziggy Stardust]

Thanks, that made it more clear. I hadn't been thinking about it as a function of time, just of light/gravity.