Time (seen by distant observer) for matter to fall to the event horizon of black hole

Jonathan Cohn
As I understand it, time (as seen by a distant observer) near event horizon of BH slows to "zero". It makes me wonder how long (as seen by a distant observer) it takes for matter to fall to the event horizon. I would guess this would be calculated via some appropriate integral (I am not confused by a Zeno's paradox here). Obviously, it depends on the distance and velocity of the mass, but one could use some "reasonable" assumptions (mass starts at rest relative to the BH at a distance of say 1000 km).

If the answer is infinite (or about the age of the universe), it would seem a black hole can't grow in mass. Presumably, this is not the answer.

ibysaiyan

If we are to say infinite then that's a paradox in itself.As far as General relativity goes to the distant, stationary observer things would appear at halt at the Event horizon due to time dilation being most apparent(never sees the object falling inwards).From my current understanding of Black holes,hawking radiation is emitted implying loss of Energy and mass of the black holes.

If the in falling object along with surrounding matter providers greater mass than the emitted h.radiation,the Black hole stays otherwise it evaporates.If two black holes collide/try to merge each other for most of the cases one recoils/kicks the other out of the newly formed galaxy.It would be amazing to find out what happens if more than two black holes collide...

Jonathan Cohn

[to ibysaiyan] Thanks for the reply, and adding "stationary" (relative to BH) observer, which is what I intended. I am interested in hearing from anyone that has done (or has reference to) an answer based on actual calculations. I suspect that arguments based on the fact that objects "appear" to halt at the event horizon, can't really address this question.

For a particle radially infalling towards a schwarzschild black hole, we have:
$$\frac{dr}{dt}=-\left(1-\frac{2M}{r}\right)\sqrt{\frac{2M}{r}}$$

You can easily integrate this to find t(r), and indeed you will find that the time it takes for an object to go from any finite distance to the event horizon (limits of integration r=x*M to r=2M), is infinite. Without integration, simply note that the quantity dr/dt is the velocity measured by our distant observer, and this goes to zero as r-> 2M.

This isn't a problem though, as it doesn't mean a black hole can never gain mass. We define the black hole's mass through the radius (or area, whichever you like) of its event horizon. The event horizon is a mathematical boundary which responds to infalling mass before it has crossed the horizon. Since the EH is the boundary which outgoing light rays will never reach infinity, it is NOT the boundary at which outgoing light rays are immediately "bent backwards"! (This is instead referred to as the Apparent Horizon).

The distinction is a bit tricky and it's difficult to explain "why" the event horizon expands prematurely. One reason is that we have certain theorems that state that the event horizon must evolve continuously. Therefore, it cannot suddenly increase in size as mass passes the old event horizon boundary. A more physical way, perhaps, is to imagine you have an outgoing light ray just above what you think is the event horizon. As the infalling mass passes the light ray, the 'gravitational pull' felt by the ray increases, and if it is enough the light ray will end up bending back in towards the black hole. Therefore, the light ray was launched from inside the horizon, i.e it expanded before the matter got there.

I realize my little example has some analogies with Newtonian physics and whatnot, but it's actually a good way of understanding the process that's going on.

Jonathan Cohn

[to Nabeshin] Thanks .. both the math and the explaination/physics .. I am now un-confused

Jonathan Cohn

The explaination still leaves me curious, and perhaps I need to re-state the question: how long does it take for matter to fall "into" a BH, where "into" means it becomes part of the mass of the BH? And is that a significant factor determining how fast BH's can grow?