Event horizon: light delay and grav. time dilation

[Searinox]

Hello,

I know that time dilates while approaching the event horizon of a black hole, but explanations failed to make me understand HOW MUCH of each phenomena causes this as an object approaches the EH.

On the one hand there is photon delay due to gravity acceleration approaching the speed of light, which stretches the light, redshifting it, and delaying its arrival, so that when an object is very close to the EH, it takes its light a very long time to reach a faraway observer, giving the illusion that the object is moving slower and slower and never crossing the EH when in fact it has already crossed it.

On the other hand there is a local gravitational time dilation, which causes an actual, and real dilation of time, which to a faraway observer makes it seem that the object's time is slowing down.

Both phenomena have the same observable effect, and I've been hit with ambiguity in explanations which I can't put to rest: is light delay causing the object to appear slowed down until it stops when, in reality it has crossed the horizon and possibly even hit singularity? Or perhaps time dilation at the EH approaches infinity and the object does indeed take forever to cross over?

Which one of these effects is an asymptote to the EH? My assumption would be that as the object crosses and the light makes it seem that it has come to a halt, the object DOES indeed undergo some extent of true time dilation, but that dilation is less than infinite? And if time dilation is less than infinite, by what order of magnitude does it cause slowdown?

Thank you.

 

[JesseM]

"Time dilation" is always expressed in terms of the rate a clock is ticking relative to coordinate time, so it depends on what coordinate system you use, there is no "true" coordinate-independent time dilation. If you're using the time coordinate in Schwarzschild coordinates then the time dilation of a falling clock does approach infinity as it approaches the horizon, but if you use some other coordinate system like Eddington-Finkelstein coordinates or Kruskal-Szekeres coordinates the time dilation will stay finite as the clock crosses the horizon.

 

[Searinox]

If time dilation is indeed infinite then there comes another question. But first let me clarify: I am talking about the rate of a clock ticking relative to an observer at "reasonable" distance from the BH, to which you can imagine any amount of miles distance where the BH's gravity is no longer as strong.

But if to external observers no time ever passes at the event horizon, then, at the moment of the BH's creation, when matter falls inwards and the singularity is created, and is still the size of a dot, how can any of the rest of the infalling matter ever fall inside and add to the BH's mass, expanding its EH and making it look like a perfect black sphere? Matter attempting to fall inward fast into the newborn EH would hit matter that is exactly at the horizon, which is almost perfectly static, and get stuck there, advancing extremely slowly inward, always stuck in the back of the innermost layer of particles surrounding the EH, which are taking forever to fall in and won't make them any room to pass. All the rest of the star's mass would be waiting on the timefrozen particle sphere around the EH to fall inside and thus BHs could never grow larger than a dot nor suck anything inside.

Another thing I can't understand is that, if the event horizon is a sphere of perfect time standstill, beyond which, to an outsider, nothing has ever gone past and time is frozen, then the contents of the EH, including infalling matter plus singularity, are also frozen in time. As far as I know gravity has a finite speed, propagating at the speed of light. If the sun were to pop out of existence right now, it would take 8 minutes before the sky suddenly got dark and Earth would be shaken by the gravity change. So even if my question above is flawed and matter CAN fall inside the BH in reasonable time and add to its mass, wouldn't it take forever for the BH's strengthened gravitational force to be "announced"?

 

[pervect]

You might try Ted Bunn's black hole FAQ:

http://cosmology.berkeley.edu/Education/BHfaq.html#q4

Ted mostly recommends the point of view that it's an illusion. It's definitely wrong to think of time stopping at the event horizon. One has to pay attention to the difference between a static observer and one who falls in.

It might help to think about what different observers see. An observer who hovers just outside the event horizon of a black hole will see from incoming light waves and/or video signals that the universe appears to age quickly. The carrier frequency of the video signals will be blueshifted - and the content will appear to be speeded up. This is consistent with the "time slowing down" idea, and in fact is its origin.

However, an observer who free-falls into a black hole won't see such an effect. So I would say the problem is generalizing from the "static" observer to the "free-falling" observer, along with the problem of some elements of the Newtonian idea of "universal time" creeping into the argument. One of the points of relativity is there is in general no such thing as universal time.

It's a more subtle example of some of the same problems that show up with the twin paradox from special relativity.

This may be a bit complicated, "it's an illusion" works too :-).

 

[JesseM]

You seem to be ignoring the main point I emphasized in my reply, namely that this is a purely coordinate-dependent effect. Even in the flat spacetime of SR you can pick non-inertial coordinate systems where time dilation "goes to infinity" (i.e. rate of clock ticking relative to coordinate time goes to zero) at some arbitrary boundary, like how time dilation goes to infinity at the Rindler horizon if you use Rindler coordinates in SR. If you want to understand what's actually going on physically, you shouldn't consider such coordinate effects, instead you should consider the coordinate-independent proper time experienced by the infalling object, and the answer here is that it crosses the horizon in a finite amount of proper time. You can show that this is true even if you use a coordinate system where the coordinate time to reach the horizon is infinite, like Schwarzschild coordinates; in this case, in the limit as coordinate time goes to infinity and the distance from the horizon approaches zero, the proper time approaches some finite value (the same value that you will calculate for the proper time of the object crossing the horizon in a coordinate system where the object crosses at a finite coordinate time and continues towards the singularity after that, like Eddington-Finkelstein coordinates or Kruskal-Szekeres coordinates).

If time dilation is indeed infinite then there comes another question. But first let me clarify: I am talking about the rate of a clock ticking relative to an observer at "reasonable" distance from the BH, to which you can imagine any amount of miles distance where the BH's gravity is no longer as strong.

And again, the only way to talk about the rate of clock ticking is either to talk about the visual appearances seen by the "reasonable distance" observer (i.e. when light from successive ticks reaches them), or the rate the clock is ticking relative to coordinate time in some coordinate system. You indicated you didn't just want to talk about visual appearances due to light delays, so the only alternative is the second, and in this case the time dilation depends completely on what coordinate system this "reasonable distance" observer chooses to use, they are free to use Schwarzschild coordinates or Kruskal-Szekeres coordinates or whatever they want.

Another thing I can't understand is that, if the event horizon is a sphere of perfect time standstill, beyond which, to an outsider, nothing has ever gone past and time is frozen, then the contents of the EH, including infalling matter plus singularity, are also frozen in time. As far as I know gravity has a finite speed, propagating at the speed of light. If the sun were to pop out of existence right now, it would take 8 minutes before the sky suddenly got dark and Earth would be shaken by the gravity change. So even if my question above is flawed and matter CAN fall inside the BH in reasonable time and add to its mass, wouldn't it take forever for the BH's strengthened gravitational force to be "announced"?

But why do you think there should be a "strengthened gravitational force" seen by external observers? If a black hole forms from a spherically symmetric collapsing sphere of matter like a star, then for any observer at a radius greater than the original radius of the collapsing star, the gravity remains constant as the star collapses (see http://cosmology.berkeley.edu/Education/BHfaq.html#q5 and here for example)