Does an event horizon ever exist?

[Nugatory]

The fact that one can reach the event horizon in a finite proper time is another firm theoretical prediction of GR.

You mean in the falling object's frame of reference?

(Be aware that "frame of reference" is a treacherous concept in GR; usually you're better off thinking in terms of local inertial frames)

Pervect specifically said "proper time"; proper time is frame-independent. Intuitively, proper time is the amount of time that passes for a single clock that travels on some path between two points in space-time. It is the same for all observers and is in no way affected by changing frames of reference.

For example, if my airplane takes off at noon according to my wristwatch and lands at 1:00 according to that same wristwatch, all observers everywhere will agree about three facts: the watch read noon at takeoff; the watch read 1:00 at landing; I experienced a one-hour journey and aged one hour between takeoff and landing. Thanks to time dilation, relativity of simultaneity, and other relativistic effects, the observers may have measured very different times for my journey, but they all agree that for me it was a one-hour journey. That's proper time.

It's important to understand that proper time only works for a single clock that's only at a single place at any moment. My wristwatch measures the proper time that I experience, but it tells me nothing about the experience of other observers outside the airplane.

Can the falling object reach the singularity?

Yes, in a finite amount of proper time according to classical GR - see #12 in this thread. It is likely that classical GR stops working very close to the singularity, in which case the answer might be different... but if so, that happens long after the infalling object has passed through the event horizon.

 

[PeterDonis]

I note that there is a statement later on on that Wiki page, in the "Alternatives" section, to the effect that a quantum gravity will not feature any event horizons either. I wasn't aware that that was a mainstream view (as I've said in this thread), but the footnote there references a review article in Annalen der Physik that I haven't read. I'll take a look at it.

Having looked at it, I don't see anything in that article that says that quantum corrections are expected to prevent a horizon from forming. I do see references to the fact that, in order to show unitarity, you have to include amplitudes for spacetime histories where a black hole does not form, as well as for histories where one does form. But it also says that macroscopic black holes have classical behavior that emerges from the underlying quantum amplitudes in the same way as for any other macroscopic object, and that classical behavior includes a horizon.

So I'm sticking with what I said earlier in this thread: it looks like the mainstream physics view is that quantum corrections will remove the singularity, but not the event horizon.

 

[photonkid]

So I'm sticking with what I said earlier in this thread: it looks like the mainstream physics view is that quantum corrections will remove the singularity, but not the event horizon.

ok, well having re-read most of this entire thread including what you said here it's hard for me to understand how an outside observer will see a falling object frozen indefinitely at the edge of the event horizon yet the object does actually reach the event horizon. It seems that this is not analogous to trying to accelerate an object of non zero mass to the speed of light. It also seems that the evidence that black holes do actually exist is strong and mainstream physics view is that event horizons are most likely real.

I suggest you don't try to explain any further but do you know if there are any authoritative articles or books that explain what happens when a star collapses e.g. that the event horizon starts out very small and gets gradually bigger or whatever happens, and why an object is frozen indefinitely at the edge of the event horizon from an outside observer's view?

Anyway, thanks for the considerable time you've spent posting in this thread, and everyone elses.

 

[PAllen]

An authoritative description of collapse including the evolution of the event horizon is the following:

http://www.aei.mpg.de/~rezzolla/lnotes/mondragone/collapse.pdf

As they discuss, the event horizon grows from the center of collapsing body, and stop growing as all the mass is encompassed.

As for what a distant observer sees for, e.g., a rod falling through a horizon, see if this helps:

- light emitted at one millimeter (say) above the horizon is received eventually as long microwaves.
- light emitted at 1/2 millimeter is received as radio waves.
- light emitted at .1 millimeter above the horizon is so long wave that no plausible detector can detect it (even ignoring being swamped by CMP radiation).

So, what you actually see is the rod getting redder and redder (including radio waves in this), finally the front disappears at radius of last detectability. Then further up the rod disappears as it reached this radius; etc. until the whole rod has disappeared. This process will actually happen relatively fast for the distant observer, despite the extreme time dilation of the near horizon region relative to the distant region. Visually, with our ideal detector, it looks for all the world like the rod has progressively vanished into a black hole in space.

 

[photonkid]

As for what a distant observer sees for, e.g., a rod falling through a horizon, see if this helps:

Thanks. It helped from the point of view that now I know what "pancaked" means. Also it helped that I no longer think the in-falling object remains visible to an outside observer indefinitely, but... at first glance my brain is still seeing a paradox because
1. it sounds like the wavelength coming from the in-falling observer approaches infinity and that to an outside observer, the wave never stops coming
and
2. the closer the in-falling object gets to the event horizon, the slower time goes, so he never quite makes it - just like trying to reach the speed of light

But for 1, it probably does stop coming because after a certain time, you have to wait an infinite amount of time to see any more "oscillations in the wave". For 2, it's impossible for me to come up with any kind of numbers that show no matter how close you get you never quite make it because the external observer can't see what's happening, whereas for accelerating to the speed of light, the object being accelerated can do the measuring and no external observer is needed.

So if it's a paradox, it's a very complicated paradox and I can't actually tell if there's a paradox or not...

Anyway, I've been wondering why this isn't in an FAQ somewhere and then today I found that it's here
http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/fall_in.html


Is there any possibility that when a star collapses, that time slows down so much that the density of the mass never goes past a certain limit and that even though the star keeps on contracting, the contraction rate gets smaller and smaller? Does the maths show this can't happen - or is it because this would look like a neutron star and black hole candidates don't look like neutron stars? (A short answer will do).

 

[PeterDonis]

Is there any possibility that when a star collapses, that time slows down so much that the density of the mass never goes past a certain limit and that even though the star keeps on contracting, the contraction rate gets smaller and smaller?

No.

Does the maths show this can't happen

Yes. This case is treated in all of the major relativity textbooks.

or is it because this would look like a neutron star and black hole candidates don't look like neutron stars?

Sort of. There is a theorem (originally due to Einstein) that says that an object in a static equilibrium, like a neutron star (or a white dwarf or anything else that holds itself up statically against its own gravity) can't have a radius smaller than 9/8 of the Schwarzschild radius for its mass. The time dilation factor at that radius is only 3 (i.e., time at the surface of such an object "flows" 1/3 as fast as it does at infinity); it certainly isn't approaching infinity.

One of the key things about black hole candidates is that at least some of them appear to be confined within a radius smaller than the above limit. (At least, that's my understanding.) So whatever is in there, it can't be something in static equilibrium like a neutron star.

 

[A.T.]

There is a theorem (originally due to Einstein) that says that an object in a static equilibrium, like a neutron star (or a white dwarf or anything else that holds itself up statically against its own gravity) can't have a radius smaller than 9/8 of the Schwarzschild radius for its mass. The time dilation factor at that radius is only 3 (i.e., time at the surface of such an object "flows" 1/3 as fast as it does at infinity); it certainly isn't approaching infinity.

What about the time dilation at the center of such an object?

 

[PeterDonis]

What about the time dilation at the center of such an object?

It depends on the object's internal structure, but it will in general be somewhat larger than at the surface. It can't go to infinity because the gradient of the time dilation factor gets smaller as you go inward from the surface, until it becomes zero at the center.

 

[photonkid]

Sort of. There is a theorem (originally due to Einstein) that says that an object in a static equilibrium, like a neutron star (or a white dwarf or anything else that holds itself up statically against its own gravity) can't have a radius smaller than 9/8 of the Schwarzschild radius for its mass. The time dilation factor at that radius is only 3 (i.e., time at the surface of such an object "flows" 1/3 as fast as it does at infinity); it certainly isn't approaching infinity.

So when we hear about the age of the universe being 13 billion years, is that 13 billion years of "earth proper time"? Since planet Earth hasn't been around all that time, where does "13 billion years" come from - is it 13 billion years of "zero gravity/ zero velocity" time?

13 billion years is presumably long enough for a black hole to form. Is it possible to calculate how long it would take (in Earth years) for an infalling object to fall the last one meter before it reaches the event horizon and if so, is this likely to be longer than 13 billion "earth years". Hopefully you can see what I'm trying to get at.

 

[PeterDonis]

So when we hear about the age of the universe being 13 billion years, is that 13 billion years of "earth proper time"?

Not really. It's the proper time that would be elapsed since the Big Bang for an observer whose current spatial location is Earth, but who has always seen the universe as homogeneous and isotropic. Such observers are called "comoving" observers. We don't see the universe as isotropic on Earth: we see a dipole anisotropy in the CMBR, for example, indicating that we are not "comoving" observers, even when the effects of the Earth's rotation and orbit about the Sun are corrected for.

13 billion years is presumably long enough for a black hole to form.

Way more than enough, yes.

Is it possible to calculate how long it would take (in Earth years) for an infalling object to fall the last one meter before it reaches the event horizon

Yes.

is this likely to be longer than 13 billion "earth years".

It's a lot shorter, even for the largest black holes that we think are likely to exist in the universe (the current estimate, I believe, is black holes of billions of solar masses at the centers of quasars). For a black hole of that size, the time for an infalling object to fall the last one meter to the horizon is much less than one Earth year. Even the time to fall from an astronomically significant distance, such as a million light-years (i.e., from well outside the quasar the black hole is at the center of), is less than 13 billion years.

Note that the "time" I'm referring to here is the proper time experienced by the infalling object, i.e., the time elapsed on that object's clock.

 

[A.T.]

It depends on the object's internal structure, but it will in general be somewhat larger than at the surface. It can't go to infinity because the gradient of the time dilation factor gets smaller as you go inward from the surface, until it becomes zero at the center.

In this previous thread:
https://www.physicsforums.com/showthread.php?p=1840160#post1840160
there were claims that proper-time at the center would become negative for R < 9/8 Rs. If that's correct, wouldn't it approach zero when R approaches 9/8 Rs from above?ETA: See also:
https://www.physicsforums.com/showthread.php?p=2430580#post2430580

Solutions of the Oppenheimer--Volkoff Equations Inside 9/8ths of the Schwarzschild Radius
http://deepblue.lib.umich.edu/bitstream/handle/2027.42/41992/220-184-3-597_71840597.pdf

We refine the Buchdahl 9/8ths stability theorem for stars by describing quantitatively the behavior of solutions to the Oppenheimer–Volkoff equations when the star surface lies inside 9/8ths of the Schwarzschild radius. For such solutions we prove that the density and pressure always have smooth profiles that decrease to zero as the radius r→ 0, and this implies that the gravitational field becomes repulsive near r= 0 whenever the star surface lies within 9/8ths of its Schwarzschild radius.

 

[PeterDonis]

In this previous thread:
https://www.physicsforums.com/showthread.php?p=1840160#post1840160
there were claims that proper-time at the center would become negative for R < 9/8 Rs. If that's correct, wouldn't it approach zero when R approaches 9/8 Rs from above?

ETA: See also:
https://www.physicsforums.com/showthread.php?p=2430580#post2430580

A brief comment: the time dilation factor given in those threads is derived from a highly idealized solution for an object with constant density, not a general solution for a static object. It has some heuristic value, but I don't think any physicist considers it physically realistic.

That said, yes, one way of stating the theorem I stated is that, if an object could be static with a radius of 9/8 of its Schwarzschild radius, the time dilation factor at its center would go to infinity (or zero, depending on how you define it). However, if you compute the proper acceleration at the center, you will see that it diverges, indicating that it's impossible for a timelike object to actually follow such a worldline. In other words, the curve R = 0 in such a solution would actually be null, not timelike. And if you do the math for a solution with R < 9/8 of the Schwarzschild radius, the curve R = 0 is spacelike (and there is a null surface at some R > 0)--to say that "time flows backwards" is a serious misstatement, since that would imply that the R = 0 curve is timelike when it's actually spacelike. So the R = 9/8 Schwarzschild radius solution is a limiting case, not an actual physically possible solution, and solutions with R < 9/8 Schwarzschild radius are not physically possible either.

What I just said does depend on some assumptions, one or more of which are violated in the solutions considered in this paper:

Solutions of the Oppenheimer--Volkoff Equations Inside 9/8ths of the Schwarzschild Radius
http://deepblue.lib.umich.edu/bitstream/handle/2027.42/41992/220-184-3-597_71840597.pdf

The paper describes the assumptions which lead to the conclusions I gave above, and what happens if they are violated. (One particular thing that makes the violations seem physically unrealistic to me: the solutions are all singular at r = 0, or, to put it another way, they require a point source at r = 0 with negative mass.) One particular thing to note: *none* of the violations lead to an infinite (or zero) time dilation factor anywhere inside the object. So even if you consider some of the solutions in the paper to be physically possible, what I said in post #68 still holds.

 

[Opti_mus]

Hi guys!

I've read all that it's said in this post, as I'm interested in clarifying a question, but I haven't been able to do it reading all the answers:

Suppose an observer A falling into a Schwarzschild's BH. It's clear he'll pass through the Horizon in a proper FINITE time. Suppose an static observer B, situated in the same vertical as A and over A, with a FINITE distance over the BH. It's clear for me too that B cannot see A passing through the Horizon, as the redshift goes to infinity. Despite that fact, I've a question I wouldn't been able to find an answer for: does A pass through the Horizon in a FINITE B's proper time or not? I'm not talking about A reaching the Horizon, I'm talking about A passing the Horizon.

Thank you, and sorry for my poor english!

 

[Nugatory]

I've a question I wouldn't been able to find an answer for: does A pass through the Horizon in a FINITE B's proper time or not? I'm not talking about A reaching the Horizon, I'm talking about A passing the Horizon.

You haven't been able to find an answer because there isn't any answer to find, both for reaching and for passing through. I'm going to try restating the situation a bit more precisely, so the problem with the question will be a bit clearer:

A and B start at the same place above the event horizon; B is somehow hovering there while A falls into the black hole. So their world lines are touching at the starting point but diverge as A and B separate, with A's world line passing through the event horizon and ending at the central singularity.

Now, when we say that some amount of an observer's proper time has passed between two events, we're talking about the time along that observer's world line between the two events. It's only meaningful if the observer's world line actually passes through the events. Thus, it's easy for us to know the proper time for A to reach the event horizon: He looks at his wristwatch at the starting point, he looks at his wristwatch again as he reaches the event horizon, the difference between the two readings is the proper time that passed between leaving B and reaching the horizon.

But what two events does B use to calculate a proper time for him? Clearly he looks at his wristwatch at the starting point, just like A; but when should he look at it again to get a second reading so that he can find the elapsed proper time between the two readings?

The problem here is that there is no point on B's world line for which we can say "this and only this is when A reached the horizon". Therefore the thing that you're asking about, the amount of B's proper time elapsed for A to reach the horizon, is undefined and your question has no good answer.

 

[A.T.]

So the R = 9/8 Schwarzschild radius solution is a limiting case, not an actual physically possible solution, and solutions with R < 9/8 Schwarzschild radius are not physically possible either.

Thanks for your reply Peter. Since a static sphere is not possible, I wonder what the collapse would look like from the center (assuming you can look through the infalling matter). How would an observer at the center see the universe when the horizon forms?

 

[photonkid]

The problem here is that there is no point on B's world line for which we can say "this and only this is when A reached the horizon". Therefore the thing that you're asking about, the amount of B's proper time elapsed for A to reach the horizon, is undefined and your question has no good answer.

ok, but in this post PeterDonis said that it's possible to calculate how long (in Earth years) it will take to fall the last one meter - so the answer to
<quote> does A pass through the Horizon in a FINITE B's proper time or not? </>

would be yes wouldn't it?

Anyhow, I don't think it makes sense to distinguish between "reaching" and "passing through" because what happens after something reaches the event horizon is just speculation.

 

[PeterDonis]

How would an observer at the center see the universe when the horizon forms?

In a spherically symmetric collapse, such as that originally modeled in the 1939 Oppenheimer-Snyder paper, the event horizon forms at the center, r = 0, and moves outward at the speed of light. (As it moves outward, the light cones get tilted inward more and more by the increasing density of the collapsing matter, so the horizon moves outward more slowly, until the horizon becomes fixed at r = 2M just as the surface of the collapsing matter reaches r = 2M and intersects the horizon.)

An observer at the center would see the density around him increasing, and would see (I believe--I haven't run a computation to check) an increasing blueshift of light coming to him from the rest of the universe; but both the density and the blueshift would still be finite (and not necessarily very large) at the moment when the horizon forms at r = 0. The density and blueshift at r = 0 would only diverge to infinity later, when the surface of the collapsing matter reaches r = 0 and the singularity forms there.

 

[PeterDonis]

so the answer to
<quote> does A pass through the Horizon in a FINITE B's proper time or not? </>

would be yes wouldn't it?

It depends on how you match up events on B's worldline with events on A's worldline, since A and B are spatially separated. In a curved spacetime there is no unique way to do that in general; the only case in which there is a unique way to do it is if both worldlines are static. B's worldline is static (he stays at the same height above the horizon forever), but A's is not; so there is no unique answer to the question of when, by B's clock, A reaches the horizon.

 

[Nugatory]

ok, but in this post PeterDonis said that it's possible to calculate how long (in Earth years) it will take to fall the last one meter - so the answer to
<quote> does A pass through the Horizon in a FINITE B's proper time or not? </>

would be yes wouldn't it?

That's still A's proper time that we're talking about, even if we're choosing to measure it in years.

 

[Opti_mus]

You haven't been able to find an answer because there isn't any answer to find, both for reaching and for passing through. I'm going to try restating the situation a bit more precisely, so the problem with the question will be a bit clearer:

A and B start at the same place above the event horizon; B is somehow hovering there while A falls into the black hole. So their world lines are touching at the starting point but diverge as A and B separate, with A's world line passing through the event horizon and ending at the central singularity.

Now, when we say that some amount of an observer's proper time has passed between two events, we're talking about the time along that observer's world line between the two events. It's only meaningful if the observer's world line actually passes through the events. Thus, it's easy for us to know the proper time for A to reach the event horizon: He looks at his wristwatch at the starting point, he looks at his wristwatch again as he reaches the event horizon, the difference between the two readings is the proper time that passed between leaving B and reaching the horizon.

But what two events does B use to calculate a proper time for him? Clearly he looks at his wristwatch at the starting point, just like A; but when should he look at it again to get a second reading so that he can find the elapsed proper time between the two readings?

The problem here is that there is no point on B's world line for which we can say "this and only this is when A reached the horizon". Therefore the thing that you're asking about, the amount of B's proper time elapsed for A to reach the horizon, is undefined and your question has no good answer.

Thanks for your answer and sorry, 'cause it's totally impossible for me to understand that facts. If I (B in this case) can't define a proper time between 2 facts, even though I can't see one of them, it seems physics are over, in the sense that determinism is over: I can't know by sure if A falls or not into the BH, so I can't make any prediction about the dinamics of A and the BH!

 

[Nugatory]

Thanks for your answer and sorry, 'cause it's totally impossible for me to understand that facts. If I (B in this case) can't define a proper time between 2 facts, even though I can't see one of them, it seems physics are over, in the sense that determinism is over: I can't know by sure if A falls or not into the BH, so I can't make any prediction about the dynamics of A and the BH!

You are confusing yourself by jumping straight into a fairly subtle general relativity problem before understanding how the concepts of proper time and coordinate time work in the much simpler (no gravitational effects, no black holes, no curvature, inertial frames cover the entire universe) case of special relativity.

I'm not sure how far back to start explaining, but I'll try...

There are two ways in which we can say that two things happened at the same time:
1) They happened at the same place. If two cars collide at a street corner we know that they both were at the street corner at the same time - that's why there was a collision.
2) They happen at different places, but they both have the same time coordinate. For example this event happened at 12:37 PM; so did that one; so they happened at the same time.

Now, have you seen Einstein's thought experiment with the lightning flashes at each end of the train, the one that demonstrates the relativity of simultaneity? That shows that there is a fundamental difference between #1 and #2 above. All observers, regardless of their state of motion, will agree that in #1 both cars were at the street corner at the same time - either there's a crash or there isn't, but it can't be that some observers see the cars destroyed in a collision while others see them slipping safely past. However, as Einstein's train experiment shows, not all observers will agree about #2 - the two events at two different locations are simultaneous using one observer's coordinates but not simultaneous using the coordinates of another observer moving with reference to the first.

Study this thought experiment until you understand it thoroughly and the paragraph above makes sense. Note that physics is by no means over: determinism is intact, there's no uncertainty about whether the cars did collide, whether Einstein's lightning flashes did strike, and what time the various observers saw the various events happen. That last is different for the different observers, but in a perfectly understandable, predictable, and deterministic way.

OK, still with me? Then we can talk about "coordinate time", which is what people mean when they say "time" without further specifying what they mean. It's what we're using any time that we say something happens at a particular time and place. You've been using coordinate time all your life; it's the only kind we ever deal with day-to-day. And because of relativity of simultaneity, different observers use different time coordinates so have a different notion of time; but at least they can talk about events that are separated in space (the #2 case above).

We also have "proper time", which has a very specific and more restricted meaning, which I gave in my post earlier:

When we say that some amount of an observer's proper time has passed between two events, we're talking about the time along that observer's world line between the two events. It's only meaningful if the observer's world line actually passes through the events. Thus, it's easy for us to know the proper time for A to reach the event horizon: He looks at his wristwatch at the starting point, he looks at his wristwatch again as he reaches the event horizon, the difference between the two readings is the proper time that passed between leaving B and reaching the horizon.

The problem with asking how much of B's proper time passes while A falls to the event horizon (your original question) is that A reaching the horizon and B looking at his watch to see what it reads are happening at different places. So if we're going to say that B looks at his watch "when/at the same time" that A reaches the event horizon, we have to use the #2 definition of "at the same time"; the event of B looking at his watch must have the same time coordinate as the event of A reaching the horizon.

And what time coordinates are we going to use to make that determination? Because of the curvature of spacetime between A and B and the impossibility of exchanging light signals between them, there's no good answer to that question.

 

[Opti_mus]

Bufff

Well, first of all, thanks for your very extense reply.

I understand you but, at the same time, it's very difficult for me to handle something like the impossibility of B to say anything about A: he can't even say if A finally falls or not into the BH! This is very difficult to assimilate for me...

 

[Passionflower]

Bufff

Well, first of all, thanks for your very extense reply.

I understand you but, at the same time, it's very difficult for me to handle something like the impossibility of B to say anything about A: he can't even say if A finally falls or not into the BH! This is very difficult to assimilate for me...

It is really not that hard:

As I hope you would see there is no way for light or anything else from the event horizon to go to observer B, then how would observer B empirically verify the time something happens on the event horizon on his clock?

Even for something extremely close to the event horizon light takes 'ages' to return to observer B, of course for B it seems it takes forever but that is simply because the information takes a long time on his watch to return while locally it already zoomed by the event horizon a very long time ago.

 

[Nugatory]

he can't even say if A finally falls or not into the BH

But he can say that A falls into the black hole... It's easy.

He calculates A's proper time to reach the black hole - this describes what A experiences - he gets a finite value, and knows from that finite value that A will experience falling into the black hole. True, B doesn't actually see it happen, but we calculate things that we don't see all the time.

 

[Opti_mus]

But he can say that A falls into the black hole... It's easy.

He calculates A's proper time to reach the black hole - this describes what A experiences - he gets a finite value, and knows from that finite value that A will experience falling into the black hole. True, B doesn't actually see it happen, but we calculate things that we don't see all the time.

That's true, but insatisfactory for B, isn't it? What is interesting for B is to know what time takes an object to enter with the time of B, and he can't calculate it!

 

[Opti_mus]

And one last question guys (and thanks for all your answers): How can you council all that facts you told to me with what Kip Thorne tells in "BH and Time Warps", referenced to an observer (Arnold) who is falling into a BH as seen by another observer? (the translation is mine, I'm sure the words are other in the original version, but I'll try to do my best):

Does it means Arnold hasn't cross yet the horizon and that he'll never do it? Not at all. Those last signals which are duplicating constantly need an infinite amount of time to escape of the Hole's gravitatory power. Arnold crossed the horizon, moving with the speed of light, many minutes ago. It keeps arriving weak signals simply due to the fact that they've been traveling a lot of time. They're relics of the past

 

[Nugatory]

And one last question guys (and thanks for all your answers): How can you council all that facts you told to me with what Kip Thorne tells in "BH and Time Warps"

That sounds pretty much like what a bunch of people have been saying here: A crosses the horizon; B never sees (receives light from) the crossing event but that doesn't mean it doesn't happen.

 

[photonkid]

ok, but in this post PeterDonis said that it's possible to calculate how long (in Earth years) it will take to fall the last one meter - so the answer to
<quote> does A pass through the Horizon in a FINITE B's proper time or not? </>

would be yes wouldn't it?

That's still A's proper time that we're talking about, even if we're choosing to measure it in years.

yikes, I'm not sure I caught on to that even though PeterDonis said it. I can't get my head round this.

Well what about this then. If I drop a stone from my hand to the ground, I can calculate within some small error margin, what the time on my clock will be when the stone hits the ground. The fact that time is going slightly slower at ground level doesn't stop me from calculating what the time on *my* clock will be when the stone hits the ground. So if I drop a stone from one meter above an event horizon, why can we not calculate what the time on my clock will be when the stone reaches the event horizon? What's so special about the event horizon?

 

[PeterDonis]

If I drop a stone from my hand to the ground, I can calculate within some small error margin, what the time on my clock will be when the stone hits the ground.

But your clock is spatially separated from the ground. So the very concept of "the time on my clock when the stone hits the ground" assumes that you have specified a way of assigning times on your clock to events that are spatially separated from you.

In general, as I've said before, there is no unique way to do that; there are many different possible ways, and none of them are picked out by any feature of the spacetime, so none of them have any physical meaning. However, in the case of your hand and the ground, assuming that your hand is at rest with respect to the ground, there does happen to be a way to do it that is picked out by a feature of the spacetime and does have physical meaning. The feature is that the spacetime has a time translation symmetry--i.e., for observers following one of a particular family of curves, the metric looks the same at every moment of time. Your hand and the ground happen to be following such curves. And observers following such curves can set up a notion of simultaneity (i.e., of events happening "at the same time") that they all can share, and which you can use to assign a time on your clock to the event of the stone hitting the ground.

However, this method breaks down at the horizon, because at the horizon, that symmetry of the spacetime is no longer a time translation symmetry. The special family of observers are the ones following worldlines of constant r (and constant theta, phi as well, but we are leaving those coordinates out of the analysis here by assuming purely radial motion). But in order for observers to follow such worldlines, they have to be timelike; and the horizon itself is a curve of constant r which is null, not timelike. (And inside the horizon, curves of constant r are spacelike.) In other words, the metric still looks the same along curves of constant r <= 2M (i.e., at or inside the horizon), but those curves aren't timelike, so observers can't follow them, and can't use them to set up a notion of simultaneity. That means that, even if you're hovering just one meter above the horizon, and you drop a stone, there is no way to assign a time on your clock to the event of the stone hitting the horizon.

 

[Dale]

That's true, but insatisfactory for B, isn't it? What is interesting for B is to know what time takes an object to enter with the time of B, and he can't calculate it!

what B considers satisfactory is a matter of personal philosophy and preference, not physics. B can calculate the physics just fine.