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How do black holes grow?

by arindamsinha
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PeterDonis
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Nov18-12, 02:13 AM
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Quote Quote by arindamsinha View Post
That would be a way of getting around the question I have asked. If ideal Black Holes never actually get fully created, but certain regions of space containing matter keep getting closer and closer to the ideal, then there is no reason for them not to be able to grow. The event horizon actually never gets created!
The word "never" in this context doesn't mean what you think it means. All it means here is "the black hole never gets created at any finite value of the Schwarzschild time coordinate". It does *not* mean "the black hole never gets created, period". That's because the region of spacetime that is covered by finite values of the Schwarzschild time coordinate is not the entire spacetime.

Quote Quote by arindamsinha View Post
Still, I have not seen this theory anywhere. Is that what the 'O-S model' states?
No. As I said, there is no alternate model of an "eternally collapsing object" in which a black hole never forms (where I'm now using "never" in the strong sense, meaning "never anywhere in the spacetime).

Quote Quote by arindamsinha View Post
What is the 'OS-model', in brief, to explain to a layman like me?
"O-S" stands for "Oppenheimer-Snyder"; in 1939 Oppenheimer and Snyder published a paper that modeled the collapse of a cloud of "dust" (which is a term for an idealized cloud of matter with zero pressure) under its own gravity, using General Relativity. Their basic model is still valid as a highly idealized (zero pressure in the matter, as I said, and perfect spherical symmetry) qualitative picture of gravitational collapse; it is discussed in most of the major GR textbooks (including Misner, Thorne, & Wheeler, which is where I first learned about it), and in the popular book Black Holes and Time Warps, by Kip Thorne.

For our purposes here, the key point is that this model predicts that the spacetime *does* contain an event horizon and a black hole region. What happens is that the outer surface of the collapsing matter, as it gets smaller and the matter gets denser, eventually becomes a "trapped surface" (this is a modern term and was not used in the original Oppenheimer-Snyder paper); that is, it is a surface from which even outgoing light (light emitted directly radially outward) does not move outward (that is, it doesn't move to a larger radius). Once this happens, the collapsing matter is doomed to continue collapsing all the way to infinite density and infinite spacetime curvature at r = 0, leaving behind an event horizon and a black hole region inside the horizon.

(Actually, the original Oppenheimer-Snyder paper, I believe, did not carry the analysis beyond the instant when the trapped surface forms; in other words, their original analysis was incomplete. But later work has confirmed their analysis and carried it to completion; the result is what I described above.)
PeterDonis
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Nov18-12, 02:18 AM
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Quote Quote by arindamsinha View Post
My contention is that current theory (GR), does not even allow "computation" of any external matter reaching an event horizon in finite time, from an external observer's point of view, thus the event horizon cannot even be "computed" to be growing from that perspective.
Then I'm a bit unclear on your definition of "compute". Read my previous post describing the Oppenheimer-Snyder model; to me, this is a "computation", done by a "distant observer" (after all, that's what we are on Earth relative to any black hole in the universe), which shows that an event horizon *does* form. Why would this not count?

(Or perhaps the problem is the phrase "from an external observer's point of view". The computation I describe shows that no light signal from at or inside the horizon will ever reach the external observer; equivalently, it shows that the region of spacetime in which the external observer's time coordinate is finite does not contain the event horizon or the black hole. If this means the EH doesn't form "from the external observer's point of view", then that's fine, but you have to be very careful not to extend that claim into "the EH doesn't form, period", which is false; the spacetime *does* contain an event horizon and a black hole, and additional matter *can* fall through the horizon and into the black hole. So adopting the "external observer's point of view" forces you to walk a very fine line, to avoid claiming too much. In my experience, most people are not able to walk that line, so it's better, IMO, to just say flat out that the event horizon and the black hole *do* form, and that the "external observer's point of view" is the wrong one to use. But your mileage may vary.)
Mike Holland
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Nov18-12, 02:33 AM
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Quote Quote by PeterDonis View Post
In order to make any sense of this, you would have to have a model of an "eternally collapsing object" that was different than the standard one, so we could verify that both models make the same predictions, at least within our current accuracy of observation. AFAIK no one has come up with such an alternate model. Otherwise you're just saying that we have only one model and therefore only one set of predictions.
When I used the term "eternally collapsing object" I meant it to describe exactly what PAllen described in his post -

"The observer outside always 'sees' a very near black hole, with an 'almost horizon' (note, this 'almost horizon' is blacker than anything else in the universe in finite time for the external observer - however, technically, it has not quite become a horizon as seen by the outside observer. When more matter falls in, the outside obsever sees the 'almost horizon' grow. So everything is always an 'almost black hole' as seen by an outside observer."

What is observed in this case, and in the case of a fully formed black hole, is a region "blacker than anything else in the universe". But perhaps I should not have used that ECO term, because it has been used elsewhere in other contexts.

You also said "Another way of putting it: claiming that the event horizon never forms because "it takes an infinite time as far as a distant external observer is concerned" is equivalent to claiming that the region of spacetime in which the distant observer's time coordinate is finite is the entire spacetime. This claim is false. ".

I understood this to be the conclusion of the Oppenherimer-Snyder calculations. Where do they allow for any distant observer actually seeing a black hole form in a finite time? OK, I accept that their ideal observer is stationary relative to the forming BH, and an infinite distance fom any space-distorting mass, but I don't believe these conditions affect the conclusion. There are other observers who see the black hole form, but they are local to it, and not distant. So I am not making a claim about all spacetime. And I'm not including spacetime inside a black hole, because that gets too complicated.

Mike
PAllen
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Nov18-12, 10:33 PM
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Quote Quote by arindamsinha View Post
I understand that. There seemed to be too many issues coming up between the terms "see", "detect" and "compute", so I was trying to combine them to state what I mean.

In some senses, "see" may be possible, as explained by the growing shadow.

"Detect" is possible through measuring the gravity growth.

"Compute" is of course possible, based on current theory.

Let us then drop "see" and "detect". My contention is that current theory (GR), does not even allow "computation" of any external matter reaching an event horizon in finite time, from an external observer's point of view, thus the event horizon cannot even be "computed" to be growing from that perspective.

I hope that makes it a little clearer on what my question is.
The problem here is 'time from an external observer's point of view'. If, in fact you talk about seeing and detecting - these are physical and there are not ambiguities. There can be no dispute about what the distant observer sees or detects.

However, as soon as we talk about 'time at a distance' there is great ambiguity. This is not physically measurable. There are perfectly reasonable definitions (and that is all their can be - definitions) of 'time at a distance' such that a distant observer computes that collapse has occurred in finite 'time at a distance' and further infallers reach the singularity in finite 'time at a distance'.

Note that SC coordinate time is a peculiar definition of time for distant observer in the following sense (in classical GR):

- there events I cause, and observers receiving signals from me that are declared never to be simultaneous with any event on my future world line, even carried to infinity.

If you simply modify SC simultaneity to include events caused by an external observer, then you attach finite times to event crossing and arrival at singularity.
PAllen
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Nov18-12, 10:52 PM
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Let me make concrete a different definition for times assigned to distant events for distant observer of a collapse leading to spherically symmetric BH. This definition will match SC time for an observer at infinity for events 'near the observer', but will differ more and more for other events. [edit: removed statement about observer not 'near infinity', that is not quite true].

We posit the world line of some static, distant obsever (reference observer). For any event E, we find the event on this observer's world line such that light will reach the chosen event (call clock time for the reference observer for this emission event T0). The chosen event E has some SC r coordinate (r1). The reference world line has some other r coordinate(r0, always). We assign to E a T coordinate of: T0 + (r1-r0)/c.

The defines a perfectly plausible alternate time coordinate for the static observer at r0. It can be combined with SC r, theta and phi coordinates.

With this modification to SC time, it is now true that r0 observer computes finite times for infall event horizon crossings, and infall times of reaching singularity.
PeterDonis
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Nov18-12, 11:12 PM
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Quote Quote by Mike Holland View Post
I understood this to be the conclusion of the Oppenherimer-Snyder calculations.
You understood what to be the conclusion of the O-S calculations? That the distant observer never sees the black hole form, or that it never forms, period? The first *is* the conclusion of the O-S calculations; the second is not.

Quote Quote by Mike Holland View Post
There are other observers who see the black hole form, but they are local to it, and not distant. So I am not making a claim about all spacetime. And I'm not including spacetime inside a black hole, because that gets too complicated.
Ok, so it looks like your understanding was the first of the two I described above.
arindamsinha
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Nov19-12, 12:30 AM
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I have been trying to follow the discussions above and make some sense.

It appears to me now there is some disagreement between interpretations of GR on this point, with three possibilities supported by different scientists, i.e. black holes/event horizons:
  1. Do form in finite time for certain observers, but not for certain other observers
  2. Do form in finite time for all observers, and
  3. Never fully form in finite time for any observer, but keeps getting closer and closer
Is it that there are different interpretations (or even follow-up theories) of GR on this aspect by different scientists?
PAllen
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Nov19-12, 12:58 AM
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Quote Quote by arindamsinha View Post
I have been trying to follow the discussions above and make some sense.

It appears to me now there is some disagreement between interpretations of GR on this point, with three possibilities supported by different scientists, i.e. black holes/event horizons:
  1. Do form in finite time for certain observers, but not for certain other observers
  2. Do form in finite time for all observers, and
  3. Never fully form in finite time for any observer, but keeps getting closer and closer
Is it that there are different interpretations (or even follow-up theories) of GR on this aspect by different scientists?
Speaking only of classical GR, it is crucial to distinguish observables from conventions. I believe there is essentially no dispute about the following observational statements (for the case of collapse, followed by possible later infall):

- An observer riding with collapsing matter or falling in later will cross an EH and reach the singularity in finite time on their watch.

- An observer remaining outside will never see an EH form, nor will they see any later infaller cross an EH (because it won't be seen to form). They will detect (gravitationally) increase in mass of 'black body' as new matter falls in, but no outside measurement will detect any information that an EH has actually formed. However, the delta between this observed almost BH and an eternal BH with actual event horizon, will grow smaller exponentially to the point where no conceivable measurement can distinguish.

As soon as you go from what observers measure to what they might consider to be true based on computation you can make the following statements:

- that the universe contains a BH according to theory (classical GR), is something any observer may compute. They all get the same result that the universe does contain such a thing (given the appropriate collapse).

- There is no objective meaning to 'time at a distance' for events not observed. There is no objective meaning to 'what is over there now'. Depending on different plausible choice for this, you can say a distant observer considers an EH never to form, or to form at some well defined finite time. In either case, it is true that (per computation), the universe as a whole contains a BH with EH.
arindamsinha
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Nov19-12, 02:15 AM
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Quote Quote by PAllen View Post
Speaking only of classical GR, it is crucial to distinguish observables from conventions. I believe there is essentially no dispute about the following observational statements (for the case of collapse, followed by possible later infall):

- An observer riding with collapsing matter or falling in later will cross an EH and reach the singularity in finite time on their watch.

- An observer remaining outside will never see an EH form, nor will they see any later infaller cross an EH (because it won't be seen to form). They will detect (gravitationally) increase in mass of 'black body' as new matter falls in, but no outside measurement will detect any information that an EH has actually formed. However, the delta between this observed almost BH and an eternal BH with actual event horizon, will grow smaller exponentially to the point where no conceivable measurement can distinguish.
This is fine. What we (the external observers) have observed in the Universe are really not actual black holes, but so close as to be measurably indistinguishable from one - that makes sense.

The only beef I have about this (not with you, but with GR), is that a 'riding' observer happens to experience an 'event' that an external observer cannot accept will ever happen, no matter how long he waits for it.

Quote Quote by PAllen View Post
As soon as you go from what observers measure to what they might consider to be true based on computation you can make the following statements:

- that the universe contains a BH according to theory (classical GR), is something any observer may compute. They all get the same result that the universe does contain such a thing (given the appropriate collapse).
This is OK, if we substitute 'may contain' for 'contains'. Any black holes that actually exist must have always existed from the Big Bang. A collapsing star since the Big Bang may get asymptotically close, but never achieve a 'hard' event horizon (for external observers).

Quote Quote by PAllen View Post
- There is no objective meaning to 'time at a distance' for events not observed. There is no objective meaning to 'what is over there now'. Depending on different plausible choice for this, you can say a distant observer considers an EH never to form, or to form at some well defined finite time. In either case, it is true that (per computation), the universe as a whole contains a BH with EH.
This part I cannot agree with. For a distant observer, he must conclude an EH never forms, by his clock, in finite time.

Also, he can only conclude that the Universe may contain objects very close to being black holes, but does not contain actual black holes, unless they were formed along with the Big Bang.
harrylin
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Nov19-12, 04:25 AM
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Quote Quote by arindamsinha View Post
That's what I was thinking after seeing Mike Holland's response.

That would be a way of getting around the question I have asked. If ideal Black Holes never actually get fully created, but certain regions of space containing matter keep getting closer and closer to the ideal, then there is no reason for them not to be able to grow. The event horizon actually never gets created!

Still, I have not seen this theory anywhere.
As a result of a little "Googling" from my part this was recently discussed in several other black hole threads, for example from here:
- http://www.physicsforums.com/showthread.php?p=4129133
and from here:
- http://www.physicsforums.com/showthread.php?p=4163727
and from here:
- www.physicsforums.com/showthread.php?p=4164194
(and also a little in here:
- www.physicsforums.com/showthread.php?t=647627)

Thus it's a matter of interpretation of models, and the interpretation that you are looking for apparently began with Einstein's 1939 paper. In that interpretation a falling observer will not experience a falling through the horizon ("The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality").

For me that topic has been sufficiently discussed now, and it's sufficiently clear, so I'll leave it up to you in your thread.

PS: I see your last remark in post #27. Just like me, your beef is not with GR but with an interpretation of GR. That is a big difference.
PeterDonis
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Nov19-12, 09:17 AM
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Quote Quote by arindamsinha View Post
The only beef I have about this (not with you, but with GR), is that a 'riding' observer happens to experience an 'event' that an external observer cannot accept will ever happen, no matter how long he waits for it.
The external observer will never directly *observe* the "event" (of the EH forming and something falling into it), but that doesn't mean he can't *accept* that it will happen. He can compute that it will happen using a well-supported scientific theory; why shouldn't that be sufficient grounds for him to "accept" that it will happen, even though he can't directly observe it?

Quote Quote by altergnostic View Post
Any black holes that actually exist must have always existed from the Big Bang.
Black holes can't be formed this way.

Quote Quote by altergnostic View Post
A collapsing star since the Big Bang may get asymptotically close, but never achieve a 'hard' event horizon (for external observers).
You have to be very, very careful about that word "never". See what I said above about the external observer "accepting" that the BH forms even though he can't directly observe it.

Quote Quote by altergnostic View Post
This part I cannot agree with. For a distant observer, he must conclude an EH never forms, by his clock, in finite time.
He can conclude that no EH forms in a finite time by his clock, yes. But you appear to be putting in an additional premise with the word "never": that *any* event, anywhere in the universe, must have a finite time, by his clock, associated with it. That is false; there is a whole region of spacetime (the EH and the BH region inside it) that *cannot* be assigned a finite time on the distant observer's clock; at least, not if he uses the most natural method of assigning times on his clock to events which are spatially distant from him. As PAllen pointed out, the real point here is that in GR, there is no unique method of doing that; there are multiple ways of assigning times on the distant observer's clock to events, and no one of them is "right". That is why we focus on things that don't depend on arbitrary decisions like how we assign times to distant events; when we focus on those things, we find that (according to our best current theoretical understanding) there *is* an event horizon and a black hole, even though the distant observer can't see it.
PAllen
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Nov19-12, 09:27 AM
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Quote Quote by arindamsinha View Post
This is fine. What we (the external observers) have observed in the Universe are really not actual black holes, but so close as to be measurably indistinguishable from one - that makes sense.

The only beef I have about this (not with you, but with GR), is that a 'riding' observer happens to experience an 'event' that an external observer cannot accept will ever happen, no matter how long he waits for it.
This is s feature of the belief system of the observer not an issue of physical theory. Why wouldn't a distant observer accept that what they calculate for an infalling clock's own experience is true? The physical theory has no conflicting events at all. It simply has a feature that no observer can detect all events in the universe. Why is this such a bizarre concept?
Quote Quote by arindamsinha View Post

This is OK, if we substitute 'may contain' for 'contains'. Any black holes that actually exist must have always existed from the Big Bang. A collapsing star since the Big Bang may get asymptotically close, but never achieve a 'hard' event horizon (for external observers).
Again, a statement that only one class of observers defines reality is at odds with GR. GR and SR say all observers measurements are meaningful. An external observer can easily compute what in infalling observer detects. There is no conflict with what an external observer detects (that is, no conflicting observation of the same event). There is, again, simply the feature that one observer experiences events that another observer cannot detect. I don't believe you generally assume, in life, that if you can't detect something it didn't happen.
Quote Quote by arindamsinha View Post

This part I cannot agree with. For a distant observer, he must conclude an EH never forms, by his clock, in finite time.

Also, he can only conclude that the Universe may contain objects very close to being black holes, but does not contain actual black holes, unless they were formed along with the Big Bang.
And this is true only if you say there is a law against the external observer computing predictions from theory about events they cannot see. That is an absurd prohibition. Note, we agree about the observational statements, which I listed first in the post you respond to. But you asked about what they may compute. Why is an external physicist prohibited from computing predicted events that they cannot observe (but someone else, e.g. an infaller, can)?

Further, independent of observations, computationally there are many ways to relate distant events in the universe as happening at your 'now'. Both GR and SR say this is a matter of convention not physics. There are numerous simultaneity conventions a distant observer may choose such that they compute an EH and singularity have formed [I gave a simple, physical, definition of one in my post #23]. This in no way contradicts that they also compute they will never detect any information from these events.
A.T.
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Nov19-12, 09:52 AM
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Quote Quote by PAllen View Post
- An observer riding with collapsing matter
- An observer remaining outside
Is it possible to present a smooth transition between those two cases?

Consider an infinite number of observers, all simultaneously starting at the same point outside the BH, but undergoing different proper accelerations ranging from a=0 (free fall into the BH) to a=a_hover (allows keeping r=const). How would the free falling one be observed by the others? Which ones would see him cross the EH and after which time on their clocks?
PAllen
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Nov19-12, 10:05 AM
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Quote Quote by A.T. View Post
Is it possible to present a smooth transition between those two cases?

Consider an infinite number of observers, all simultaneously starting at the same point outside the BH, but undergoing different proper accelerations ranging from a=0 (free fall into the BH) to a=a_hover (allows keeping r=const). How would the free falling one be observed by the others? Which ones would see him cross the EH and after which time on their clocks?
As long as an observer remains outside the EH, they will not see any object cross the EH. If they stop madly accelerating (it takes asymptotically infinite acceleration as measured by an accelerometer to avoid crossing the EH as you get close to it; and this, by the way, is trivial for a distant observer to compute using only external SC coordinates) and fall through the EH, they will see prior infallers as of when they crossed the EH.

The behavior is very similar to a uniformly accelerating rocket and its Rindler horizon. If the rocket drops a series of probes, it will see them freeze, one after the other at its Rindler horizon (this is all SR). Only if it stops accelerating, so that the Rindler horizon overtakes it, will it see these dropped probes crossing the horizon, and see their subsequent history.
DrGreg
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Nov19-12, 02:54 PM
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Quote Quote by A.T. View Post
How would the free falling one be observed by the others? Which ones would see him cross the EH and after which time on their clocks?
There is only one way for you to visually observe something crossing a perpetual event horizon, and that is for you to cross the horizon yourself. That moment is precisely when you will see it, as the horizon itself is a wavefront of light travelling away from the hole. But there's no flagpole to mark the location of the horizon so you wouldn't notice it had happened; you'd see the other object moving continuously from outside to inside the horizon.

Anyone moving radially with insufficient constant proper acceleration to hover will fall through the horizon in finite proper time (at which moment they'll see falling through anyone else who fell through before them).
DrGreg
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Nov19-12, 03:34 PM
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Quote Quote by arindamsinha View Post
For a distant observer, he must conclude an EH never forms, by his clock, in finite time.
The problem is that you can only use your own clock to measure things that happen right next to you. To measure things a distance away from you, you have to invent a definition of simultaneity and maybe use someone else's clock that has been synchronised using your definition. Some events may lie outside the region you have chosen to apply your definition. Does this mean those events don't exist?

Consider the following example from special relativity. A ("Born rigid") rocket has constant proper acceleration in empty space, far from any gravity. At any instant in time, you can define distance and simultaneity according to an inertial observer who is momentarily at rest relative to the rocket. You can "glue" all these observers together to form a valid coordinate system ("Rindler coordinates") for the rocket. Here's a spacetime diagram showing the distance X and time T of an inertial frame, and the pink grid lines show the rocket's Rindler coordinates x and t. The curved line x=1 represents one point on the rocket.


Dr Greg, Wikimedia Commons, CC BY-SA 3.0

The white area above this chart is not covered by the pink grid. Even after waiting an infinite time t, none of the events in that area become simultaneous with an event on the rocket as measured by the rocket. Yet these events do have a finite inertial T coordinate. Would you say that the events in those areas "do not exist"?
arindamsinha
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Nov19-12, 09:51 PM
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Quote Quote by DrGreg View Post
The problem is that you can only use your own clock to measure things that happen right next to you. To measure things a distance away from you, you have to invent a definition of simultaneity and maybe use someone else's clock that has been synchronised using your definition. Some events may lie outside the region you have chosen to apply your definition. Does this mean those events don't exist?...

...none of the events in that area become simultaneous with an event on the rocket as measured by the rocket. Yet these events do have a finite inertial T coordinate. Would you say that the events in those areas "do not exist"?
You have outlined the situation quite well. Then the question is, when, by our own clock, does the event happen?

I feel the event I am talking about (matter crossing EH) is always in the future, getting asymptotically closer to the EH, but never reaching it. Yes, by our own clock, and my question is based on our own clock (can black hole EH grow for external observers?).

I understand that the event may actually happen for an observer falling into the black hole, but by our clock, this falling observer also never reaches the event horizon! So I stand by the statement that the event "does not exist" or come to pass ever, by our clock.

This is where I see a conflict. From our point of view, drawing a geometric parallel, two lines are asymptotic and only meet at infinity, and never cross over. For the observer falling into the BH, not only do the two lines meet, but they even cross over.

I am getting the feeling that there is still some lack of appropriate interpretation of GR in this area, or perhaps, GR may have to be further generalized in this area for a proper interpretation of Universal events (If we accept the astronomical conclusion that black holes exist, and grow in finite time of external observers' clocks).

That is, unless we accept the other possible explanation that black holes never really fully form, but get aymptotically closer to forming all the time.
PeterDonis
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Nov19-12, 10:31 PM
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Quote Quote by arindamsinha View Post
I understand that the event may actually happen for an observer falling into the black hole, but by our clock, this falling observer also never reaches the event horizon! So I stand by the statement that the event "does not exist" or come to pass ever, by our clock.
You can look at it this way, as long as you only draw valid conclusions from your statement. For example, it is valid to conclude that events at or inside the event horizon can never causally affect you (because no causal influence can travel faster than light), so in that sense you can behave as if they don't "exist". But it isn't valid to conclude that *nobody* can ever feel any causal influence from those events, because someone could always choose to fall into the black hole.

Quote Quote by arindamsinha View Post
This is where I see a conflict. From our point of view, drawing a geometric parallel, two lines are asymptotic and only meet at infinity, and never cross over.
You have to be careful interpreting what "only meet at infinity" means. You appear to be picturing it the way it would work on a flat Euclidean plane: two parallel lines on a plane "only meet at infinity", meaning that you can extend them to any finite length you like and they will never meet.

This is *not* true for the worldlines of two infalling objects that meet inside the horizon. "Length" along worldlines in spacetime means proper time, and the two objects will meet in a *finite* amount of proper time. You already agree with this, but you apparently haven't fully comprehended what it means. It means that the two lines are *not* "infinitely long" before they meet below the horizon, in the way that parallel lines on a Euclidean plane are "infinitely long" before they meet. You can only extend the two worldlines for a finite length before they meet, even though doing so covers an infinite range of the distant observer's time coordinate.

In other words, when you have extended the two lines "to infinity" according to your clock, you have only extended them to a finite length in geometrically invariant terms. You have chosen a time coordinate that is so distorted at the horizon that it extends finite lengths (i.e., finite proper times) so they look like infinite lines. The analogy you are trying to draw with "infinite lines" in ordinary plane geometry does not work; the lines that "look infinite" to the distant observer because of his choice of time coordinate are *not infinite*.

Quote Quote by arindamsinha View Post
I am getting the feeling that there is still some lack of appropriate interpretation of GR in this area
No, it is just that you don't fully understand what the standard GR picture says. The above may help.

Quote Quote by arindamsinha View Post
That is, unless we accept the other possible explanation that black holes never really fully form, but get aymptotically closer to forming all the time.
No, this "explanation" does not work; it amounts to claiming that the lines that "look infinite" in your time coordinate really are infinite, in the way parallel lines on the Euclidean plane are infinite. That is not correct. See above.


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