Black hole matter accumulation

[A-wal]

Largely because you refuse to even attempt to learn the math which expresses all of these concepts in a single coherent, logical, and rigorous framework. Your anti-math prejudice is preventing you from learning, so there is little anyone else is going to be able to do for you here.

WannabeNewton's explanation in Schwarzschild coordinates goes along with my earlier explanation to you for the same in Rindler coordinates. It is all there. But you won't even make an effort. Each time you receive an answer you close your eyes and mind and look away and go back to claiming that you have not received an answer.

Answer me this. How can you reasonably expect anyone to be able to present a logical picture to you if you refuse to look at the math which expresses the logic when it is presented to you?

I'm trying to build a logical picture of what the maths says. Looking at the maths itself won't help because I can't build a logical picture from that. I don't think there's a logically consistent way of picturing what you're describing.

Where exactly is the contradiction?

The same event happens, it just happens at two different times depending on the observer's FoR.

It's identical to a scenario in SR where one observer is traveling at, for all intents and purposes, c. While a split second passes for him before his dropped spoon hits the deck of his rocketship, to an external observer, the entire universe ages and dies before the spoon hits the deck.

Same event two different timescales.


Explicitly, where is the contradiction?


From one frame of reference, sure. From another it takes split second. Why is this so difficult?

This is a definite yes/no situation that can't be Lorentzed away. The instant that object reaches the event horizon it causes a paradox. Then there's the fact that no object can cross until you do and then everything crosses at the same time. And the front of an object won't be able to reach the horizon before the back of it, even on the sub-atomic scale.

And can you prove that they can always escape, without *assuming* that they can? Every argument you've offered has had a hidden assumption that's equivalent to your conclusion.

They never reach the horizon from the outside.

They say different things because they're *about* different things. One is about how a hovering observer far away from the hole assigns space and time coordinates to events. The other is about how an observer free-falling into the hole assigns space and time coordinates to events. Neither of them is saying anything that contradicts the other. The first simply can't assign coordinates to events inside the horizon, while the second can. That's not a contradiction, just a limitation of the first observer's coordinates.

That's just another way of saying the object never reaches the horizon from the outside.

Have you looked at a Kruskal chart or a Penrose diagram of Schwarzschild spacetime? Those are easy ways to visualize exactly what we've been saying. In fact, I've referred to them repeatedly in the other thread.

No. I'll do it before my next post. Schwarzschild spacetime? That says nothing can reach a horizon doesn't it? I'll have a look.

 

[DaveC426913]

This is a definite yes/no situation that can't be Lorentzed away. The instant that object reaches the event horizon it causes a paradox. Then there's the fact that no object can cross until you do and then everything crosses at the same time. And the front of an object won't be able to reach the horizon before the back of it, even on the sub-atomic scale.

I think you have some misunderstandings about this. Your statements above don't make a lot of sense.

The instant that object reaches the event horizon it causes a paradox.

What paradox? In one FoR that event happens after a split second, in another it happens after the universe is old. Exactly what is the paradox?

Then there's the fact that no object can cross until you do and then everything crosses at the same time.

You must understand the distortions we're talking about. Remember, we're saying it never crosses from a distant external observer's PoV. If you're near the EH it's a different story. Kind of like when talking about SR, we were talking about moving at .99999c and you're now introducing someone moving at .999999999c. The distortions are magnified.

And the front of an object won't be able to reach the horizon before the back of it, even on the sub-atomic scale.

No. You are seeing this is an absolute, black and white thing. It is a scale, and we are looking at smaller and smaller increments.

From near the EH, it is perfectly possible to see objects fall into the EH.

 

[PeterDonis]

I don't think there's a logically consistent way of picturing what you're describing.

Too bad. There is. What there isn't is a way of picturing it that is logically consistent *with your assumptions*. You refuse to let go of your assumptions, even though they are not valid for all situations.

This is a definite yes/no situation that can't be Lorentzed away. The instant that object reaches the event horizon it causes a paradox. Then there's the fact that no object can cross until you do and then everything crosses at the same time. And the front of an object won't be able to reach the horizon before the back of it, even on the sub-atomic scale.

They never reach the horizon from the outside.

That's just another way of saying the object never reaches the horizon from the outside.

You haven't given a valid argument for any of this. Basically you just keep saying "I can't see how this can work", which may be a valid statement about your state of mind, but says nothing about the actual physics.

Schwarzschild spacetime? That says nothing can reach a horizon doesn't it?

No. Schwarzschild *spacetime* is just the complete solution to the Einstein Field Equation that we've been talking about. It's a geometric object, not a coordinate system. (The Schwarzschild *coordinate system* doesn't say nothing can reach the horizon either, just that an external observer won't ever see light rays from someone reaching and crossing the horizon. But I wanted to be clear what I meant by "Schwarzschild spacetime".)

 

[Dale]

I'm trying to build a logical picture of what the maths says. Looking at the maths itself won't help because I can't build a logical picture from that. I don't think there's a logically consistent way of picturing what you're describing.

On the contrary, math is the only way to form a logical picture. Math is the language of logic. If you want a logical picture then you need to learn the relevant math. The mathematical framework of Riemannian geometry is what guarantees the logical self-consistency of GR.

Your anti-math stance is exactly the thing which will guarantee that you will continue to be unable to understand the logical picture.

 

[DrGreg]

A-wal: your problem is that you can't see how the two statements

1. "the free-falling object reaches the event horizon and passes through it in a finite time according to the object's own clock"
2. "the free-falling object takes an infinite time to reach the event horizon according to a distant object hovering at a fixed height"

can both be true. And you won't listen to any mathematical argument to prove this.

Well, let's forget black holes and see how the above statements can both be true in the absence of any gravitation at all. Let's consider flat spacetime, i.e. no gravity, and apply the equivalence principle to the above statements. They now become

1. "the inertially-moving object reaches the 'horizon' and passes through it in a finite time according to the object's own clock"
2. "the inertially-moving object takes an infinite time to reach the 'horizon' according to a distant object accelerating with constant proper acceleration"

At this stage I haven't specified what the 'horizon' is in this context.

Here is a diagram that I first posted in the thread about the Rindler metric, post #9.

Ignore the diagram on the right. The diagram on the left is a standard spacetime diagram from the point of view of an inertial observer. Vertical distances represent time and horizontal distances represent distance, both in the frame of an inertial observer. The curved black line represents an observer who is experiencing constant proper acceleration.

Each curved red line is a constant distance from the black observer as measured by himself. Each green line represents simultaneity as measured by the black observer. So the red and green lines form a grid which the black observer can use to assign space and time coordinates to every event in the triangular white region. Note there are no lines at all in the blue and yellow regions: the black observer's coordinate system never reaches these regions.

On this diagram, the speed of light is represented by any line drawn at 45 degrees. Note that it is impossible for light from any event in the blue region to ever reach the black observer. The light would have to pass from the blue region to the white region faster than light, which is impossible. Thus the border between blue and white behaves as an "apparent horizon" to the black observer.

Now suppose at the event (t=0, x=0) the black observer drops an apple from his spaceship. The apple will subsequently travel inertially, straight up the diagram along the line x=0. After 10 years of its own time it will reach the point (t=10, x=0). It will then continue uninterrupted through the blue region.

From the black observer's coordinate point of view we must measure the motion using the red and green gridlines.

Distance is obtained from the red lines so we see that when the apple reaches the blue/white horizon it has crossed 10 red lines, so the distance coordinate is −10 light years.

Time is obtained from the green lines so we see that when the apple reaches the blue/white horizon it has crossed an infinite number of green lines, so the time coordinate is ∞ years.

And what does the black observer actually see with his eyes? Well once the apple is in the blue zone, no light from the apple can reach the black observer. So the black observer never sees the apple cross into the blue zone. As it approaches the blue/white 'horizon' the apple appears to slow down and takes an infinite time to reach the 'horizon'. This should be clear if you consider light being emitted at an angle 45 degrees upwards by the apple as it travels vertically upwards on the diagram.

So we see that an inertial observer and a non-inertial observer can disagree over whether something takes a finite or infinite time.

(If there's anyone reading this who doesn't share A-wal's allergy to mathematics, you can get more details by looking up "Rindler coordinates" or "Rindler metric", or follow the blue link earlier in this post and the rest of the thread it is in.)

 

[Dale]

This isn’t just relative simultaneity. This is a direct contradiction.

There are two statements here:
1) The proper time for a free falling object to cross the event horizon is finite
2) The coordinate time for a free falling object to cross the event horizon is infinite

These two statements are in no way contradictory. They are different statements, and both are true. I showed how both were true for a Rindler horizon on the other thread, and WannabeNewton showed how 1 is true for a Schwarzschild horizon (and you are well aware of 2 for Schwarzschild).

You have been shown logically, your question has been answered clearly. Now it is simply up to you to decide if you really want to know the answer to your question or if you want to continue to pretend that it is illogical.

At this point, your continued ignorance is a deliberate choice that no one besides you can resolve.

 

[DaveC426913]

A-wal, I understand your aversion to the mathematics. I am not advanced enough to follow it. And I understand the expectation that the universe should logically be able to be made sense of without it.

But here's the problem: all conceptual, logical models based on what we already know must be flawed. Our conceptual models depend on using concepts we already know. But we are in new territory here, concept-wise.

Take the electron. We say it has a property called spin. The math of QM describes accurately what really happens but we had no conceptual model of this kind of spin. We couldn't have had one. The best we could do was say "This is how it really works, the math shows it and predicts correctly the behaviour. Now that we see that, we can only then form a conceptual model of an electron that has this thing called spin" which we still don't have a conceptual model of. (at least I don't. How does a particle go around a half time and come back to the same place?)

The same is true with something like Einsteinian SR. We had no mental model for time being a dimension that can be distorted just like space. Only after the math was shown do we accept it and build a conceptual model, breaking the old one where time and space were absolute backgrounds against which events happened.

You accept those. But your conceptual model of them came after the math.

Same with the EH of a BH. We have no precedent for a conceptual model of how this works. Any time we try to make one it must be wrong because it doesn't actually work like anything else. We must first satisfy ourselves with the math that it really is this way, and only then can we build a conceptual model.

The math describes how it really works. No conceptual model you currently have can do so. Full stop. You have yet to build that new conceptual model.

 

[zonde]

A free falling observer physically crosses the event horizon however no information can reach an outside observer about this. Outsiders simply cannot see and measure it.

Does it means that outside observer have no means how to test the statement that "a free falling observer physically crosses the event horizon"?

 

[PAllen]

Does it means that outside observer have no means how to test the statement that "a free falling observer physically crosses the event horizon"?

Yes, this is true (absolutely in classical GR). That's why, in my view, while it is correct to say the horizon singularity is coordinate artifact, the horizon itself is a physical phenomenon for a class of observers (all those who will live). Similarly, the Rindler Horizon is a physical phenomenon for a much more specialized class of observers in flat spacetime (the non-existent eternally accelerating observers). So, observer dependent - yes, coordinate dependent - no.

 

[PeterDonis]

Does it means that outside observer have no means how to test the statement that "a free falling observer physically crosses the event horizon"?

Yes, if by "test" you mean "test by actually receiving physical signals from events inside the horizon", as opposed to "calculate the free-falling observer's worldline, based on what you *can* observe of it outside the horizon, to infer the portion of it inside the horizon that you can't observe". An outside observer can certainly do the latter.

 

[PAllen]

Yes, if by "test" you mean "test by actually receiving physical signals from events inside the horizon", as opposed to "calculate the free-falling observer's worldline, based on what you *can* observe of it outside the horizon, to infer the portion of it inside the horizon that you can't observe". An outside observer can certainly do the latter.

Yes, but if we're talking about validating theories, there is no way to distinguish GR + a new theory that takes over at the event horizon (and I'm sure you're aware that there are peer reviewed proposals of this type), from classical GR. This is an in principle limitation, more fundamental than the practical issues of 'planck energy' predictions. It's sort of unfortunate: GR predicts the un-testability of some of its predictions.

The only way I see around this is if conditions were realized for a naked singularity. Then you might observe that GR holds sway at least until very close to the singularity, and you could rule out theories of 'new physics at the horizon' as implausible.

[EDIT: let me state my personal belief is against 'new physics at the horizon' type theories. I don't see this as a plausible boundary of validity for GR. I do see the Rindler horizon and the event horizon as physically equivalent. Thus, I would expect GR breakdown to be unrelated to the horizon.]

 

[MrNerd]

From what I've read on this page, the problem seems to be conceptualizing the differences in time perception. Maybe I can help(hopefully).

When an object is in a gravitational field, time slows down. Why is this? I'm not completely sure myself, but I believe one way to think of it is because the light moving from the body is sucked in, making it harder to escape. The light is redshifted leaving the body, having the same effect as if the radiating source were slowed down in time. This leaves an outside observer perceiving time slowing down.

In math terms, the time passing in the slowed down time object can be given by:

dt$_{shell}$ = (1 - $\frac{2M}{r}$)$^{1/2}$dt

dt$_{shell}$ = time for slowed down time object
M = mass of the object(in meters, convert from kg to meters by multiplying by $\frac{G}{c^{2}}$
r = reduced radius from the object(Given by dividing circumference by 2$\pi$)
t = far away time(time to the outside observer, not slowed down by gravity)

Note that the (1 - $\frac{2M}{r}$)$^{1/2}$ will always be less than 1, hence time flows slower relative to the outside observer. Also, as r approaches 2M, the time of an outside observer passes faster and faster. This means you would never see the object cross the event horizon, as too much time passes on your side. Additionally, the light is redshifted more and more as r approaches 2M. So you couldn't physically see the object cross the event horizon with your eyes, anyway, since the light would be redshifted out of the visible light range. BUT the falling object does eventually fall in, using the falling object's perspective. Time is not felt to be faster/slower on yourself if you were to fall into the black hole. In the falling object's time, nothing changes; it just keeps zooming, passing the event horizon without so much as a bump. It is only the perceival of time by an outside observer that will prevent the observation of the falling object reaching the horizon. The key word here is perceived time.

Perhaps a book can help you. I've been reading Exploring Black Holes by Taylor and Wheeler. There's no super advanced math like tensors in it, which are essential for a totally complete understanding of General Relativity. There is some calculus, but it's mostly basic calculus like derivatives and integrals of a single variable, so you can get a fairly good idea of what is going on. It is only an introduction to General Relativity, after all. If you truly wish to understand physics, you cannot just ignore the math. You MUST learn it if you want to get anywhere in physics like this.

 

[PeterDonis]

Yes, but if we're talking about validating theories, there is no way to distinguish GR + a new theory that takes over at the event horizon (and I'm sure you're aware that there are peer reviewed proposals of this type), from classical GR. This is an in principle limitation, more fundamental than the practical issues of 'planck energy' predictions. It's sort of unfortunate: GR predicts the un-testability of some of its predictions.

Yes, as long as the new theory left everything outside the horizon the same, an outside observer wouldn't be able to tell the difference.

In fact, I believe that, since Schwarzschild spacetime is exactly spherically symmetrical, it is unstable against small perturbations, so it could not describe an actual black hole (which would not be exactly spherically symmetrical), even if no non-standard GR physics came into play. In Black Holes and Time Warps, Kip Thorne says that a real black hole spacetime, close to the singularity, would look like a BKL singularity:

http://en.wikipedia.org/wiki/BKL_singularity

I believe such a spacetime could still look pretty close to spherically symmetrical far away from the hole, so we wouldn't be able to know for sure exactly what spacetime inside the horizon looked like (the BKL family of solutions is chaotic, so even if we could somehow know that the spacetime inside the horizon was a BKL solution, we wouldn't be able to tell which one).

This comes into play even more for the case of a Kerr black hole, since that spacetime, mathematically speaking, allows closed timelike curves in the region inside the inner horizon. But again, the Kerr solution is (I believe) unstable against small perturbations, so an actual rotating hole might have a quite different, chaotic, BKL-like spacetime inside the horizon, while looking the same outside. I found some interesting papers on arxiv about this some time back; if I can find the links again I'll post them.

 

[zonde]

From what I've read on this page, the problem seems to be conceptualizing the differences in time perception. Maybe I can help(hopefully).

When an object is in a gravitational field, time slows down. Why is this? I'm not completely sure myself, but I believe one way to think of it is because the light moving from the body is sucked in, making it harder to escape. The light is redshifted leaving the body, having the same effect as if the radiating source were slowed down in time. This leaves an outside observer perceiving time slowing down.

I believe that this picture is rather popular. Otherwise I can't understand why it is believed that simultaneity of a body inertially moving toward gravitating body is more meaningful than for example simultaneity of a body that is inertially moving away from gravitating body.

But I have tried to check consistency of this picture and it appears inconsistent. If simultaneity is sloped toward gravitating body then we can speak about flux of space-time toward gravitation source. But this flux is not conserved. So I have scraped the idea.

Maybe there are even testable predictions of this model. Then it can be tested (and possibly falsified) experimentally or demonstrated that it does not agree with observations that we already have.

 

[Dale]

This is an in principle limitation, more fundamental than the practical issues of 'planck energy' predictions. It's sort of unfortunate: GR predicts the un-testability of some of its predictions.

They are not untestable. They are just unpublishable.

 

[A-wal]

I think you have some misunderstandings about this. Your statements above don't make a lot of sense.


What paradox? In one FoR that event happens after a split second, in another it happens after the universe is old. Exactly what is the paradox?


You must understand the distortions we're talking about. Remember, we're saying it never crosses from a distant external observer's PoV. If you're near the EH it's a different story. Kind of like when talking about SR, we were talking about moving at .99999c and you're now introducing someone moving at .999999999c. The distortions are magnified.

No. You are seeing this is an absolute, black and white thing. It is a scale, and we are looking at smaller and smaller increments.

From near the EH, it is perfectly possible to see objects fall into the EH.

Because it is an absolute, black and white thing. Are you sure you can observe an object reaching an event horizon if you're close enough? So you could watch it happen and then move away because you haven't reached the horizon yet? Then what happens when you move out to a point in space-time where the object hasn't crossed yet?

Too bad. There is. What there isn't is a way of picturing it that is logically consistent *with your assumptions*. You refuse to let go of your assumptions, even though they are not valid for all situations.

That's exactly how I feel about the assumptions of GR.

You haven't given a valid argument for any of this. Basically you just keep saying "I can't see how this can work", which may be a valid statement about your state of mind, but says nothing about the actual physics.

The physics is the same as SR.

No. Schwarzschild *spacetime* is just the complete solution to the Einstein Field Equation that we've been talking about. It's a geometric object, not a coordinate system. (The Schwarzschild *coordinate system* doesn't say nothing can reach the horizon either, just that an external observer won't ever see light rays from someone reaching and crossing the horizon. But I wanted to be clear what I meant by "Schwarzschild spacetime".)

They won't see light rays from someone reaching and crossing the horizon because of time dilation and length contraction. It's not just that the information is late.

On the contrary, math is the only way to form a logical picture. Math is the language of logic. If you want a logical picture then you need to learn the relevant math. The mathematical framework of Riemannian geometry is what guarantees the logical self-consistency of GR.

Your anti-math stance is exactly the thing which will guarantee that you will continue to be unable to understand the logical picture.

Maths is one form of logic. Conceptualised what if questions are another.

There are two statements here:
1) The proper time for a free falling object to cross the event horizon is finite
2) The coordinate time for a free falling object to cross the event horizon is infinite

These two statements are in no way contradictory. They are different statements, and both are true. I showed how both were true for a Rindler horizon on the other thread, and WannabeNewton showed how 1 is true for a Schwarzschild horizon (and you are well aware of 2 for Schwarzschild).

You have been shown logically, your question has been answered clearly. Now it is simply up to you to decide if you really want to know the answer to your question or if you want to continue to pretend that it is illogical.

At this point, your continued ignorance is a deliberate choice that no one besides you can resolve.

Those two statements are completely contradictory. I don't understand how you can claim that they're not. The only possible way that could work is if black holes last forever. Is that what you're claiming?

A-wal, I understand your aversion to the mathematics. I am not advanced enough to follow it. And I understand the expectation that the universe should logically be able to be made sense of without it.

But here's the problem: all conceptual, logical models based on what we already know must be flawed. Our conceptual models depend on using concepts we already know. But we are in new territory here, concept-wise.

Take the electron. We say it has a property called spin. The math of QM describes accurately what really happens but we had no conceptual model of this kind of spin. We couldn't have had one. The best we could do was say "This is how it really works, the math shows it and predicts correctly the behaviour. Now that we see that, we can only then form a conceptual model of an electron that has this thing called spin" which we still don't have a conceptual model of. (at least I don't. How does a particle go around a half time and come back to the same place?)

The same is true with something like Einsteinian SR. We had no mental model for time being a dimension that can be distorted just like space. Only after the math was shown do we accept it and build a conceptual model, breaking the old one where time and space were absolute backgrounds against which events happened.

You accept those. But your conceptual model of them came after the math.

Same with the EH of a BH. We have no precedent for a conceptual model of how this works. Any time we try to make one it must be wrong because it doesn't actually work like anything else. We must first satisfy ourselves with the math that it really is this way, and only then can we build a conceptual model.

The math describes how it really works. No conceptual model you currently have can do so. Full stop. You have yet to build that new conceptual model.

The conceptual model of SR makes so much sense that's it's very difficult to imagine it working any other way. It's simple, elegant and it in no way contradicts itself. This mixing and matching of coordinate systems to produce what you think is a coherent model is really just a mixing of two different models that don't go together. This distant observers perspective is the only one. Nothing reaches an event horizon.

The conceptual model I have of a black hole is very simple. Use SR and the event horizon is the point when gravity would accelerate an object all the way to c, and the singularity is the point when it would accelerate it to infinite speed. Now we remember that the harder you accelerate the more time dilated and length contracted you become and that’s what a distant observer sees. The falling object can never reach c no matter how hard they’re accelerated.


DrGreg: That’s exactly the kind of thing I need. Thankyou. I haven’t got time to reply now but I will next time.

 

[PeterDonis]

Well, it looks like we're moving the discussion from the other thread to here.

That's exactly how I feel about the assumptions of GR.

Yes, that's obvious. But there's a difference. We are adopting assumptions that imply something is possible; you are adopting assumptions that require that something to be impossible. That means you have a much harder burden of proof. All we have to show is that there is *some* consistent model that uses our assumptions, which we have. You would have to show that *any* consistent model *must* use your assumptions, and you haven't (and I don't see how you can, since we have a consistent model that violates your assumptions).

The physics is the same as SR.

No, it isn't. We already had this discussion in the other thread. SR does not allow for tidal gravity, and there is tidal gravity outside the hole. So the physics can't be the same as SR.

They won't see light rays from someone reaching and crossing the horizon because of time dilation and length contraction. It's not just that the information is late.

"Time dilation and length contraction" are not different causes of the delay of the light rays; they're just different names for the same cause ("the information is late"). What you're saying here is perfectly consistent with what I've been saying, and does not rule out the possibility of a free-faller reaching and crossing the horizon.

 

[DrGreg]

Are you sure you can observe an object reaching an event horizon if you're close enough? So you could watch it happen and then move away because you haven't reached the horizon yet?

Actually, the only way you can see something fall through the horizon is if you fall through it yourself.

The only possible way that could work is if black holes last forever. Is that what you're claiming?

This is an important point. I think most participants in this thread have been talking about a Schwarzschild black hole, which is mathematical model built on certain assumptions about a black hole, including

1. it is spherically symmetric
2. it is not rotating (i.e. has zero angular momentum)
3. its mass is constant, has been forever in the past, and will be forever in the future

A real black hole won't satisfy any of those conditions, but the Schwarzschild solution is still a good approximation and is much easier to derive than other solutions. In particular, if the Schwarzschild model says you have to wait an infinite time for something to happen, with a real black hole you are likely to have to wait for a very long time (e.g. until the hole has completely evaporated due to Hawking radiation).

DrGreg: That’s exactly the kind of thing I need. Thankyou. I haven’t got time to reply now but I will next time.

Bear in mind that the Rindler horizon is also a model built on the assumption of an observer who maintains constant proper acceleration forever, which would be impossible in the real Universe in which your rocket would eventually run out of fuel.

 

[Dale]

There are two statements here:
1) The proper time for a free falling object to cross the event horizon is finite
2) The coordinate time for a free falling object to cross the event horizon is infinite

Those two statements are completely contradictory. I don't understand how you can claim that they're not.

I not only claim that they are not contradictory, I proved it through a rigorous derivation in the previous thread*. If you disagree then either point out the specific error in my derivation or post a correct derivation of your own showing the contradiction. If you cannot do either then you have no grounds for claiming that there is a contradiction.

*https://www.physicsforums.com/showpost.php?p=3316839&postcount=375

 

[PeterDonis]

This is an important point. I think most participants in this thread have been talking about a Schwarzschild black hole, which is mathematical model built on certain assumptions about a black hole, including

1. it is spherically symmetric
2. it is not rotating (i.e. has zero angular momentum)
3. its mass is constant, has been forever in the past, and will be forever in the future

A real black hole won't satisfy any of those conditions, but the Schwarzschild solution is still a good approximation and is much easier to derive than other solutions. In particular, if the Schwarzschild model says you have to wait an infinite time for something to happen, with a real black hole you are likely to have to wait for a very long time (e.g. until the hole has completely evaporated due to Hawking radiation).

One clarification here, since this point came up in the other thread where A-wal, I, and others were discussing whether the event horizon can be reached and passed. A real black hole, which would eventually evaporate due to Hawking radiation (other things being equal), still allows a *long* period of time where free-falling objects can reach and pass the horizon. In other words, even in a "real" black hole spacetime where the hole eventually evaporates, there is still a genuine region of spacetime inside the horizon, bounded by the event of formation of the hole (more precisely, of its horizon), the horizon itself, the singularity at r = 0, and the event of the hole's final evaporation. A-wal is claiming that this region of the spacetime does not exist.

 

[Passionflower]

Perhaps some of the confusion is due to the fact that people are talking about different things.

The Schwarzschild solution is a valid solution of the EFE. We can calculate all kinds of things with it, including things passed the event horizon (although preferably not with Schwarzschild coordinates).

However a completely different question is for instance how long it takes for black holes to fully form, or how black holes form at all etc, this is something the Schwarzschild solution does not tell us. There are other solutions (with rather simplified pressure assumptions) that can give us hints but for instance I do not think it is possible for any black hole not to rotate and as of now we do not have analytic interior solutions of rotating collapsing masses.

 

[PeterDonis]

However a completely different question is for instance how long it takes for black holes to fully form, or how black holes form at all etc, this is something the Schwarzschild solution does not tell us. There are other solutions (with rather simplified pressure assumptions) that can give us hints but for instance I do not think it is possible for any black hole not to rotate and as of now we do not have analytic interior solutions of rotating collapsing masses.

This is true, but it doesn't affect the main point, which is that in all such solutions, there is an event horizon and a region of spacetime inside it, which can be reached by observers traveling on timelike worldlines; and once an observer is inside that region, they can't escape back out to the exterior. If the spacetime is that of a collapsing star (or other massive object), then the event horizon forms at some finite time instead of existing infinitely far into the past, and if the Hawking evaporation of the hole is included, then the event horizon disappears at some finite time after it forms instead of existing infinitely far into the future. But there is still plenty of room, so to speak, for observers to fall into the black hole and be trapped during the period when it exists.

 

[Passionflower]

This is true, but it doesn't affect the main point, which is that in all such solutions, there is an event horizon and a region of spacetime inside it, which can be reached by observers traveling on timelike worldlines; and once an observer is inside that region, they can't escape back out to the exterior. If the spacetime is that of a collapsing star (or other massive object), then the event horizon forms at some finite time instead of existing infinitely far into the past, and if the Hawking evaporation of the hole is included, then the event horizon disappears at some finite time after it forms instead of existing infinitely far into the future. But there is still plenty of room, so to speak, for observers to fall into the black hole and be trapped during the period when it exists.

So assume we have a rotating star, on which solution do you base your conclusions?

I am not saying you are wrong, but I do not place theories with simplified models above experimental verification.

 

[PeterDonis]

So assume we have a rotating star, on which solution do you base your conclusions?

On the exterior Kerr (or Kerr-Newman, if the collapsing rotating star has a nonzero electric charge) solution. Once the collapsing rotating star is small enough in radius, and a horizon forms around it, the exterior solution is enough to show that there are timelike geodesics which cross that horizon in a finite proper time, and that the curvature there is finite so there's no singularity or anything similar that would result in the geodesics just stopping at the horizon. As far as I know, Kerr (or Kerr-Newman) works basically the same as Schwarzschild in that respect.

I agree that the exterior solution can't tell us the details of the interior, and the interior may well be a chaotic mess rather than anything close to the interior Kerr solution. (I believe numerical simulations have been done that suggest this, but I'll have to do some link digging to verify that, unless some of the experts here can point us in the right direction. I also believe the same would be true even in the case of a non-rotating collapsing star, since perfect spherical symmetry is unstable against small perturbations.) But the chaotic mess still has to be *somewhere*, and that somewhere has to be inside the horizon.

 

[Passionflower]

I think you are missing my point.

One can say whether something is true with respect to a particular solution as it is simply a case of mathematics.
One can only surmise that something is true based on the existence of a solution that sets proper boundaries for all real situations found in nature.

What I think one cannot do is to claim something is true based on models that do not set those boundaries. All one can do is speculate it is true.

In my opinion, as long as we do not have a solution for a collapsing rotating star (both analytic or numerical) that does not show clear boundary conditions such that a black hole must form in all real cases all we can do is speculate and make educated guesses and conjectures. Those guesses might make total sense and even be true, but it is a disrespect to the scientific method to call it more than that in my opinion.

 

[PeterDonis]

Passionflower, I don't disagree with your general point; we shouldn't be claiming more than what the actual models we have show. My understanding is that, as you say, we don't have an analytic solution for a collapsing rotating star (i.e., a solution analogous to the collapsing FRW solution in the non-rotating case), but that we do have numerical models of such collapses and they all show a horizon forming. I'm not aware of any known models of collapse that *don't* show a horizon forming. But I admit I am not very familiar with the literature in this area.

 

[zonde]

I not only claim that they are not contradictory, I proved it through a rigorous derivation in the previous thread*. If you disagree then either point out the specific error in my derivation or post a correct derivation of your own showing the contradiction. If you cannot do either then you have no grounds for claiming that there is a contradiction.

*https://www.physicsforums.com/showpost.php?p=3316839&postcount=375

You can prove that proper time of reaching EH for object falling into the Black Hole is finite.
But that does not prove that object will cross EH.
That is your error.

Obviously you can enumerate infinite dimension using hyperbolic units. But that does not prove that hyperbolic units are meaningful beyond the point where finite value in hyperbolic units correspond to infinite value in linear units.
In similar fashion you can use hyperbolic units to reach infinite value at finite value of linear units. That does not prove that linear units don't extend beyond that point.

So the question is if we can we distinguish what should be considered linear unit and what should be considered hyperbolic unit. And we can't do that within mathematical model. The only meaningful way how we can do that is by establishing some correspondence to physical world.

 

[PeterDonis]

You can prove that proper time of reaching EH for object falling into the Black Hole is finite.
But that does not prove that object will cross EH.

We can prove more than just that the proper time of reaching the EH is finite. We can prove that, at that point (when the EH has been reached), the spacetime manifold is still continuous, the curvature of spacetime is finite, and the outgoing side of the light cone at that point stays at the same radius, so any timelike worldline has to continue to decrease in radius. That means one of two things must be the case: either (1) the timelike worldline on which the object is traveling when it reaches the EH continues inside the EH (meaning there is a region of spacetime in there); or (2) a timelike worldline can abruptly just *stop*, and an object traveling on it can abruptly cease to exist, without any reasonable physical cause--no infinite curvature, no break in the continuity of spacetime, nothing. You are right that we are implicitly assuming that (1) is the case, not (2), but unless you're prepared to defend (2) as an actual alternative, I don't see what other choice there is.

 

[Dale]

Also, we can prove that the object will cross the EH in a finite time in any coordinate chart which covers any open ball containing the intersection of the object's worldline and the horizon. The Schwarzschild coordinates do not cover the EH itself nor the interior, but many other charts do.

 

[zonde]

We can prove more than just that the proper time of reaching the EH is finite. We can prove that, at that point (when the EH has been reached), the spacetime manifold is still continuous, the curvature of spacetime is finite, and the outgoing side of the light cone at that point stays at the same radius, so any timelike worldline has to continue to decrease in radius. That means one of two things must be the case: either (1) the timelike worldline on which the object is traveling when it reaches the EH continues inside the EH (meaning there is a region of spacetime in there); or (2) a timelike worldline can abruptly just *stop*, and an object traveling on it can abruptly cease to exist, without any reasonable physical cause--no infinite curvature, no break in the continuity of spacetime, nothing. You are right that we are implicitly assuming that (1) is the case, not (2), but unless you're prepared to defend (2) as an actual alternative, I don't see what other choice there is.

Of course (2) is an alternative. Object approaches EH asymptotically. Idea that it just cease to exist comes from bad choice of coordinate system.

And ... hmm why do you think that infinite curvature and break in the continuity of spacetime are physical things? I suppose that these are the things that you have at central singularity where objects abruptly cease to exist just the same.

 

[zonde]

Also, we can prove that the object will cross the EH in a finite time in any coordinate chart which covers any open ball containing the intersection of the object's worldline and the horizon. The Schwarzschild coordinates do not cover the EH itself nor the interior, but many other charts do.

Seems interesting. What coordinate chart would you prefer for discussion purposes as an alternative to Schwarzschild coordinates?
I have looked at Gullstrand–Painleve coordinates and I suppose that I understood more or less what they are doing but I don't know how popular they are.

 

[PeterDonis]

Of course (2) is an alternative. Object approaches EH asymptotically. Idea that it just cease to exist comes from bad choice of coordinate system.

No, the idea that the object only approaches the EH asymptotically comes from a bad choice of coordinate system, and from confusing coordinate time with proper time. The proper time for an object to fall to the horizon is finite, and that can be calculated using Schwarzschild coordinates. See below.

And ... hmm why do you think that infinite curvature and break in the continuity of spacetime are physical things? I suppose that these are the things that you have at central singularity where objects abruptly cease to exist just the same.

Yes, GR predicts infinite curvature at the central singularity. But the curvature is *not* infinite at the horizon. There is *no* "break" in spacetime there. And yet we have timelike geodesics that reach the horizon in a finite proper time and can't go back outward there because the outgoing side of the light cone there stays at the same radius. So to maintain anything except (1), that the timelike geodesics continue inward from the horizon (meaning that there has to be spacetime inside the horizon for them to continue into) would require you to maintain that timelike geodesics, and objects traveling on them, could abruptly cease to exist when there is *not* infinite curvature.

This is not a matter of choice of coordinates. As I noted above, you can calculate that the proper time along a timelike geodesic to the horizon is finite in Schwarzschild coordinates. Those coordinates are singular at the horizon, so you can't, technically, model the extension of a timelike infalling worldline across the horizon in those coordinates; but you *can* tell that there must be such an extension, because you can compute that the curvature is finite at the horizon, and that the outgoing side of the light cone stays at the same radius there, using a limiting process in those coordinates. Picking a better coordinate system makes all this easier to do, but it doesn't affect the physical result.

 

[PeterDonis]

Seems interesting. What coordinate chart would you prefer for discussion purposes as an alternative to Schwarzschild coordinates?
I have looked at Gullstrand–Painleve coordinates and I suppose that I understood more or less what they are doing but I don't know how popular they are.

Painleve would be fine, or ingoing Eddington-Finkelstein, or Kruskal. All of those are non-singular at the horizon and meet the requirement that DaleSpam stated.

 

[zonde]

No, the idea that the object only approaches the EH asymptotically comes from a bad choice of coordinate system, and from confusing coordinate time with proper time.

Can we agree that we disagree?
Our judgment what to consider good or bad coordinate system is different. The same about which observer should be chosen for global bookkeeping either distant observer that is not going to fall into particular black hole or the one that is falling towards particular black hole.

The proper time for an object to fall to the horizon is finite, and that can be calculated using Schwarzschild coordinates. See below.

Did I claim something different?

Yes, GR predicts infinite curvature at the central singularity. But the curvature is *not* infinite at the horizon. There is *no* "break" in spacetime there. And yet we have timelike geodesics that reach the horizon in a finite proper time and can't go back outward there because the outgoing side of the light cone there stays at the same radius.

But ingoing side of light cone continues inwards? Basically you are saying that light can cross EH and therefore massive object can do that as well.
Then the question is - why do you think that light can cross EH?

So to maintain anything except (1), that the timelike geodesics continue inward from the horizon (meaning that there has to be spacetime inside the horizon for them to continue into) would require you to maintain that timelike geodesics, and objects traveling on them, could abruptly cease to exist when there is *not* infinite curvature.

This is not a matter of choice of coordinates. As I noted above, you can calculate that the proper time along a timelike geodesic to the horizon is finite in Schwarzschild coordinates. Those coordinates are singular at the horizon, so you can't, technically, model the extension of a timelike infalling worldline across the horizon in those coordinates; but you *can* tell that there must be such an extension, because you can compute that the curvature is finite at the horizon

You can compute that the curvature is finite at the horizon in Schwarzschild coordinates?
How is that?

and that the outgoing side of the light cone stays at the same radius there, using a limiting process in those coordinates. Picking a better coordinate system makes all this easier to do, but it doesn't affect the physical result.

And still you haven't answered my question - why do you think that infinite curvature and break in the continuity of spacetime are physical things?

 

[?????]

(e.g. until the hole has completely evaporated due to Hawking radiation).

Could you clarify how long it would take for this to happen? In other words, I think I am reading that all the posters on this thread agree that to the outside universe, time goes very slowly (maybe even stops) at the EH. This applies to this falling towards the BH, so I assume it applies to things going out again. So the rest of the universe is sitting on our hovering spaceships and watching the Hawking radiation coming out past the EH. But how does it get out? The rest of the universe is long gone by the time this radiation manages the trip.

It's the same argument as things falling into the BH. Sure, the proper time on the falling object proceeds at the normal rate (whatever that is). But the rest of the universe outside the EH has lived out it's lifetime and no longer exists by the time the falling observer crosses the EH. All of the Hawking radiation will be frozen at the EH going the other way, just as all of the falling objects will be frozen at the EH. The BH will never lose any mass, as view by those of us who are in the rest of the universe, it will only exponentially gain mass until it consumes everything.