Black Holes - the two points of view.

[Mike Holland]

I wish to take up a discussion between Elroch and DrStupid in RossiUK’s topic “First Post – a question about Black Holes and Gravity”. My post is essentially an exposition of Elroch’s view, which I have shared for many years. Elroch’s exposition was very sedate, and I feel it needs to be shouted from the rooftops - “There are no Black Holes in this universe”. Well, not quite, anyway!

When discussing black holes, there are basically two points of view, that of a remote observer and that of the poor spaceman who falls into one. The difference is caused by gravitational time dilation. From the remote viewer’s point of view (or in his time frame, if you prefer), the passage of time is retarded near the black hole, and comes to a complete stop at the Schwarzschild Radius. So as far as this remote viewer is concerned, a falling spaceman would never reach the Schwarzschild radius, but would hover just outside it gradually edging closer and closer. But the spaceman, in turn, will have a very different experience, falling past the SR in a very short period of time according to his clock.

The consequence of this is that as far as outside observers are concerned, the spaceman never enters the black hole. And neither does any other falling matter. Nothing has ever fallen into a black hole as far as our clocks are concerned!

But extreme time dilation would exist for a collapsing star even before it reaches the black hole state. A super-massive collapsing object which is nearly a black hole would itself be highly time dilated (by our clocks), and the collapse process itself would slow down and come to a complete stop just as it reaches black hole status - which would only happen when our clocks read infinity.

NB. Schwarzschild radius and Event Horizon are not the same thing. Every mass has a Schwarzschild radius within it, and only when all the mass is compressed within this radius would an Event Horizon form.




Many prominent astrophysicists who have performed the calculations support these conclusions:

“What would happen if you fall in? As seen from the outside, you would take an infinite amount of time to fall in, because all your clocks – mechanical and biological – would be perceived as having stopped’”
- Carl Sagan “Cosmos”, 1981

“ .. a critical radius, now called the “Schwarzschild radius,” at which time is infinitely dilated.”
- Paul Davies “About Time”, 1995

“From the standpoint of an outside observer, time grinds to a halt at the event horizon.”
- Timothy Ferris “The Whole Shebang”, 1997

“The closer we are to the event horizon, the slower time ticks away for the external observer. The tempo dies down completely on the boundary of the black hole.”
- Igor Novikov “The River of Time”, 1998

“When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. This contraction will continue indefinitely till the radius of the star approaches asymptotically its gravitational radius.”
- Oppenheimer and Snyder “Phys.Rev. 56,455” 1939

“According to the clocks of a distant observer the radius of the contracting body only approaches the gravitational radius as t -> infinity.”
- Landau and Lifschitz “The Classical Theory of Fields”, 1971

“What looks like a black hole is “in reality” a star frozen in the very late stages of collapse.”
- Paul Davies “About Time”, 1995

“At the stage of becoming a black hole, time dilatation reaches infinity.”
- Jayant Narlikar

In all his writings, Fred Hoyle referred to them as “near black holes”, while the Russians called them “frozen stars”..


All the mathematicians who have solved Einstein’s equations for a collapsing super-massive body have come to the same conclusion - in the reference frame of any external observer, it takes an infinite time for a Black Hole to form. This means that there are no Black Holes in the universe, and won’t be until the age of the universe is infinity!

I have seen arguments that these calculations were all done for a distant observer in the “proper time” of the Black Hole. Proper time means that the observer is motionless relative to the BH, and nowhere near any gravitational mass which could affect his clock. But this condition was used simply to simplify the mathematics. We can calculate the effect of our relative motion, which is hardly relativistic, and Earth’s gravity, which is so infinitesimal it can only be measured with atomic clocks, and these factors have no significant effect on the results of the calculations.

The time dilation around a collapsing super-massive object only becomes significant extremely close to the Schwarzschild radius and so for all intents and purposes such an object would be indistinguishable from a Black Hole. But perhaps one difference is the magnetic fields that have been observed around some supposed Black Holes in other galaxies, indicating that they are not quite there yet.

What we end up with is an object collapsing more and more slowly as it tries to fit within its Schwarzschild radius, and this almost Event Horizon area becomes extended as more material falls onto it. The almost-EH is not a surface, but a whole volume of the collapsing mass, with never enough mass within its Schwarzschild Radius to actually form an event horizon. So we don’t have an expanding Event Horizon as matter falls in, we have an expanding region of “almost Event Horizon”, with the inner regions being compressed ever closer to forming a Black Hole.

But what about the other point of view, that of the poor spaceman who is falling into such a super-massive object as at collapses into a Black Hole? He will see an almost-Black –Hole ahead of him as he approaches. It only becomes a BH for him when he arrives there. If he could hover close to the object (rockets blasting like anything to keep him there), then he would see the outside universe speeded up, just as we see clocks in orbit above the Earth running faster. But as he is accelerating under the gravitational attraction, the converse happens, and he will actually see our clocks slowed down. Counter-acting the gravitational speed-up of our clocks, from his point of view, are apparent time dilation effects due to the time our photons take to reach him as he speeds up.

From his point of view, he will approach the speed of light as he approaches the Black Hole to be. But our view is different. We see him accelerating until he is about twice the Schwarzschild Radius away, and then time dilation takes over and he slows down and in fact never gets there. If he was hovering, we would simply see him gravitationally time dilated. But as he approaches the SR, photons take longer and longer to escape and this gives rise to another, optical, time dilation. This apparent time dilation is added to the GR dilation making him appear even more frozen in time.

When he reaches the Schwarzschild radius, along with all the other collapsing matter, he does not travel any further because space and time are distorted in such a way that the distance between him and the centre becomes a time dimension. The singularity is in his future, not in any space direction. In effect, he is already at the centre and all the surrounding matter is collapsing in on him (OK, I expect a lot of controversy about this description!).

I have written this as though we could observe events all the way into the forming event horizon. But of course this would be impossible. Time dilation creates such a red shift that visible light will be stretched to into radio waves and beyond, making observation impossible. Also, any such collapsing mass would probably be surrounded by in-falling matter and by the radiation that it emits. So as far as observations are concerned, all the above probably makes no difference,.

My one concern with this description of events is that the dilation only becomes significant extremely close to the SR, and I don’t know what happens when one gets down to quantum dimensions. At one Plank length away from an Event Horizon of 10 km radius, the time dilation factor is about 10**19 to 1. Which rules at this scale? Quantum uncertainty or gravity? My money is on gravity, but I think a Theory of Quantum Gravity is required to resolve this issue.

Mike

 

[pervect]

There isn't any particular reason to favor the observer at infinity over the one who falls into the black hole.

THis becomes clearer if you consider the closely related example of event horizons, the Rindler horizon, which is caused by acceleration and is formally very similar to that of a black hole (except it's flat, not curved).

Suppose a rocketship accelerates at 1 gravity. About 1 year into their journey, they will see the Earth appear to fall into an event horizon, called the Rindler horizon.

The Earth will get redder and dimmer, and their clocks on Earth will appear to slow and stop according to the accelerating observer.

If we take the viewpoint of the accelerating observer as representing some "universal truth", we would say that "time stops on the Earth" and we might add "It stops in the year xxxx", where xxx is the year the Earth falls behind the horizon.

Which should be obviously silly, because the person on Earth won't even know anything happened.

Applying the same argument in this only slightly different situation shows how silly it is to give one particular observer "priveleged status" as far as existence goes. The observer at infinity might not be able to see certain events, but that hardly means that they don't happen, just as the rocketship observer's inability to see anything after some specific date on Earth doesn't mean that "it never happens".

 

[Mike Holland]

Pervect, I don't recall saying at any stage that one observer is "privileged". My title says "two points of view", and that's what they are. All I am pointing out is that in "our" reference frame, some events take an infinite time, according to all the GR mathematicians, and therefore as far as we are concerned, they haven't happened. Doesn't mean they won't happen (after an infinite time).

We can only say something "has" happened when we can prove that it occurred before our present.

I read up on Rindler horizons several years ago, but don't remember much about them. Will have to look them up again. But do they prove that Oppenheimer, Snyder, Landau, etc are all wrong? If not, how do you make sense of my quotes from those guys?

Mike

 

[Mike Holland]

OK, I just calculated that 1 years acceleration at 1g = c. So the receding Earth's relativistic mass will have reached infinity, and it must have become a black hole shortly before that. But you cannot accelerate up to c, Special Relativity prevents that with time dilation, lorentz contraction, relativistic mass increase, etc, so the problem should never arise!
Mike

 

[Dale]

in the reference frame of any external observer, it takes an infinite time for a Black Hole to form. This means that there are no Black Holes in the universe, and won’t be until the age of the universe is infinity!

This seems to be your key thesis, and there are several things wrong with it.

First, the initial statement is not true for "any external observer", as claimed. It is only true for observers using Schwarzschild coordinates. External observers using other coordinates may disagree.

Second, the reference to "in the universe" is a coordinate independent reference to the manifold. The coordinate-dependent reasoning presented cannot be used to justify the coordinate-independent conclusion asserted. Just because something is not in a particular coordinate chart does not imply it is not in the manifold.

Third, the "age of the universe" is usually associated with the FLRW spacetime, not the Schwarzschild spacetime, so I am not sure what you actually intended to refer to there.

 

[PeterDonis]

Many prominent astrophysicists who have performed the calculations support these conclusions

You should be extremely careful about how much you read into pop-science statements about black holes, or indeed about any counterintuitive aspect of physics, even when they are written by world-class physicists. English, or any other natural language, is not well adapted to expressing scientific conclusions; it is very difficult to avoid drawing incorrect deductions from the English statements (see below for an example). That's why the real descriptions of our scientific theories, the ones we actually use to make predictions and test them, are written in mathematics, not natural language.

All the mathematicians who have solved Einstein’s equations for a collapsing super-massive body have come to the same conclusion - in the reference frame of any external observer, it takes an infinite time for a Black Hole to form.

Within the limitations of English, this is one way of stating what the math says. But you go on to draw an incorrect deduction from it:

This means that there are no Black Holes in the universe, and won’t be until the age of the universe is infinity!

This is *not* what the math says, and it is not correct. If you disagree, then please post the actual math (not English statements) that you are using to justify your claims.

Also, even given the limitations of English, you have some of the terminology wrong:

Proper time means that the observer is motionless relative to the BH, and nowhere near any gravitational mass which could affect his clock.

This is not what "proper time" means, not even in Special Relativity, let alone in General Relativity.

 

[Dale]

So the receding Earth's relativistic mass will have reached infinity, and it must have become a black hole shortly before that.

This is a common misconception. See the FAQ
http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html

 

[Naty1]

All the mathematicians who have solved Einstein’s equations for a collapsing super-massive body have come to the same conclusion - in the reference frame of any external observer, it takes an infinite time for a Black Hole to form.

Hey Mike, interesting quotes...lots of perspectives...

Dalespam posts:

[The coordinate-dependent reasoning presented cannot be used to justify the coordinate-independent conclusion asserted. Just because something is not in a particular coordinate chart does not imply it is not in the manifold.
/QUOTE]

I'd even generalize this a bit further: Someone else mentioned Schwarzschild and FLRW coordinates and neither of those are EXACT models for our observations...so I would doubt
what appears as 'infinite time' in any such idealized model should be taken too literally.


The description I like is from Kip Thorne's in BLACK HOLES AND TIME WARPS:

Finkelstein's reference frame was large enough to describe the star's implosion ...simultaneously from the viewpoint of far away static observers and from the viewpoint of observers who ride inward with the imploding star. The resulting description reconciled...the freezing of the implosion as observed from far away with the continued implosion as observed from the stars surface...an imploding star really does shrink through the critical circumference without hesitation...That it appears to freeze as seen from far away is an illusion...General relativity insists that the star's matter will be crunched out of existence in the singularity at the center of the black...

I suspect we have lots more to learn about geometry, spacetime, horizons and particles!

 

[Naty1]

My one concern with this description of events is that the dilation only becomes significant extremely close to the SR, and I don’t know what happens when one gets down to quantum dimensions. ... Which rules at this scale? Quantum uncertainty or gravity? ...I think a Theory of Quantum Gravity is required to resolve this issue.

likely that could provide additional insights... but the singularity at the horizon is a coordinate singularity different from the singularity at the center of a BH where both relativity and QM diverge...In other words, the horizon time divergence is Schwarzschild dependent and disappears in other coordinates...This is analogous to the apparent horizon
of a constantly accelerating observer in Rindler coordinates in Minkowski space.

But as I posted in the prior note, it doesn't seem certain to me exactly what conclusions can be drawn from these idealized models.

 

[pervect]

OK, I just calculated that 1 years acceleration at 1g = c. So the receding Earth's relativistic mass will have reached infinity, and it must have become a black hole shortly before that. But you cannot accelerate up to c, Special Relativity prevents that with time dilation, lorentz contraction, relativistic mass increase, etc, so the problem should never arise!
Mike

A better way to describe what happens is to use the relativistic rocket equation to plot the course of the rocket.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html has the formulae, converting them to latex they are:

$$t = \frac{c}{a} sinh \frac{aT}{c} = \sqrt{ \left(\frac{d}{c}\right) ^2 + \frac{2d}{a} }$$
$$d = \frac{c^2}{a} \left( cosh \frac{aT}{c} - 1\right) = \frac{c^2}{a} \sqrt{1 + \left(\frac{aT}{c}\right)^2 - 1}$$Here t is time as measured on earth, d is distance as measured on earth, and T is the proper time aboard the rocket.

You can consider a = 1 light year / year^2 to be 1 g - it's quite close, and the approximation I used when I made my remark.

If you plot the course of the rocket, you'll see though it never reaches the speed of light, it does accelerate fast enough that light signals emitted at a certain time from the Earth (the time when the asymptote of the hyperbola crosses the origin) will never catch up to it.

The problem with saying that the relativistic mass goes to infinity is that a) a correct relativistic analysis of the rocket reveals it never gets to 'c' and b) you can't compute gravity by putting "relativistic mass" into Newtonian formulae.

[add]Plus the point that Doctor Gregg made.

If you want to adopt the rocketship's point of view, you say that the metric coefficient of g_00 goes to zero and forms and event horizon, the Rindler horizon, behind the rocketship.

Greg Egan has a webpage on the Rindler Horizon for sure, it might be a bit advanced though.

 

[DrGreg]

THis becomes clearer if you consider the closely related example of event horizons, the Rindler horizon, which is caused by acceleration and is formally very similar to that of a black hole (except it's flat, not curved).

Suppose a rocketship accelerates at 1 gravity. About 1 year into their journey, they will see the Earth appear to fall into an event horizon, called the Rindler horizon.

The Earth will get redder and dimmer, and their clocks on Earth will appear to slow and stop according to the accelerating observer.

OK, I just calculated that 1 years acceleration at 1g = c. So the receding Earth's relativistic mass will have reached infinity, and it must have become a black hole shortly before that. But you cannot accelerate up to c, Special Relativity prevents that with time dilation, lorentz contraction, relativistic mass increase, etc, so the problem should never arise!

Mike, this has nothing to do with the Earth's mass or it's gravity. The same effect occurs even if Earth is not there, i.e. if you accelerate at 1 g in empty space, with no gravitational sources nearby, an "apparent horizon" forms (immediately) at about 1 light-year behind you which behaves almost identically to the event horizon of a black hole as observed by a hovering observer. Nothing, not even light, can pass through the horizon towards you, and objects "dropped" towards the horizon take an infinite time, according to you, to get there. However, from the point of view of any inertial observer, the location of your "horizon" is no different than anywhere else in empty space, and the dropped objects pass through it with no problem.

 

[pervect]

Pervect, I don't recall saying at any stage that one observer is "privileged". My title says "two points of view", and that's what they are. All I am pointing out is that in "our" reference frame, some events take an infinite time, according to all the GR mathematicians, and therefore as far as we are concerned, they haven't happened. Doesn't mean they won't happen (after an infinite time).

No, assigning an event an infinite time coordinate doesn't mean that it "didn't happen". In general having an infnite coordinate value is reason for concern, but it doesn't "prove" anything.

There is an opportunity here though, inspired by your title "two points of view". The opportunity here is for you to learn that the concept of "Now" depends on the observer - that it is not a universal concept.

This is just a rather extreme example, whereby one observer's notion of "now" is at future infinity for another observer.

 

[DrGreg]

But extreme time dilation would exist for a collapsing star even before it reaches the black hole state. A super-massive collapsing object which is nearly a black hole would itself be highly time dilated (by our clocks), and the collapse process itself would slow down and come to a complete stop just as it reaches black hole status - which would only happen when our clocks read infinity.

NB. Schwarzschild radius and Event Horizon are not the same thing. Every mass has a Schwarzschild radius within it, and only when all the mass is compressed within this radius would an Event Horizon form.

The Schwarzschild coordinate description is an idealised version of something that doesn't happen in practice. It's a good approximation in many ways but has some flaws. In particular it assumes a black hole's mass remains constant, implying it has existed for an infinite time in the past and will continue to exist for an infinite time in the future. It doesn't account for a black hole gaining mass from in-falling matter or losing mass via Hawking radiation. Also any analysis of the behaviour of matter near a Schwarzschild black hole ignores any gravitational effects due to the matter itself.

To a distant observer, a black hole surrounded by a shell of matter (outside its Schwarzschild radius) behaves identically to a black hole that has absorbed that matter. Strange though it may seem, matter outside the Schwarzschild radius can cause the Schwarzschild radius to increase. I think I am right in saying that matter can actually be absorbed by an expanding event horizon within finite time according to a distant hovering observer.

 

[Mike Holland]

Well, that’s a whole lot of responses to reply to. Guess I asked for it! So here goes…

Dalespam, to your knowledge has the collapse of a super-massive object ever been computed using a different coordinate system? And if so, did the results disagree? You are just supposing that they “may” disagree. Please let me know of other coordinate systems that have been used for the calculation.

OK, I used the term “universe” rather loosely. Using your term, I should say that there are no black holes in our coordinate chart. One cannot convert an event (x,y,z,t) at a black hole to an (x,y,z,t) in our coordinate system without infinities appearing.

I thought the age of the universe was 13.7 billion years. Is it different in a Schwarzschild coordinate system?

I haven’t read the FAQ on Rindler horizons yet. Give me a few days to catch up.

PeterDonis, I am surprised you regard “The Classical Theory of Fields” and Physical Review as pop-science articles. I did not draw incorrect deductions. I simply quoted the deductions of the mathematicians. Please show me where I have misinterpreted the conclusions of Landau and Lifschitz, for example, quoted above.

“This is *not* what the math says, and it is not correct. If you disagree, then please post the actual math (not English statements) that you are using to justify your claims.”
Rubbish. The maths is written out in Landau and Lifschitzs’ book on pages 297 to 299. Please tell me where the error is between their maths and their conclusions.

Naty1, I have Kip Thorne’s book next to me, and he says that in Finklestein’s solution, the geometry outside the imploding star is that of Schwarzschild (top of page 246), so as far as the external observers are concerned, the results will be the same. You suggest that Schwarzschild and FLRW coordinates are not EXACT models. What evidence? And are there any more exact formulations!

Pervect, I had assumed that as the rocket man never reaches c, so the Rindler horizon would never quite form. You are telling me that this is incorrect, so as I mentioned to Dalespam, I have some reading to do.

‘No, assigning an event an infinite time coordinate doesn't mean that it "didn't happen". In general having an infnite coordinate value is reason for concern, but it doesn't "prove" anything.’

Not true. Just go back a step to before he reached the EH. According to GR, when we look at the falling spaceman today, he is a meter away from the EH. We come back in a year’s time and he is a centimeter away. Another thousand years and he’s down to 1mm, etc. OK, we can’t really see him when he is this close, but GR gives us the equation to calculate his position. We don’t have to use the word “infinity”, Just the fact that he approaches the EH asymptotically in time means it hasn’t happened yet in our timeframe.

I know what is meant by “now” for an observer. In a theoretical sense it is a line drawn vertical to his world line, but in a practical sense it is the light cone that he is at the apex of.

DrGreg, I understand the effect of acceleration. It just the same as standing in a gravitational field and clocks above you go faster than yours while those below go slower (except that it would have to be a linear grav field to be fully equivalent). I am a great believer in Einstein’s equivalence principle, and always try to look at thing from both points of view. But I’m going to have a hard time trying to reconcile all my quotes above with Rindler horizons forming. Give me a while to work on it.

“To a distant observer, a black hole surrounded by a shell of matter (outside its Schwarzschild radius) behaves identically to a black hole that has absorbed that matter. Strange though it may seem, matter outside the Schwarzschild radius can cause the Schwarzschild radius to increase. I think I am right in saying that matter can actually be absorbed by an expanding event horizon within finite time according to a distant hovering observer.” - DrGreg

“What would happen if you fall in? As seen from the outside, you would take an infinite amount of time to fall in, because all your clocks – mechanical and biological – would be perceived as having stopped’” - Carl Sagan “Cosmos”, 1981

“From the standpoint of an outside observer, time grinds to a halt at the event horizon.” - Timothy Ferris “The Whole Shebang”, 1997

Who do I believe? The only ones who have maths to back up their claims are Sagan and Ferris. All the maths I have seen disagrees with you.


Thanks all of you for your responses and attempts to educate me. Please keep throwing stuff at me – it keeps the old mind active!

Mike

 

[TMM]

I think it's the case that time dilation prevents you from ever seeing something reach the singularity. After all, as the mass M increases, the field strength at the event horizon is ~1/M, so we can make it as small as we like.

 

[Dale]

Dalespam, to your knowledge has the collapse of a super-massive object ever been computed using a different coordinate system? And if so, did the results disagree? You are just supposing that they “may” disagree. Please let me know of other coordinate systems that have been used for the calculation.

As far as I know, the quotes you referred to are about the Schwarzschild coordinates, which can describe a small amount of matter falling into an already existing static black hole. In Schwarzschild coordinates a falling object goes to the event horizon as t goes to infinity. There are alternative coordinates for the Schwarzschild spacetime such as Gullstrand-Painleve, Eddington-Finkelstein, Kruskal-Szekeres, and Lemaitre coordinates.

http://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates
http://en.wikipedia.org/wiki/Eddington–Finkelstein_coordinates
http://en.wikipedia.org/wiki/Kruskal-Szekeres_coordinates
http://en.wikipedia.org/wiki/Lemaitre_coordinates

I think that all of these remove the coordinate singularity in different ways, and they all have a falling object cross in a finite coordinate time.

OK, I used the term “universe” rather loosely. Using your term, I should say that there are no black holes in our coordinate chart. One cannot convert an event (x,y,z,t) at a black hole to an (x,y,z,t) in our coordinate system without infinities appearing.

Sure, but rather than "our" coordinate chart I would say "the Schwarzschild" coordinate chart. There is no reason that we have to pick any of the above charts as "ours".

I thought the age of the universe was 13.7 billion years. Is it different in a Schwarzschild coordinate system?

The age of the universe is a feature of the FLRW spacetime. It is a different spacetime than the Schwarzschild space-time, so you cannot get from one to the other with just a coordinate transform. The Schwarzschild spacetime is static, so there is nothing corresponding to an age.

http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

 

[Nugatory]

Strange though it may seem, matter outside the Schwarzschild radius can cause the Schwarzschild radius to increase.

Can you point me to some references on this mechanism? It seems altogether plausible to me, but I'd expect that it takes more than the Schwarzschild solution (stationary solution doesn't leave much room for an increasing anything, vacuum solution only valid for negligible test masses outside the central singularity) to describe properly.

I think I am right in saying that matter can actually be absorbed by an expanding event horizon within finite time according to a distant hovering observer.

That also sounds plausible - it's hard to imagine what else could be meant by "an expanding event horizon".

 

[Naty1]

Here are some other explanations and points of view...I have saved these in my notes from other discussions in these forums.

As Wald says,

"there appears to be no natural notion of a black hole in a closed Robertson-Walker universe which re-collapses to a final singularity", and further, "there seems to be no way to define a black hole in a closed universe, because it requires going to infinity, but there is no infinity in a closed universe."

a Schwarzschild singularity in a coordinate system doesn't necessarily represent a pathology of the manifold. (Consider traveling due East at the North Pole). Nevertheless, the fact that no true black hole can exist in a finite universe shows that the coordinate singularity at r = 2m is not entirely inconsequential, because it does (or at least can) represent a unique boundary between fundamentally separate regions of spacetime,
Let's take the simplest case, where the black hole is in asymptotically flat space-time. This will happen automatically if one uses the usual Schwarzschild metric

different frames see different things...analogous to length contraction and as mentioned above, and time dilation…. The proper time for a freely-falling observer to reach the event horizon is finite, yet the free-fall time as measured at infinity is infinite…The acceleration is due to depth of gravitational well - a global feature. The tidal forces due to gradient - a local feature.

, when we mention velocity, acceleration, etc, we need to be clear what is being measured relative to what, and whether it is a "proper" invariant measurement, or a local or remote coordinate measurement.

.

Hence it would appear that, in the falling frame, the observer should encounter an infinite amount of radiation in a finite time, and so be destroyed. On the other hand, the event horizon is a global construct, and has no local significance, so it is absurd to0 conclude that it acts as physical barrier to the falling observer.
Quantum Fields in Curved Space by Birrell and Davies, pages 268-269

I posted this previously ...I believe it's Brian Greene or Kip Thorne

We found earlier that the Schwarzschild metric has a coordinate singularity at the event horizon, where the coordinate time becomes infinite. Recall that the coordinate time is approximately equal to the far away observer's proper time. However, a calculation using transformed coordinates shows that the infalling observer falls right through the event horizon in a finite amount of time (the infalling observer's proper time). How can we interpret solutions in which the proper time of one observer approaches infinity yet the proper time of another observer is finite?

The best physical interpretation is that, although we can never actually see someone fall through the event horizon (due to the infinite redshift), he really does. As the free-falling observer passes across the event horizon, any inward directed photons emitted by him continue inward toward the center of the black hole. Any outward directed photons emitted by him at the instant he passes across the event horizon are forever frozen there. So, the outside observer cannot detect any of these photons, whether directed inward or outward.

There's no coordinate-independent way to define the time dilation at various distances from the horizon—a clock is ticking relative to coordinate time, so even if that rate approaches zero in Schwarzschild coordinates which are the most common ones to use for a nonrotating black hole, in a different coordinate system like Kruskal-Szekeres coordinates it wouldn't approach zero at the horizon,

I believe this to be a precise description of an idealized model. It seems inconsistent with DrGreg's post :

I think I am right in saying that matter can actually be absorbed by an expanding event horizon within finite time according to a distant hovering observer.

which I believe is correct in a real world...a lumpy,curved spacetime not in our idealized
models...but that is a GUESS on my part.

 

[DrGreg]

DrGreg, I understand the effect of acceleration. It just the same as standing in a gravitational field and clocks above you go faster than yours while those below go slower (except that it would have to be a linear grav field to be fully equivalent). I am a great believer in Einstein’s equivalence principle, and always try to look at thing from both points of view. But I’m going to have a hard time trying to reconcile all my quotes above with Rindler horizons forming. Give me a while to work on it.

I think you would gain much by studying Rindler coordinates and Rindler horizons. Virtually all of the weird properties of a black hole's event horizon are also properties of a Rindler horizon that is caused simply by the acceleration of an observer in empty space. In my view, Rindler horizons are easier to understand than black hole horizons because if you get confused you can always transform back into standard Minkowski SR coordinates to see what is "really" happening, so to speak. (Not that I am suggesting there is anything "unreal" about using other coordinates.)

Others have already given you several places to look. If those aren't enough you could also look at my own contributions in previous threads, e.g.
Stupider-er Twins Question
about the Rindler metric
Questions about acceleration in SR, post #13 onwards

“To a distant observer, a black hole surrounded by a shell of matter (outside its Schwarzschild radius) behaves identically to a black hole that has absorbed that matter. Strange though it may seem, matter outside the Schwarzschild radius can cause the Schwarzschild radius to increase. I think I am right in saying that matter can actually be absorbed by an expanding event horizon within finite time according to a distant hovering observer.” - DrGreg

“What would happen if you fall in? As seen from the outside, you would take an infinite amount of time to fall in, because all your clocks – mechanical and biological – would be perceived as having stopped’” - Carl Sagan “Cosmos”, 1981

“From the standpoint of an outside observer, time grinds to a halt at the event horizon.” - Timothy Ferris “The Whole Shebang”, 1997

Who do I believe? The only ones who have maths to back up their claims are Sagan and Ferris. All the maths I have seen disagrees with you.

Sagan and Ferris are correctly describing the mathematical model for an object of negligible mass (compared to a black hole) falling into a black hole of constant mass (i.e. whose mass doesn't increase due to absorption of other matter or decrease due to Hawking radiation). That wasn't what I was talking about.

 

[Naty1]

Mike...really good discussions so far here...

Naty1, I have Kip Thorne’s book next to me, and he says that in Finklestein’s solution, the geometry outside the imploding star is that of Schwarzschild (top of page 246), so as far as the external observers are concerned, the results will be the same. You suggest that Schwarzschild and FLRW coordinates are not EXACT models. What evidence? And are there any more exact formulations!

My posts above and others have already answered..but I can offer a bit more. Schwarzschild coordinates include a flat asymptotic spacetime [that's not realistic]; FLRW assumes a perfectly homogeneous and isotropic spacetime and everyone here agrees the FLRW model does NOT apply to galactic scales...One has to also wonder how precise it is on cosmological scales...but that is not especially important for this discussion.

My reading SO FAR leads me to conclude there are not more exact formulations...we don't know how to solve EFE equations in an irregular, curved and lumpy spacetime. Mike: You might find this in Wikipedia an interesting adjunct to Kip Thorne's description {I looked it up to get insight on what Kip Thorne meant}:

http://en.wikipedia.org/wiki/Eddington-Finkelstein_coordinates
where it points out:

...In both these coordinate systems the metric is explicitly non-singular at the Schwarzschild radius

...

So while there is a type of time 'singularity' at the Schwarzschild radius, uniqueto those coordinates, I can think of three cases where it is NOT present: a free falling observer in those SAME coordinates, in the Eddington-Finklestein coordinates, and as I think has already been mentioned in this discussion, Kruskal-Szekeres coordinates.

So my own {novice} view is that between the different coordinate dependent descriptions and local versus global considerations, I have not yet come across any single, universal
all encompassing perspective that is absolute.

 

[DrGreg]

Strange though it may seem, matter outside the Schwarzschild radius can cause the Schwarzschild radius to increase.

Can you point me to some references on this mechanism? It seems altogether plausible to me, but I'd expect that it takes more than the Schwarzschild solution (stationary solution doesn't leave much room for an increasing anything, vacuum solution only valid for negligible test masses outside the central singularity) to describe properly.

I have to confess that the dynamic formation of black holes isn't my area of expertise -- there are others on this forum who have explained this in greater depth in previous posts -- but I believe it is a consequence of Birkhoff's theorem.

 

[Naty1]

Mike,
There is a closely related discussion here which you might find interesting:

Unruh effect and lessons regarding reality
https://www.physicsforums.com/showthread.php?t=625633

[I'm pretty sure this was previously blocked...anyway, now is open again as I post here.
"reality" is a not a good word to bring up as it quickly devolves into philosophy.]

The essence of this discussion revolves around that fact an an inertial observer and an
accelerating observer have different spacetimes,one flat, one curved, hence different apparent degrees of freedom, hence different observations. What this actually means appears open to some debate...

Sound familar?

 

[Naty1]

Strange though it may seem, matter outside the Schwarzschild radius can cause the Schwarzschild radius to increase.

Can you point me to some references on this mechanism?

You can get some insights into this, although no mathematics, from BLACK HOLES AND
TIME WARPS, by Kip Thorne...

I'll post if I can find it...

As I recall from other sources, the original horizon description is now called 'apparent' Horizon as discovered by Roger Penrose; Stephen Hawking did not like that coordinate dependent description, especially it's instantaneous discontinuous jumps when matter/energy was absorbed, and developed a complementary viewpoint, I believe the one routinely utilized today, the absolute horizon...

Others in these forum have discussed further distinctions, and one of them is that the [apparent] horizon jumps in anticipation of matter crossing the horizon. I believe this is analogous to the instantaneous appearance of the apparent horizon during the initial formation of a BH when it appears and cloaks the singularity; in contrast, Hawking's absolute horizon is created at the center of a new forming BH and moves smoothly to the stars surface as it impodes meeting the apparent singularity at the Schwarzschild radius.

 

[Mike Holland]

Naty1 and Greg, I really appreciate the time you have put into try and educate me, but I remain unconvinced. I have read Kip Thorne’s book, and am very aware of his dilemma - after mentioning all the calculations I quoted, he concludes that Black Holes would take an infinite time to form, and then he gets on to Wheeler, and decides to suppress that thought and go along with majority opinion.

I am very aware of the relativity of realities. The fact is that a spaceman falling into a Black Hole REALLY DOES fall in in a short time, in HIS time frame, but he REALLY DOESN’T in OUR distant observer frame. Neither view is an illusion. Neither “appears to”. They are both valid descriptions of what REALLY happens.

Birkoff’s Theorem proves that the space outside a spherically symmetrical Black Hole follows Schwarzschild’s metric, and EVERY calculation using Schwarzschild’s metric has given the same result. The calculation has also been done for spinning Black Holes, but I cannot recall the reference.

Eddington-Finklestein coordinates also resolve to Schwarzschild coordinates outside the event horizon, so they make no difference.

As an example of where Kip Thorne gets it wrong,
The best physical interpretation is that, although we can never actually see someone fall through the event horizon (due to the infinite redshift), he really does. As the free-falling observer passes across the event horizon, any inward directed photons emitted by him continue inward toward the center of the black hole. Any outward directed photons emitted by him at the instant he passes across the event horizon are forever frozen there. So, the outside observer cannot detect any of these photons, whether directed inward or outward.

“Really does”? This assumes one frame is valid and the other an illusion, and that is rubbish. He then describes the observed redshift as purely resulting from the difficulty photons have escaping, and totally ignores the gravitational time dilation.

Naty1, you say

Schwarzschild coordinates include a flat asymptotic spacetime [that's not realistic]; FLRW assumes a perfectly homogeneous and isotropic spacetime and everyone here agrees the FLRW model does NOT apply to galactic scales...One has to also wonder how precise it is on cosmological scales...but that is not especially important for this discussion.

So many replies say the Schwarzschild solution is not accurate, but no one has proved that it gives the wrong answer, or shown me another one which can be proved to be more accurate and gives a different result for gravitational collapse. As I understand it, Schwarzschild only requires that the spacetime be flat and asymptotic at infinity. I doubt that Earth’s gravity and movement would affect the calculation. I’m beginning to think the only reason for rejecting the Schwarzschild calculations is that they say Black Holes don’t exist (at least not yet, in our time frame).

When describing Event Horizons expanding, one must always remember what space-time frame you are using. These events may occur locally, but gravitational time dilation means that in a remote time frame time is stopped at the Event Horizon. This means nothing happens. If you don't accept this, then you need to provide another equation relating time dilation to gravity near a Black Hole.

Mike

 

[PeterDonis]

Sorry, I lost track of this thread for a while so I am catching up with my responses:

PeterDonis, I am surprised you regard “The Classical Theory of Fields” and Physical Review as pop-science articles. I did not draw incorrect deductions. I simply quoted the deductions of the mathematicians.

No, you didn't, at least not when you claim that black holes do not exist. That is not what the mathematicians said.

Please show me where I have misinterpreted the conclusions of Landau and Lifschitz, for example, quoted above.

Here's what Landau and Lifschitz said, that you quoted:

“According to the clocks of a distant observer the radius of the contracting body only approaches the gravitational radius as t -> infinity.”

That does *not* say that the black hole does not exist. It only says something about the clocks of the distant observer. If you look at all the other quotes you gave in context, they all say the same thing. None of them say that the black hole does not exist. In fact, the Oppenheimer-Snyder paper from Physical Review, that you quoted, explicitly says, IIRC, that there is a region of spacetime inside the horizon, and that the collapsing matter falls through that region to a curvature singularity at r = 0.

Rubbish. The maths is written out in Landau and Lifschitzs’ book on pages 297 to 299. Please tell me where the error is between their maths and their conclusions.

The error isn't between their math and their conclusions, it's between their math and *your* conclusions. *They* didn't conclude that the black hole doesn't exist. Only you are concluding that.

 

[PeterDonis]

I have read Kip Thorne’s book, and am very aware of his dilemma - after mentioning all the calculations I quoted, he concludes that Black Holes would take an infinite time to form

Please give the exact chapter and verse for this. I've read Thorne's book too, multiple times, and I don't remember reading this. I remember him saying that, *from the viewpoint of the distant observer*, the BH takes an infinite time to form; but that is *not* the same as saying the BH takes an infinite time to form, period. Nor is it the same as saying the BH does not exist.

and then he gets on to Wheeler, and decides to suppress that thought and go along with majority opinion.

Again, please give exact, specific quotes and references. I don't know what you are referring to here; AFAIK Thorne's opinion about BH spacetimes has not changed significantly since the publication of MTW in 1973, at least, and probably well before that. The book you're referring to was published in 1993.

I am very aware of the relativity of realities. The fact is that a spaceman falling into a Black Hole REALLY DOES fall in in a short time, in HIS time frame, but he REALLY DOESN’T in OUR distant observer frame. Neither view is an illusion. Neither “appears to”. They are both valid descriptions of what REALLY happens.

I don't agree with this way of putting it; or at least, it seems like a very unusual use of the words "real" and "reality". The BH spacetime is a single, geometric object; either it includes a region below the horizon, or it doesn't. The fact that the distant observer can't *see* the region below the horizon doesn't mean it isn't there.

Birkoff’s Theorem proves that the space outside a spherically symmetrical Black Hole follows Schwarzschild’s metric, and EVERY calculation using Schwarzschild’s metric has given the same result.

Yes, if you mean the result that there is a region of the spacetime below the horizon. Every calculation has indeed shown that.

As an example of where Kip Thorne gets it wrong,
...

“Really does”? This assumes one frame is valid and the other an illusion

No, it doesn't. It means the distant observer can't *see* the region below the horizon. That's all it means. Why is that a problem?

He then describes the observed redshift as purely resulting from the difficulty photons have escaping, and totally ignores the gravitational time dilation.

"Gravitational time dilation" is just another way of saying that the photons take a long time escaping.

When describing Event Horizons expanding, one must always remember what space-time frame you are using. These events may occur locally, but gravitational time dilation means that in a remote time frame time is stopped at the Event Horizon. This means nothing happens.

No, it doesn't. It means the coordinates used by the distant observer can't *describe* what happens (because they are singular at r = 2m), but that doesn't mean nothing happens. For example, Eddington-Finkelstein coordinates, which you have mentioned, are not singular at r = 2m, and they say things *do* happen there. Which, btw, is perfectly consistent with the fact that E-F coordinates give the same results as Schwarzschild coordinates when r > 2m, i.e., in the region where Schwarzschild coordinates are not singular.

 

[Dale]

My reading SO FAR leads me to conclude there are not more exact formulations...we don't know how to solve EFE equations in an irregular, curved and lumpy spacetime.

This is correct. There are relatively few exact solutions to the EFE. But you can always solve them numerically for irregular, curved, and lumpy space times.

 

[harrylin]

I think that the following has been cited twice here, but it doesn't make any sense to me:

“What would happen if you fall in? As seen from the outside, you would take an infinite amount of time to fall in, because all your clocks – mechanical and biological – would be perceived as having stopped’' - Carl Sagan “Cosmos”, 1981

I would think that if I (being "outside") perceive that your clocks stop due to your speed and gravitational potential far away from me, this has no effect whatsoever on my clocks. Thus it can have no effect on the Earth time that I estimate it will take for you to fall in. And inversely, for you it will look as if you faster and faster accelerate into the black hole - the final descent happens in nearly no proper time.

If anyone can explain my misunderstanding, I would be very grateful.

 

[DrGreg]

I think that the following has been cited twice here, but it doesn't make any sense to me:

“What would happen if you fall in? As seen from the outside, you would take an infinite amount of time to fall in, because all your clocks – mechanical and biological – would be perceived as having stopped’' - Carl Sagan “Cosmos”, 1981

I would think that if I (being "outside") perceive that your clocks stop due to your speed and gravitational potential far away from me, this has no effect whatsoever on my clocks. Thus it can have no effect on the Earth time that I estimate it will take for you to fall in. And inversely, for you it will look as if you faster and faster accelerate into the black hole - the final descent happens in nearly no proper time.

If anyone can explain my misunderstanding, I would be very grateful.

If I'm falling into a black hole in a finite time according to my own clock, let's say my clock reads exactly 4 pm at the moment I cross the event horizon. If you, hovering at a great constant height, are watching me, you'll see my clock approaching 4 pm, but it will keep slowing down and never actually reach 4 pm. And if you haven't seen my clock reach 4 pm, then you can't have seen me cross the event horizon.

Unless I've misunderstood your question, that's all there is to it, isn't it?

 

[harrylin]

If I'm falling into a black hole in a finite time according to my own clock, let's say my clock reads exactly 4 pm at the moment I cross the event horizon. If you, hovering at a great constant height, are watching me, you'll see my clock approaching 4 pm, but it will keep slowing down and never actually reach 4 pm. And if you haven't seen my clock reach 4 pm, then you can't have seen me cross the event horizon.

Unless I've misunderstood your question, that's all there is to it, isn't it?

OK I misunderstood what Hawkins meant with "“what would happen" - so he was talking about what, in theory, a distant observer literally might see - thanks for the clarification!

 

[DrGreg]

OK I misunderstood what Hawkins meant with "“what would happen" - so he was talking about what, in theory, a distant observer literally might see - thanks for the clarification!

Well, it's not clear whether Sagan was referring to the time that you see an event or the time coordinate that you assign to the event, but the same logic applies either way.

 

[harrylin]

Well, it's not clear whether Sagan was referring to the time that you see an event or the time coordinate that you assign to the event, but the same logic applies either way.

Ah yes, Sagan and not Hawkins! However, the difference between what one sees (an astronaut ever slowly disappearing near a black hole?) and what one infers from that (an astronaut fell into a black hole?) can be huge.

 

[Mike Holland]

Harrylin, you cannot at any stage "infer" that DrGreg fell into the black hole. After carrying out his researches very close to the Event Horizon he might have fired his rockets and come back to join us for tea and to discuss his observations.

Only when he falls through the EH can he say it has happened. But we cannot translate this event into our coordinate system (time frame) because we land up with t = infinity. So we external observers can never say something has fallen into a Black Hole, only that it will, and that it does in its local timeframe.

If I'm falling into a black hole in a finite time according to my own clock, let's say my clock reads exactly 4 pm at the moment I cross the event horizon. If you, hovering at a great constant height, are watching me, you'll see my clock approaching 4 pm, but it will keep slowing down and never actually reach 4 pm. And if you haven't seen my clock reach 4 pm, then you can't have seen me cross the event horizon.

Yes, but be careful with "seen me". It can lead to confusion. If you hover close to the EH, we will see you time dilated, and this has nothing to do with the time photons take to escape the gravity - it depends purely on your distance from the EH. As you fall in photons will take longer and longer to escape, and this causes an additional "apparent" redshift superimposed on the gravitational redshift. Many writers get mixed up with these two redshifts, and conclude that the time dilation is an illusion.

Now if you take this a step or two further, we find that Xwatl, from the planet Wortl, who started falling intro a Black Hole 10,000 years ago, also hasn't reached the EH in our time frame, and neither has that gas cloud that the BH started eating a billion years ago. In fact, nothing has ever falllen into a Black Hole, in our timeframe, so we can never say it has happened.

Mike

 

[PAllen]

Only when he falls through the EH can he say it has happened. But we cannot translate this event into our coordinate system (time frame) because we land up with t = infinity. So we external observers can never say something has fallen into a Black Hole, only that it will, and that it does in its local timeframe.

"our coordinate system" is a convention. Numerous times it has been explained there is nothing physically preferred about SC coordinates. Using a different simultaneity convention, an outside observer can specify a specific time of event horizon crossing even though they never 'see' the crossing. Do you really think a rocket accelerating at 1 g must conclude that much of the universe has ceased to exist? But you say they can stop accelerating. Well, any accelerating hovering observer can choose to stop at any time - and find the part of the universe on the other side of the horizon.

Yes, but be careful with "seen me". It can lead to confusion. If you hover close to the EH, we will see you time dilated, and this has nothing to do with the time photons take to escape the gravity - it depends purely on your distance from the EH. As you fall in photons will take longer and longer to escape, and this causes an additional "apparent" redshift superimposed on the gravitational redshift. Many writers get mixed up with these two redshifts, and conclude that the time dilation is an illusion.

This is pure and simply wrong. There are not two red shifts - period; mathematical fact. The time dilation and the slow speed of photon escape and the redshift are all manifestations of exactly the same factor in the metric, not additive phenomena. All of the authors you misinterpret understand this. Find one author or any mathematical justification of additive redshifts for this situation.

Now if you take this a step or two further, we find that Xwatl, from the planet Wortl, who started falling intro a Black Hole 10,000 years ago, also hasn't reached the EH in our time frame, and neither has that gas cloud that the BH started eating a billion years ago. In fact, nothing has ever falllen into a Black Hole, in our timeframe, so we can never say it has happened.

Mike

In GR, frames are local and coordinates are arbitrary. This is fundamental fact of GR that you reject - and despite your misinterpretation, all the authors you cite did understand this.

 

[Mike Holland]

There are two different redshift phenomena taking place. If a clock is hovering near the Event Horizon, we will see it ticking slowly, at a rate dependent on the size of the BH and its distance from it. Any photons leaving it will be delayed as they escape the gravity, but ALL the photons will be delayed the same amount as the clock is hovering at a fixed point in the gravitational field. So there is no redshift due to the photons taking a long time to get to us. They all take the same long time to reach us. The only redshift in this case is due to the clock slowing, ie. gravitational time dilation. I agree that in this case the gravitatiional time dilation and the slowing of escaping photons are all part of the metric.

If the clock falls into the gravity field, then successive photons take longer and longer to escape the increasing gravity, and this creates a redshift in addition to that of a hovering clock.

Do you believe that there would be a photon redshift when the clock is not moving? Or do you not accept that as the clock descends into the gravity field, emitted light gets delayed more and more?

OK, I agree that a complete mathematical analysis of the descent should cover both redshifts, but there are still two processes involved. For a falling body, successive photons take longer to be emitted because of the time slowing, and then, in addition, take longer to reach us when they have been emitted. Two steps which both cause redshift.

Mike