B How do Black Holes Grow? A Far-Away Observer's Perspective

[Rene Dekker]

TL;DR Summary

According to General Relativity, objects falling towards a black hole will never actually appear to fall in. That bags the question, how do they grow? And if they cannot grow, how can they exist in the first place?

Black holes are everywhere in astrophysics. There are numerous discussion about how black holes look like, what happens to gas falling into black holes, how light bends around black holes, whether there is loss of information when mass or energy falls in, etc. There is thought to be a black hole at the centre of almost every galaxy.

At the same time, If we look into General Relativity, we can discuss what appears to happen to objects falling towards black holes, from the viewpoint of a far-away observer. Then I understand that they actually never appear to fall in. They appear to slow down, their time appears to slow down, they get red-shifted until they are almost not visible anymore, but never actually appear to pass the event horizon.

That bags the question: if nothing ever falls in, how do black holes grow?

The question has been asked before on various fora, here as well, and the answer is often in the trend of “when you look at it from the point of the view of the object falling in, then there is no problem; it does not even notice the event horizon”. That is a valid response, and true as far as know, but that does not help much with describing black holes from the viewpoint of a far-away observer. For such an observer, the object will not fall in before the end of the universe (if I understand it correctly). And to us on Earth, that viewpoint is much more applicable than the viewpoint of the unlucky object.

So in this question, I am really interested in the viewpoint of a far-away observer.

To such an observer, if nothing ever falls into the black holes, then how can they exist in the first place? How do they start? I can imagine two particles getting pressed together so hard that their combined mass is within their event horizon (if that is even possible according to QM), creating a micro black hole. But how does that black hole ever grow further if no other particles can fall in?

Is it fair to say that when astrophysicist use the term “black hole”, that they actually mean “a very massive, very dense object that is almost a real black hole, but not quite yet”.
Maybe that is justified because there is no real difference in influence of such very massive objects, and real black holes?

Or am I wrong in my understanding somewhere?

 

[martinbn]

According to General Relativity, objects falling towards a black hole will never actually appear to fall in.

Why do you think this is the case and why do you think it matters?

 

[Rene Dekker]

That comes from my limited understanding of General Relativity. Is that incorrect?
It matters, because of the reasons I state in the question.

 

[phinds]

Or am I wrong in my understanding somewhere?

Yes, you are. Think about the word "appear" in your first sentence and consider that perhaps the faraway observer understands physics. Also, you might do a forum search since the topic has been discussed here many times.

 

[Rene Dekker]

I did do a forum search, and all answers I found where paraphrases of what I mentioned in the question already. That, when you look at it from the reference point of the object falling in, there is no problem.
If you can point me to an answer that discusses the point of view of a far-away observer in details, that would be great.

It would also be great if you could explain how an object falling towards a black hole would appear to a far-away observer, if it differs from what I mentioned.

 

[phinds]

You are missing the point. If appearance and reality don't match, there's something wrong with the appearance. The object falls in. Period. The fact that it does not appear to do so to a remote observer is irrelevant.

 

[Rene Dekker]

I understand that the event horizon of a black hole is similar to a Rindler horizon of an accelerated observer. I understand Rindler coordinates better than General Relativity, and in Rindler coordinates a free-falling object will definitely never appear to cross the horizon. That is, the free-falling object never leaves the Rindler coordinate system in finite coordinate time. That is what I mean with "appear".

"If appearance and reality don't match, there's something wrong with the appearance"
As far as I understand, in the reality of the far-away observer, it never happens (or, equivalently, it happens in the infinite future). Is that an incorrect understanding?

 

[phinds]

As far as I understand, in the reality of the far-away observer, it never happens (or, equivalently, it happens in the infinite future). Is that an incorrect understanding?

Yes, it is incorrect, unless the far-away observer does not understand science. In the reality of the far-away observer, he knows that although he can't SEE the object fall it, it has non-the-less fallen in. If he insists that it has NOT fallen in than he simply doesn't understand physics or is being willfully obtuse.

 

[Rene Dekker]

"he can't SEE the object fall it"
With the term "appear", I don't mean seeing the object falling in, in the sense that light from the object falling in will never reach him.
I am talking about the object not falling in, in the sense of the Rindler horizon for accelerated observers: the object never leaves the coordinate system of the observer. That is, in the reality of the observer, it never falls in.

Maybe a black hole is not comparable to a Rindler horizon, or maybe it does not make a difference for the physics, but it would great if you had some pointers on where I can get more information on that.

 

[Nugatory]

If you can point me to an answer that discusses the point of view of a far-away observer in details, that would be great.

This thread https://www.physicsforums.com/threa...orever-approaching-a-bh-and-bh-growth.921644/

I’ll dig up a few more in a moment

 

[TeethWhitener]

You might want to take a look at the following insight (and its continuations):
https://www.physicsforums.com/insig...-model-of-gravitational-collapse-an-overview/

 

[Nugatory]

If you can point me to an answer that discusses the point of view of a far-away observer in details, that would be great.

Also post #57 of https://www.physicsforums.com/threads/where-is-the-matter-in-a-black-hole.836527/

 

[Rene Dekker]

Thanks for that, Nugatory. One of your responses in that thread cleared it up for me. I quote the text here:

Back up for a moment and consider a black hole of mass M, surrounded by a spherically symmetrical shell of dust with total mass m at a great distance from the black hole and falling/collapsing into the black hole. Consider R' the Schwarzschild radius of a hypothetical black hole of mass M + m, and R the Schwarzschild radius of the black hole. Clearly R' is greater than R, so is outside the event horizon; thus the in-falling shell of dust will reach R' in finite time according to you and other external observers. But once it gets there... we have a spherically symmetric mass distribution all inside its Schwarzschild radius R', and that is a black hole with radius R'. So our initial state is a black hole of radius R and our final state is a black hole of radius R', and we get from one to the other in a finite external time. We just can't use the Schwarzschild solution to describe what's happening in between.

I will study the other good responses and links as well.
Thanks.

 

[Dale]

As far as I understand, in the reality of the far-away observer, it never happens (or, equivalently, it happens in the infinite future). Is that an incorrect understanding?

There are not multiple realities. There are multiple coordinates, but coordinates do not define reality.

 

[George Jones]

I am talking about the object not falling in, in the sense of the Rindler horizon for accelerated observers: the object never leaves the coordinate system of the observer. That is, in the reality of the observer, it never falls in,

What about according to the object, and the object's clock?

 

[phinds]

coordinates do not define reality

Thank you Dale. That's a much better way of saying what I've been trying to explain to him.

 

[Nugatory]

One of your responses in that thread cleared it up for me.

Glad it helped, but note that it doesn't explain what the "infinite time to cross the horizon" statement does imply, and that's actually more important to overall understanding. So please be sure to work through the second link as well.....

 

[Dale]

TL;DR Summary: According to General Relativity, objects falling towards a black hole will never actually appear to fall in. That bags the question, how do they grow? And if they cannot grow, how can they exist in the first place?

To such an observer, if nothing ever falls into the black holes, then how can they exist in the first place? How do they start? I can imagine two particles getting pressed together so hard that their combined mass is within their event horizon (if that is even possible according to QM), creating a micro black hole. But how does that black hole ever grow further if no other particles can fall in?

In addition to the issues about overemphasizing coordinates that have already been covered, I would point out that the Schwarzschild solution is a vacuum spacetime. So when you are talking about a black hole growing you are no longer talking about the Schwarzschild solution, at least not in the region where there is matter falling in to make it grow.

A better spacetime to represent a growing black hole would be the Oppenheimer Snyder spacetime, which describes a collapsing ball of pressureless dust. Any more complicated growing black hole model will require a numerical solution, but such numerical solutions have been performed for decades and the general features are well known. While there is matter outside the horizon the horizon can still grow and will increase its surface area, even in the usual coordinates of an external observer.

Edit: I believe that this is the model that @Nugatory had in mind in his post from the previous thread

 

[Ibix]

Any more complicated growing black hole model will require a numerical solution

There's also the ingoing Vaidya metric, for an exact solution approximating a black hole absorbing radiation. It's not trivial...

 

[Rene Dekker]

There are not multiple realities. There are multiple coordinates, but coordinates do not define reality.

Maybe they don't define reality, but they should at least describe reality, right? Coordinates are a generalisation of the measurements that we can make about reality. If the real measurements don't correspond to the coordinate system that we have chosen, then we need to change our coordinate system.
Measurements are our only way to observe and describe reality. If something cannot be measured, then it is not really part of reality. Hence if it is outside our coordinate system (carefully chosen such that all possible measurements can be described with them), it is not really part of our reality.

That is very philosophical, of course, and it is not a practical basis for physics, because it would assign a personal reality to everybody, which would change based on your actions.

 

[Rene Dekker]

Glad it helped, but note that it doesn't explain what the "infinite time to cross the horizon" statement does imply, and that's actually more important to overall understanding. So please be sure to work through the second link as well.....

Thanks, will do that. Is it OK to ask questions about that here, as the original thread is closed?

 

[Nugatory]

Thanks, will do that. Is it OK to ask questions about that here, as the original thread is closed?

Here or in a new thread. New thread might be best if the questions will be primarily about how the outside observer's wristwatch time is related to points on the timelike worldline of a hypothetical infalling test mass (which is what the "never reaches the horizon" misunderstanding is mostly about).

 

[PeterDonis]

Maybe they don't define reality, but they should at least describe reality, right?

Only in a very weak sense: a valid coordinate chart will assign a distinct 4-tuple of real numbers to every event in spacetime, subject to some conditions about continuity, etc. But there is no guarantee that those numbers will correspond to any actual measurements that anyone can make.

Coordinates are a generalisation of the measurements that we can make about reality.

Many coordinate charts work this way, but there is no requirement in General Relativity that all coordinate charts must work this way. So you need to look at the specific case to see whether the coordinates being used are intended to correspond to actual physically meaningful measurement numbers, or not.

If the real measurements don't correspond to the coordinate system that we have chosen

Any valid measurement result will be capable of being represented as an invariant, which means a number that works out the same regardless of which coordinate chart you use to compute it. So it's not really possible for a real measurement result to "not correspond to the coordinate system".

However, there is no guarantee that the calculation of the invariant representing a particular measurement result will look simple in a given coordinate chart. I suspect that "look simple" is closer to what you had in mind when you said "correspond"--that there will be some simple relationship between coordinate numbers and measurement numbers. That will not always be the case.

 

[Rene Dekker]

Many coordinate charts work this way, but there is no requirement in General Relativity that all coordinate charts must work this way. So you need to look at the specific case to see whether the coordinates being used are intended to correspond to actual physically meaningful measurement numbers, or not.

Thanks, I did not expect such an insightful answer to my philosophical daydreams
Are there physical situations where it is not possible at all to assign any coordinate system that reflects the measurements? That is, where it is not possible to define a coordinate system where spacetime distances between events are identical to the distance in the coordinate system?

 

[Ibix]

That is, where it is not possible to define a coordinate system where spacetime distances between events are identical to the distance in the coordinate system?

All of them except Cartesian coordinates on Euclidean space, I think, since I think what you are asking requires that the metric is the identity matrix.

 

[PeterDonis]

Are there physical situations where it is not possible at all to assign any coordinate system that reflects the measurements?

It depends on how strict you are about what "reflects the measurements" means. As I have said, any valid coordinate system will allow you to calculate measurement results correctly. But the calculations might not be simple.

where it is not possible to define a coordinate system where spacetime distances between events are identical to the distance in the coordinate system?

"Spacetime distances between events" (more precisely, spacetime distances along particular curves) are invariants, so any valid coordinate system will calculate the same numbers for them. But, as noted, the calculations might not be simple.

If your question is whether there are cases where it is not possible for "raw" coordinate distances (i.e., just subtract one set of coordinates from the other, without using the spacetime metric) to directly reflect actual spacetime distances (the ones you get by calculating using the metric), the answer is that every curved spacetime is such a case; in any curved spacetime it is impossible to find a coordinate chart in which "coordinate distances" (just subtract one set of coordinates from the other) are the same as spacetime distances (calculated using the metric).

 

[Rene Dekker]

Here or in a new thread. New thread might be best if the questions will be primarily about how the outside observer's wristwatch time is related to points on the timelike worldline of a hypothetical infalling test mass (which is what the "never reaches the horizon" misunderstanding is mostly about).

Quick question then (about post #57 of https://www.physicsforums.com/threads/where-is-the-matter-in-a-black-hole.836527/)
You talk about bouncing light signals off the object that is falling into the black hole, and measuring how much time it takes for those signals to return, in order to find out at what (observer) time the object falls in (similar to an Einstein clock synchronisation protocol).

Those signals take longer and longer to return, the closer the object comes to the horizon, and eventually they will not return at all anymore. So, from the observers naive point of view, that would make it appear as if the object never falls in. Do I understand that correctly?

Given the constant speed of light, it could be surprising that the light would take longer and longer to return.
Is it a correct assumption that in Schwartzchild coordinates, that is described as space stretching more and more closer to the black hole, and that the event horizon is virtually infinitely far away?

I am not familiar with Kruskal coordinates, and a quick glance shows that it is probably above my pay grade. Is it possible to summarise shortly how Kruskal coordinates describe the same phenomenon?

Let me know if I am being too naive.

 

[Rene Dekker]

If your question is whether there are cases where it is not possible for "raw" coordinate distances (i.e., just subtract one set of coordinates from the other, without using the spacetime metric) to directly reflect actual spacetime distances (the ones you get by calculating using the metric), the answer is that every curved spacetime is such a case; in any curved spacetime it is impossible to find a coordinate chart in which "coordinate distances" (just subtract one set of coordinates from the other) are the same as spacetime distances (calculated using the metric).

Thanks. That is clear.

 

[Ibix]

Is it a correct assumption that in Schwartzchild coordinates, that is described as space stretching more and more closer to the black hole, and that the event horizon is virtually infinitely far away?

No, the horizon is a finite distance away (or, at least, $\int\sqrt{ g_{rr}}dr$ is finite). It's just that the horizon is never at a time you'd call "now" using those coordinates.

 

[Rene Dekker]

No, the horizon is a finite distance away (or, at least, $\int\sqrt{ g_{rr}}dr$ is finite). It's just that the horizon is never at a time you'd call "now" using those coordinates.

I need to mull that over. Isn't that just another way of saying the same thing? At a moment you'd call "now", the horizon is infinitely far away?

 

[Ibix]

I need to mull that over. Isn't that just another way of saying the same thing? At a moment you'd call "now", the horizon is infinitely far away?

No, the value of the integral is finite and constant at all times. It's just that no time coordinate can be assigned to any event on the horizon in this coordinate system. Trying to do it is like asking what's east of the north pole on Earth. The coordinate system just doesn't work in a way that lets the question make sense.

 

[Dale]

Maybe they don't define reality, but they should at least describe reality, right?

Sure, but the end of a description does not imply the end of the thing described. Similarly, a description that goes forever does not imply the thing goes forever.

Coordinates are a generalisation of the measurements that we can make about reality.

No. Coordinates are just an assignment of a set of numbers to events such that the mapping is smooth and invertible. Coordinates are not a generalization of measurements. In particular, measurements are invariant and coordinates are not.

If the real measurements don't correspond to the coordinate system that we have chosen, then we need to change our coordinate system

No. That is not a requirement for coordinates. Coordinates are only required to be smooth and invertible. Using any such coordinates the measurements can be expressed as invariants.

Measurements are our only way to observe and describe reality. If something cannot be measured, then it is not really part of reality.

Yes.

Hence if it is outside our coordinate system (carefully chosen such that all possible measurements can be described with them), it is not really part of our reality.

No. The correct “hence” is: “Hence coordinates are not part of reality”.

This should not be a surprising conclusion since we began by saying that coordinates just describe reality. So the idea that a mere description of reality is not reality, should not be objectionable.

Are there physical situations where it is not possible at all to assign any coordinate system that reflects the measurements? That is, where it is not possible to define a coordinate system where spacetime distances between events are identical to the distance in the coordinate system?

Any time there is spacetime curvature (tidal gravity) it is impossible.

 

[jbriggs444]

Any time there is spacetime curvature (tidal gravity) it is impossible.

A trip to Google finds this post from Physics Forums back in 2009 by @George Jones in #5.

Chris Clarke* showed that every 4-dimensional spacetime can be embedded isometically in higher dimensional flat space, and that 90 dimensions suffices - 87 spacelike and 3 timelike. A particular spacetime may be embeddable in a flat space that has dimension less than 90, but 90 guarantees the result for all possible spacetimes.

* Clarke, C. J. S., "On the global isometric embedding of pseudo-Riemannian
manifolds," Proc. Roy. Soc. A314 (1970) 417-428

This would suggest that it is possible to have a flat space time (with a lot of dimensions) with an ordinary pseudo-Riemanian metric so that an embedded four dimensonal, intrinsically curved subspace can share the same metric.

Likely the required embedding is so convoluted that the conclusion is rendered meaningless for all practical purposes.

 

[PeterDonis]

This would suggest that it is possible to have a flat space time (with a lot of dimensions) with an ordinary pseudo-Riemanian metric so that an embedded four dimensonal, intrinsically curved subspace can share the same metric.

"The same metric" on the much higher dimensional space, yes. But "distance" in the higher dimensional space is not the same as "distance" restricted to the 4-D subspace. So the fact that the metric on the higher dimensional space is flat is useless, since it doesn't give you the "spacetime distance" that you are looking for.

 

[Dale]

But "distance" in the higher dimensional space is not the same as "distance" restricted to the 4-D subspace.

Well, “distance” between two separate events is not the same. But “distance” along a path is the same in the embedding space as in the curved space.

 

[PeterDonis]

“distance” along a path is the same in the embedding space as in the curved space.

Yes, but the flat metric "distance" that the OP is interested in (what the OP calls "coordinate distance") is the distance along a geodesic of the flat metric, not along a curve lying in the embedded curved space.

 

[Demystifier]

You are missing the point. If appearance and reality don't match, there's something wrong with the appearance. The object falls in. Period. The fact that it does not appear to do so to a remote observer is irrelevant.

The problem is that one cannot say when the object falls in, in terms of time that makes sense to the outside observer.

 

[renormalize]

The problem is that one cannot say when the object falls in, in terms of time that makes sense to the outside observer.

I'm not sure I agree. For example, imagine an external observer watching an electric charge fall toward a neutral Schwarzschild black hole. When the charge is at any finite height above the hole, the observer sees the charge's electric field as a complicated pattern of superposed multipole moments arising from the distorted spacetime around the hole. But as the charge approaches the horizon, the field smoothly transitions to the electric monopole field of a charged Reissner-Nordstrom black hole. So even though the observer never sees the charge actually pass the horizon, by monitoring the distribution of the E-field outside the hole could she not reasonably declare the time of in-fall to be the moment the external field is sufficiently close to a coulomb field?

 

[Ibix]

The problem is that one cannot say when the object falls in, in terms of time that makes sense to the outside observer.

I don't entirely agree with this. You can't say when it crosses the horizon in Schwarzschild coordinates, I agree, because it left the region covered by Schwarzschild coordinates. But that isn't the only definition of "now" available to an external observer - merely one of the simplest, as it corresponds to radar time.

There is a time for any external observer when the event "infalling object crosses the horizon" is no longer in their future light cone. Up until this time they might do something that stops the infalling object from entering the hole, but after this time it is too late. Even if all they need do is send a light pulse to the object asking it to turn on its arbitrarily powerful rocket motor, they are too late - the object crossing the horizon is no longer in their causal future. At any later event on the external observer's worldline there is an acausal hyperplane that passes through that event and the horizon crossing event. So you can call any time after the horizon crossing leaves your future light cone "when" the object crossed the horizon if you want.

 

[PeroK]

I don't entirely agree with this. You can't say when it crosses the horizon in Schwarzschild coordinates, I agree, because it left the region covered by Schwarzschild coordinates. But that isn't the only definition of "now" available to an external observer - merely one of the simplest, as it corresponds to radar time.

There is a time for any external observer when the event "infalling object crosses the horizon" is no longer in their future light cone. Up until this time they might do something that stops the infalling object from entering the hole. But after this time it is too late. Even if all they need do is send a light pulse to the object asking it to turn on its arbitrarily powerful rocket motor, they are too late - the object crossing the horizon is no longer in their causal future.

There's a calculation of this here:

https://www.physicsforums.com/threa...into-a-black-hole.1012103/page-3#post-6599762

 

[Nugatory]

I'm not sure I agree…. could she not reasonably declare the time of in-fall to be the moment the external field is sufficiently close to a coulomb field?

I don't entirely agree with this…. There is a time for any external observer when the event "infalling object crosses the horizon" is no longer in their future light cone.

That’s two different but equally reasonable definitions, which I think demonstrates @Demystifier’s point.

 

[Ibix]

That’s two different but equally reasonable definitions, which I think demonstrates @Demystifier’s point.

I'd say it illustrates the opposite - there are many definitions that make sense.

 

[Demystifier]

The outside observer can reason as follows. I define time as the thing measured by my own local clock. With this definition of time, it takes an infinite time that a falling object reaches the horizon, so the falling object can never (where "never" means not after a finite time) enter the black hole interior. Moreover, according to classical general relativity, there is no experiment that I (the outside observer) can perform that would prove me wrong.

 

[martinbn]

The outside observer can reason as follows. I define time as the thing measured by my own local clock. With this definition of time, it takes an infinite time that a falling object reaches the horizon, so the falling object can never (where "never" means not after a finite time) enter the black hole interior. Moreover, according to classical general relativity, there is no experiment that I (the outside observer) can perform that would prove me wrong.

That is not clear. He can define time by his watch only along his world-line. For these questions he needs time for much more than his world line. And he can define a time function, whose level surfaces will constitute his convention of nows and at the events of his world line will have values equal to his watch readings, and it can cover parts of the space-time inside the black hole. This way he can say when the object entered the black hole.

 

[Rene Dekker]

No, the horizon is a finite distance away (or, at least, $\int\sqrt{ g_{rr}}dr$ is finite). It's just that the horizon is never at a time you'd call "now" using those coordinates.

The earlier question was about light pulses that are sent from an outside observer, reflect back from the object falling towards the black hole, and then return back at the observer again. They apparently would take longer and longer time to return back, the closer the object comes to the event horizon, until they take infinitely long.
If the distance to the horizon is finite in Schwarzschild coordinates, then how do they describe these increasing return times? Is the coordinate speed of light close to the black hole less than c?

 

[Dale]

The outside observer can reason as follows. I define time as the thing measured by my own local clock. With this definition of time, it takes an infinite time that a falling object reaches the horizon

This reasoning does not follow. That definition of time only assigns time to events on your worldline. To extend it anywhere else you must also adopt a simultaneity convention.

 

[Demystifier]

This reasoning does not follow. That definition of time only assigns time to events on your worldline. To extend it anywhere else you must also adopt a simultaneity convention.

Fair enough. Is there a reasonable simultaneity convention which I, as an outside observer, can apply to the black hole interior?

 

[martinbn]

Fair enough. Is there a reasonable simultaneity convention which I, as an outside observer, can apply to the black hole interior?

Depends on what you mean by reasonable.

 

[Dale]

Fair enough. Is there a reasonable simultaneity convention which I, as an outside observer, can apply to the black hole interior?

Sure. You can use the simultaneity convention of any of the following:

https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

https://en.m.wikipedia.org/wiki/Lemaître_coordinates

https://en.m.wikipedia.org/wiki/Eddington–Finkelstein_coordinates

https://en.m.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

Or any other simultaneity convention you like. Your simultaneity convention is arbitrary.

 

[Dale]

Is the coordinate speed of light close to the black hole less than c?

Not only is the coordinate speed of light less than c, it is also anisotropic in Schwarzschild coordinates.