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.