# Black hole growth paradox question

• Grinkle
In summary: Again, this is not an illusion - they will reach equilibrium in finite time no matter how fast they are moving.If it matters how one infers black hole mass in the answer, please help me understand how.
Grinkle
Gold Member
Are the two following hypothetical observations contradictory according to GR?

1. Particles fall towards a black hole but never cross the event horizon
2. After observing this for a few million years, the mass of the black hole is observed to have grown over that few million years. Or whatever time period less than infinity one chooses.

I am having trouble understanding how a black hole can be observed to grow in finite time if one can never actually observe it obtain an additional particle in finite time.

The best answer I can come up with is that GR does not prohibit an particle crossing the absolute event horizon in finite time, only the apparent event horizon. But I don't know if that is a correct statement about the math, I am just left without anything else to suspect.

Particles don't even NOTICE the event horizon ... only external observers do. The particles just keep right on falling into the black hole. Particles ARE affected by the EH but only in the sense that once they cross it, they can't get back out again.

Particles taking an infinite coordinate time to pass the EH is only a feature of schwarzchild coordinates of the schwarzchild metric. You can eliminate that altogether by swapping to say kruskal coordinates. Particles fall through the EH in finite proper time however for any of the coordinate systems.

I'm not interested in what happens from the perspective of the particle. I don't disagree with what you said.

I am the observer. If I observe a black hole grow in finite time, would that be contradictory to me observing that no mass has entered the black hole in finite time, is my question.edit:
I have to make both observations from my own co-ordinate system - I have no choice.

I may conclude that if I were in a different co-ordinate system I could observe black hole growth, but I am not in such a co-ordinate system.

W-Newton -

What do you mean by -

Particles fall through the EH in finite proper time however for any of the coordinate systems.

?

This is not my understanding. From our earthbound perspective, if we had some means of directly observing a black hole, we would not see particles cross its apparent event horizon in finite time.

Finite proper time means that the particle itself passes the EH in finite time in its own reference frame regardless of the coordinate system. The infinite coordinate time you are talking about is because of schwarzchild coordinates which behave badly at the EH.

Grinkle said:
1. Particles fall towards a black hole but never cross the event horizon

Particles cross the event horizon. From a distance, you don't see the particles going into the event horizon but they do.

As you put more matter in one area, the event horizon moves outward.

Grinkle said:
I'm not interested in what happens from the perspective of the particle. I don't disagree with what you said.

I am the observer. If I observe a black hole grow in finite time, would that be contradictory to me observing that no mass has entered the black hole in finite time, is my question.

edit:
I have to make both observations from my own co-ordinate system - I have no choice.

I may conclude that if I were in a different co-ordinate system I could observe black hole growth, but I am not in such a co-ordinate system.

If you measure the gravity of the black hole, you will see it increase (in YOUR frame of reference) as the mass increases. Also, as twofish said, you'll see the event horizon increase in diameter.

You have in fact NOT observed that no mass has entered the black hole, although I understand that this is a tricky semantic argument. You have observed a property of light/time at the event horizon that is not an actual reflection of what is happening to the stuff that falls in.

I have a lot difficulty with that line of reasoning - that my postulated observation of no particles entering the event horizon is a trick of light.

GR says that if I observe a watch moving very quickly, I observe it ticking slowly. If it were moving at the speed of light, I would never observe it to tick in finite time. This is not an illusion - it would never, in fact, tick in finite time in my frame of reference, according to GR. Of course GR says a watch can only approach c and never actually reach it, but we can let the watch move as close to c as we like to approximate 'never' to however long we want.

Radioactive particles moving very quickly as observed from our frame of reference take longer to decay than they would take if they were at rest in our frame of reference - this is not an illusion.

Unless there is some fundamental difference between this kind of time dilation and the kind of time dilation experienced as a particle approaches an event horizon, I don't see that the latter can be dismissed as a trick of the light.

If we say the above stuff I wrote is over my head to the tune of me standing at the bottom of a 12 foot pool, then the below is me diving a quarter mile below the surface. Bear with me if its intolerably amateurish.

I read about absolute and apparent horizons, and I think maybe the answer is that at some point the matter gets so close to the event horizon that no external observer can tell if it has entered or not. GR tells us it has not, but its mass is not distinguishable from the mass of the singularity that is inside the apparent event horizon so we now call the sphere containing the matter at the edge of the EH the new EH.

This leads me to a think that singularities don't actually come to be, what we observe are collapsing / slowing down pieces of very dense stuff. There never is an actual apparent EH, there is only a growing absolute EH, the EH that would exist anyway if the actual collapse / singularity formation could occur in finite time, which it cannot.

Grinkle said:
GR says that if I observe a watch moving very quickly, I observe it ticking slowly.

That's not what SR says.

Radioactive particles moving very quickly as observed from our frame of reference take longer to decay than they would take if they were at rest in our frame of reference - this is not an illusion.

But they don't take any longer to decay from their reference frame, and there reference frame is just as good as yours. Also, it will help if you don't think in terms of "real" or "illusion". Person A sees this. Person B sees this other thing.

If you try to think about which is "real" then it will just confuse you. Part of it is that people are using different rulers, and my measurements are no more "real" or "less real" than someone else's.

This leads me to a think that singularities don't actually come to be, what we observe are collapsing / slowing down pieces of very dense stuff. There never is an actual apparent EH, there is only a growing absolute EH, the EH that would exist anyway if the actual collapse / singularity formation could occur in finite time, which it cannot.

You aren't the first person to think of this, but the problem is that when you work everything out it doesn't work. The person that enters the black hole does so in finite time so he or she gets crushed by the singularity.

People first came up with GR in the 1920's. Until the late 1960's, most people assumed that the scenario you mentioned would happen so there wouldn't be a black hole. It took a few years of working through the math around 1970 to convince most people that things really don't get frozen when an event horizon forms.

One other way of thinking about it is that something doesn't actually have been be seen to cross the event horizon to make the event horizon grow. I have an event horizon. I put something near the event horizon and, I now have a new event horizon that is outside of the original horizon. All of the math that has things appearing to freeze when they hit the event horizon assumes that everything is static. If you put something near the old event horizon, then what will happen is that you'll get a new event horizon that absorbs the object.

But they don't take any longer to decay from their reference frame, and there reference frame is just as good as yours. Also, it will help if you don't think in terms of "real" or "illusion". Person A sees this. Person B sees this other thing.

Certainly. However, in my hypothetical, I am observer A both times.

I see 'x' as observer A.
I see 'y' as observer A.

The response - if only you allow yourself to be observer B when observing 'y' everything is ok - is not a satisfying response for me. I am always only observer A.
The person that enters the black hole does so in finite time so he or she gets crushed by the singularity.

I think that his finite time needn't also be my finite time. Our respective spacetime cones needn't intersect, am I right? Maybe I am wrong. And if I am making both observations, then my problem still remains.

One other way of thinking about it is that something doesn't actually have been be seen to cross the event horizon to make the event horizon grow.

I think you are talking about absolute event horizons vs apparent event horizons, and this concept is much more intuitive to me. I don't think it contradicts my current (I acknowledge I am not the first person to think this, and I acknowledge it is many decades old thinking) thinking, though. The absolute event horizon may be all that exists, and it keeps expanding as matter approaches the very dense point that is trying to but never can quite become a singularity.

If it has been mathematically shown that earthbound observers can and will see particles cross the absolute event horizon in finite time, then even though I likely won't follow the math, I will feel closer to comfortable with the whole question. Do you know of any citations?

I don't think Hawking et al are wrong - my ego is not nearly so big as that. I am just stuck on this point and having trouble getting past it ...

Grinkle said:
I think you are talking about absolute event horizons vs apparent event horizons, and this concept is much more intuitive to me. I don't think it contradicts my current (I acknowledge I am not the first person to think this, and I acknowledge it is many decades old thinking) thinking, though. The absolute event horizon may be all that exists, and it keeps expanding as matter approaches the very dense point that is trying to but never can quite become a singularity.

If it has been mathematically shown that earthbound observers can and will see particles cross the absolute event horizon in finite time, then even though I likely won't follow the math, I will feel closer to comfortable with the whole question. Do you know of any citations?

I don't get your distinction between absolute vs apparent event horizons. There are coordinate versus intrinsic singularities, but so far as I know, there are just event horizons. An event horizon may be associated with a coordinate singularity in some (ill chosen) coordinate system; you can also have a coordinate singularity that has nothing to do with an event horizon (e.g. the pole in polar coordinates).

A horizon in GR is simply boundary dividing causal influence. It is one way: an event on one side (call it the inside, but not all horizons are closed, e.g. the Rindler horizon) of the horizon is unable to influence anything on the other side; while events on the outside can influence the inside. Put yet another way, the backward pointing null cones of events inside the horizon include portions of the outside. But no events inside the horizon are in the backward null cones of any event outside the horizon.

Thus, a horizon is more than an optical illusion, and has really nothing to do with light, but instead causal influence. Yet this same causal structure also explains why it is expected that matter can cross from outside to inside the horizon - if it couldn't, the causal structure would be wrong - you would have that the two sides are disjoint. That would be a separation of the manifold into disjoint universes, effectively, not a horizon.

Grinkle said:
I see 'x' as observer A.
I see 'y' as observer A.

The response - if only you allow yourself to be observer B when observing 'y' everything is ok - is not a satisfying response for me. I am always only observer A.

If you are observing something locally, then you never notice any time changes. If you are traveling with the thing that you are observing, the clocks stay the same.

I think you are talking about absolute event horizons vs apparent event horizons, and this concept is much more intuitive to me.

Unfortunately it's wrong. An event horizon is an event horizon. An observer falling into a black hole will hit the singularity in a finite time.

If it has been mathematically shown that earthbound observers can and will see particles cross the absolute event horizon in finite time

Here is a diagram that explains what is going on.

http://www.phy.syr.edu/courses/modules/LIGHTCONE/schwarzschild.html

Here is another

http://www.astro.ucla.edu/~wright/bh-st.html

One thing about relativity is that a lot of things start making sense if you draw pictures.

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PAllen said:
I don't get your distinction between absolute vs apparent event horizons. There are coordinate versus intrinsic singularities, but so far as I know, there are just event horizons. An event horizon may be associated with a coordinate singularity in some (ill chosen) coordinate system; you can also have a coordinate singularity that has nothing to do with an event horizon (e.g. the pole in polar coordinates).

A horizon in GR is simply boundary dividing causal influence. It is one way: an event on one side (call it the inside, but not all horizons are closed, e.g. the Rindler horizon) of the horizon is unable to influence anything on the other side; while events on the outside can influence the inside. Put yet another way, the backward pointing null cones of events inside the horizon include portions of the outside. But no events inside the horizon are in the backward null cones of any event outside the horizon.

Thus, a horizon is more than an optical illusion, and has really nothing to do with light, but instead causal influence. Yet this same causal structure also explains why it is expected that matter can cross from outside to inside the horizon - if it couldn't, the causal structure would be wrong - you would have that the two sides are disjoint. That would be a separation of the manifold into disjoint universes, effectively, not a horizon.

Since I wrote this, I have found there is a sense in which some authors define an apparent event horizon versus an actual event horizon. The latter is the causal horizon I describe. The apparent event horizon is actually inside the causal event horizon and is roughly what the event horizon would be if the black hole never grew. However, which events close to the apparent event horizon *now* are ever able to influence distant observers depends on the total future growth of the black hole. Strange as it seems, the causal event horizon *now* must factor in the total future growth of the black hole, and grows *in anticipation* of infalling mass/energy (or a black hole merger).

Remember, GR is a theory that treats the whole of spacetime as an object - all past and future equally manifest. It obviously cannot be considered a theory of time evolution because every observer's time is different.

PAllen -

I don't get your distinction between absolute vs apparent event horizons.

Ref - Black Holes and Time Warps, Kip Thorne, Norton (pub), chapter 12, box 12.2. Thorne attributes the concept to Hawking.

Thorne has a very good reference list with lots of Hawking entries, but I couldn't find one for this particular concept, so I don't know if in what publications Hawking might have discussed it.

An observer falling into a black hole will hit the singularity in a finite time.

Isn't it true that not all observers need necessarily agree on what happens in finite time? If I am mistaken on that then I need to move backwards a couple steps and re-structure my thinking.

PAllen -

We posted at about the same time.

Your description is roughly as well as I am able to picture it also. Especially your statement that the apparent horizon is the horizon that would exist if the hole never grew.

This concept is what put me at the point of thinking that there is really no such (apparent) horizon, because where does that 'hole never grew' reasoning end? If the singularity never forms in finite time, but is always getting closer and closer to forming, then the horizon we see is what would exist if we waited around for inifinite time to tick by, is where my thinking has cornered me.

All matter falling towards the forming singularity enlarges the effect-before-cause EH that we observe, and its the EH that would truly exist if only the collapse could actually occur.

I don't think I am correct - but I haven't been able to think my way past this picture.

Grinkle said:
PAllen -
Ref - Black Holes and Time Warps, Kip Thorne, Norton (pub), chapter 12, box 12.2. Thorne attributes the concept to Hawking.

Thorne has a very good reference list with lots of Hawking entries, but I couldn't find one for this particular concept, so I don't know if in what publications Hawking might have discussed it.
Since I wrote that, I have seen and read descriptions of apparent event horizons. See my later post. However, I don't know whether all authors use it the same. It is not a fundamental concept compared to actual or causal event horizons.

Grinkle said:
Isn't it true that not all observers need necessarily agree on what happens in finite time? If I am mistaken on that then I need to move backwards a couple steps and re-structure my thinking.

Of course that's true. My point is that what is seen from Earth (or any single vantage point) is a silly definition of what 'is'. Depending on expansion rate, two sufficiently distant galaxies that collide will never be seen by us (in principle - their light will never reach us). Does that mean we should deny they collided? The fact that we don't see something reach the singularity should no more be taken to mean it doesn't happen (as long as one is accepting GR).

Also, note that definitions of event horizons are in terms of null cones from event considered to have no mass. You can't go from here to conclusions about a massive body approaching a black hole, where there is mutual influence. I think what you would see during the inspiral and merger of two black holes is not very different from what you would see for two neutron stars. All the light you see would be from outside the evolving event horizon, but accreting matter lighting things up would make it look similar.

[EDIT: Note, when two stars collide, you don't see what happens deep inside. So you see the whole process via what happens outside the event horizon (whose spatial shape varies over time for some given observer). ]

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Grinkle said:
PAllen -

We posted at about the same time.

Your description is roughly as well as I am able to picture it also. Especially your statement that the apparent horizon is the horizon that would exist if the hole never grew.

This concept is what put me at the point of thinking that there is really no such (apparent) horizon, because where does that 'hole never grew' reasoning end? If the singularity never forms in finite time, but is always getting closer and closer to forming, then the horizon we see is what would exist if we waited around for inifinite time to tick by, is where my thinking has cornered me.

All matter falling towards the forming singularity enlarges the effect-before-cause EH that we observe, and its the EH that would truly exist if only the collapse could actually occur.

I don't think I am correct - but I haven't been able to think my way past this picture.

I think the key thing to let go of is the identity between what you see and what is. I accelerate away from Earth with uniform acceleration. A Rindler horizon forms, and I never see anything from Earth past a certain time. Has Earth stopped evolving because I can't see it? What happens inside the BH event horizon is similar.

What if I still observe earth, even though I predict I shouldn't given my acceleration? That is a reasonable analogy to my OP.

If I do in fact observe black hole growth, and I do in fact observe that no matter has entered a black hole, are these two observations made by be in a single reference frame contradictory?

I am fine with the concept that black holes may grow in finite time and I just couldn't observe them to grow due of my own particular spacetime reference being exclusive from any possibility of seeing the growth that is happening.

But if I DO see them grow, then I should also be able to see particles enter them, is my point.

Unless one allows for this effect-before-cause growth, and then I get back to whether any collapse has actually yet occurred, if I am now postponing the cause.

Grinkle said:
What if I still observe earth, even though I predict I shouldn't given my acceleration? That is a reasonable analogy to my OP.

If I do in fact observe black hole growth, and I do in fact observe that no matter has entered a black hole, are these two observations made by be in a single reference frame contradictory?

I am fine with the concept that black holes may grow in finite time and I just couldn't observe them to grow due of my own particular spacetime reference being exclusive from any possibility of seeing the growth that is happening.

But if I DO see them grow, then I should also be able to see particles enter them, is my point.

Unless one allows for this effect-before-cause growth, and then I get back to whether any collapse has actually yet occurred, if I am now postponing the cause.

OK, I think I see some of your confusion. Consider two cases:

1) A big planet collides with a neutron star. The star grows a little and you see a slightly bigger surface. You don't see anything from below the surface. You can suppose there is a center and something happens there.

2) A big planet collides with a black hole. The event horizon itself is getting bigger as the object approaches. However, you can't see this or any event horizon itself. You see the planet colliding with a visible surface outside the event horizon. You see this visible surface grow, quite similar to the neutron star case. At all times, the true event horizon is below the visible surface. You can't see what happens deeper, but if you believe GR, you can suppose what happens.

Grinkle said:
Are the two following hypothetical observations contradictory according to GR?

1. Particles fall towards a black hole but never cross the event horizon
2. After observing this for a few million years, the mass of the black hole is observed to have grown over that few million years. Or whatever time period less than infinity one chooses.

I am having trouble understanding how a black hole can be observed to grow in finite time if one can never actually observe it obtain an additional particle in finite time.
...

There is one little point to make which I didn't see made when I took a quick look at the thread.

Your paradox is that you as an observer see mass collect on this spherical shell. You don't see it collect at the center of the shell. So how do you see the mass increase?

Well, wouldn't it be the same if the mass DID collect on the spherical shell, as long as it was evenly spread out?

Doesn't a uniform massive hollow sphere curve spacetime the same as an equal pointmass at center?

And what you see as an observer is that a small bit of mass dropped onto the hole does spread out evenly. Because the lightwaves you see it with are redshifted. For you as an observer, it does not retain a fixed location on the horizon.

So you the observer can calculate what the increased mass should be after the particle is dropped in based literally on what you can see. You can believe what you see as an observer (even though you might know better) and calculate with that and still get the right answer.

That may be wrong. Hopefully someone will correct, if it is. But if it is right then that helps to resolve the "Grinkle paradox": the apparent contradiction which you present to us in your lead post.

This may be a simplistic answer to this question, but regardless to any observer and the event horizon one thing is quite clear; when a particle falls into a black hole gravitation increases, in spite of spacetime effects and immediately being that gravity is not effected by gravitation. An outside observer may see what appears like a particle or other mass smeared on the event horizon but when that mass merges with the singularity increased gravitation is immediately felt. The light is effected by the temporal effects but gravity appears not. I know of no mathematics that would describe that effect for I haven't found ether in Special or General Relativity. Could be described by Quantum Gravity.

Therefore: mass enters event horizon; temporal effects makes mass appear to hover forever on the event horizon; mass actually falls to the singularity; gravitation increases; event horizon and various radii get larger; black hole becomes larger.

This does not conflict with Special or General Relativity to my knowledge.

## 1. What is the Black Hole Growth Paradox?

The Black Hole Growth Paradox is a question about how black holes can grow to their observed sizes in such a short amount of time, given the limited amount of matter available in the universe.

## 2. How do black holes grow?

Black holes grow by accreting matter from their surrounding environment. As matter falls into the black hole's gravitational pull, it becomes heated and emits radiation, which can be detected by telescopes.

## 3. Why is the Black Hole Growth Paradox considered a paradox?

The paradox arises from the fact that black holes can grow to enormous sizes in a relatively short amount of time, even though the amount of matter available for accretion is limited. This seems to contradict the laws of physics, which state that matter cannot be created or destroyed.

## 4. What are some proposed solutions to the Black Hole Growth Paradox?

Some proposed solutions include the existence of primordial black holes that formed in the early universe, or the possibility of black holes merging and growing in size. It is also possible that our current understanding of the universe is incomplete and there may be other processes at play.

## 5. Why is the Black Hole Growth Paradox important?

Studying the growth of black holes can provide insights into the formation and evolution of galaxies, as black holes are thought to play a crucial role in shaping the structures of galaxies. Understanding the paradox can also push the boundaries of our current understanding of physics and the universe.

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