I Being at the position of a singularity before it is formed

[Ibix]

For a stationary shell, it would. But this shell is not stationary. It's collapsing.

OK, here's my argument.

Inside the shell we have Minkowski spacetime, with a particular simultaneity convention picked out by the symmetry of the shell. Outside we have Schwarzschild spacetime. So I have two regions with well defined notions of simultaneity.

Consider a shell of finite thickness. I can establish a notion of simultaneity between a clock just above the shell and infinity. I cannot establish unique simultaneity between a clock just inside the shell and infinity. However, any definition of simultaneity I do define between the clock just inside and just outside the shell requires simultaneous events to be outside each others' light cones.

If I consider the same scenario with a thinner shell, I can have my just-inside and just-outside clocks closer together, so I have a narrower range of events I can consider simultaneous.

So if I consider a shell of delta-function thickness I can have my clocks arbitrarily close together. So do I have any choice about which event just inside and just outside the shell are simultaneous? You seem to think so, if I understand you correctly, but I don't see why.

 

[HansH]

I have another point bothering me since I saw a video (sorry I don't know the link anymore) probably not completely on topic but could help to get the picture more clear.

in the video it was told that in our universe you can calculate the amount of matter inside a certain volume and that the conclusion is that for a certain sphere of volume of the universe the corrsponding event horzon is within that shpere, drawing the conclusion that we could live inside a black hole. But if that would make sense, why is all the matter inside that event horizon than not forming a singularity?

or is what they say therefore simply not true?

 

[Ibix]

in the video it was told that in our universe you can calculate the amount of matter inside a certain volume and that the conclusion is that for a certain sphere of volume of the universe the corrsponding event horzon is within that shpere,

They (or you) are mixing up the Schwarzschild radius and the event horizon. You can calculate the Schwarzschild radius for any lump of matter, but it doesn't mean there's an event horizon there. For example, the Schwarzschild radius for the Sun is about 3km, but there is not an event horizon in the solar core.

Yes, if you pick a large enough region of the universe it's wholly contained within its Schwarzschild radius, but that doesn't automatically make it a black hole. On cosmological scales there is matter everywhere with more or less the same density. That is very different from a black hole spacetime, which has the mass concentrated in one region. So no, we don't live in a black hole (or at least, the line of argument you cite does not lead to that conclusion).

 

[PeterDonis]

you have to be able to exit the shell before it crosses its own horizon.

No, it's more stringent than that. You have to avoid crossing the horizon while you are inside the shell; that can happen even if the shell itself has not yet crossed the horizon. And there is no way to know, locally, where the horizon is inside the shell; you can't even use the local spacetime geometry to estimate it because the local spacetime geometry is flat everywhere inside the shell.

 

[PeterDonis]

Inside the shell we have Minkowski spacetime, with a particular simultaneity convention picked out by the symmetry of the shell.

Yes, this part is fine.

Outside we have Schwarzschild spacetime.

Yes, but bounded on the "inside" by the shell, whose areal radius $r$ is decreasing.

So I have two regions with well defined notions of simultaneity.

Not quite. For the region occupied by simultaneity surfaces that cross the shell before the shell crosses the horizon, the Schwarzschild region is entirely exterior to the horizon, so yes, it will have a well-defined set of simultaneity surfaces picked out by orthogonality to the timelike Killing vector field in the exterior region. But you still have to match the surfaces, and there is an issue there that you do not appear to have recognized; see below.

But once the shell crosses the horizon, the Schwarzschild region includes the horizon and a portion inside the horizon, and Schwarzschild spacetime is not stationary there (the KVF is null on the horizon and spacelike inside it), so there are no well-defined surfaces of simultaneity defined by the KVF.

if I consider a shell of delta-function thickness I can have my clocks arbitrarily close together. So do I have any choice about which event just inside and just outside the shell are simultaneous? You seem to think so, if I understand you correctly

Sort of. The issue I am thinking of is that, heuristically, the Schwarzschild simultaneity surfaces outside the shell and the Minkowski simultaneity surfaces inside the shell are not parallel--except possibly at one particular instant of time. Heuristically, you can see this by imagining trying to match Rindler simultaneity surfaces with Minkowski (inertial frame) simultaneity surfaces, across a boundary at some finite value of Minkowski $x$ to the right (positive $x$ direction) of the origin. Except for the surface at Minkowski $t = 0$, the surfaces are not parallel. Schwarszschild simultaneity surfaces, heuristically, behave like Rindler simultaneity surfaces, so the same issue will arise with them.

You could argue that the "kink" in each simultaneity surface is due to the idealization of a shell of zero thickness, and that for a shell of finite thickness the kink would be smoothed out into a curve. But then there is ambiguity about how to do the smoothing.

 

[HansH]

if you pick a large enough region of the universe it's wholly contained within its Schwarzschild radius, but that doesn't automatically make it a black hole.

Here I cannot follow you anymore. As far as I understand is that for any mass a Schwarzschild radius can be calculated and when alle this mass is compressed withing this Schwarzschild radius then the Schwarzschild radius becomes an event horizon. For the sun this is of course not the case because within this 3km radius only a small fraction of the solar mass is in. sou you first must compress the sun to 3km radius to get that done.
For me however it is not clear if it is possible to have a region of normal space (which has on average a density of almost nothing) with all its mass within its own Schwarzschild radius. I understand you indicate that this is not possible as I also would expect.
But why does someone then tell such thing in a video Is there some equation that shows that the larger the volume, the lower the average mass density required to form a black hole that could trigger such person to tell something like that?

 

[Ibix]

As far as I understand is that for any mass a Schwarzschild radius can be calculated and when alle this mass is compressed withing this Schwarzschild radius then the Schwarzschild radius becomes an event horizon.

If it's in an otherwise empty spacetime, sure. Cosmological spacetime is filled with matter everywhere, though. So by your argument you can pick any sufficiently large spherical volume and argue that it "ought to be a black hole". Even overlapping volumes "ought to be black holes", and you are simultaneously in infinitely many such volumes. Which direction do you imagine you will fall if you are simultaneously trapped inside infinitely many black holes?

For me however it is not clear if it is possible to have a region of normal space (which has on average a density of almost nothing) with all its mass within its own Schwarzschild radius.

The Schwarzschild radius is $R_S=2GM/c^2$. The mass contained in a spherical volume is $\frac 43\rho r^3$. You want to know if it's possible for the radius containing a mass to be less than the Schwarzschild radius associated with that mass:$$\begin{eqnarray*}
r&<&R_S\\
&<&\frac{2G}{c^2}\frac 43\rho r^3\\
\frac{3c^2}{8G\rho}&<&r^2
\end{eqnarray*}$$so, as I said, any sufficiently large radius is larger than the Schwarzschild radius of the mass contained within it if $\rho$ is constant throughout space, no matter how small that value is. But the region still won't turn into a black hole because "all the mass inside the Schwarzschild radius" may be a necessary condition for a black hole to form, but it is not a sufficient condition on its own.

 

[PeroK]

Here I cannot follow you anymore. As far as I understand is that for any mass a Schwarzschild radius can be calculated and when alle this mass is compressed withing this Schwarzschild radius then the Schwarzschild radius becomes an event horizon. For the sun this is of course not the case because within this 3km radius only a small fraction of the solar mass is in. sou you first must compress the sun to 3km radius to get that done.
For me however it is not clear if it is possible to have a region of normal space (which has on average a density of almost nothing) with all its mass within its own Schwarzschild radius. I understand you indicate that this is not possible as I also would expect.
But why does someone then tell such thing in a video Is there some equation that shows that the larger the volume, the lower the average mass density required to form a black hole that could trigger such person to tell something like that?

Understanding the nature of the Schwarzschild spacetime is not trivial. It helps to understand the mathematics to some extent, at least. Understanding the formation of a black hole from a collapsing star is yet more complex. And it's almost impossible to convey this adequately using videos or non-mathematically in general.

From your posts generally, I suspect you haven't fully digested the nature of spacetime as a four-dimensional manifold; but, rather, are thinking of space and then a separate, dissociated time dimension. For example, the singularity is not a point in space where you can sit and wait while universal time ticks by until that point in space becomes a singularity. This is not the right way to think about things at all! In fact, singularity really means a failure of the mathematical model. One could argue there is no such thing as a singularity, other than a deficiency in the mathematical model (i.e. GR as we know it today).

We all hope that eventually a theory of quantum gravity will resolve what really happens in the case of a collapsing star.

 

[Nugatory]

But why does someone then tell such thing in a video Is there some equation that shows that the larger the volume, the lower the average mass density required to form a black hole that could trigger such person to tell something like that?

That part is indeed right - as the mass increases the "density" required to fit all that mass inside the event horizon decreases. However, there are a bunch of complexities here, starting with the scare-quotes around the word "density" - that quantity does not correspond to the classical notion of density as mass per unit volume. However, this is somewhat of a distraction when considering the question about whether the universe could be the interior of a black hole.

More important is that everything you see about black hole even horizons is based on the the Schwarzschild and other solutions of the Einstein field equations for a mass sitting in an otherwise empty and asymptotically flat universe. This is not an exactly accurate description of anything, but it turns out to be a really good approximation for stellar black holes - there's nothing else nearby and the gravitational effects of more distant objects can be ignored (this is somewhat analogous to the way that we don't worry about distant starts when we're calculating planetary orbits inside the solar system).

However, it is a completely inaccurate description of an entire universe filled with a large-scale uniform, small scale lumpy distribution of mass. Thus none of the Schwarzschild spacetime stuff applies to the universe as a whole. It's a different solution of the field equations; we can plug the mass into the formula for the Schwarzschild radius but that calculation isn't telling us anything useful.

 

[HansH]

This is not an exactly accurate description of anything, but it turns out to be a really good approximation for stellar black holes - there's nothing else nearby and the gravitational effects of more distant objects can be ignored

Do you mean that the formation of a black hole in this way is applicable to the situation where the mass density in the region where the black hole forms is much larger than the mass density outside that region? such as valid for a collapsing star, but not the case in a more or less equal distribution of mass in the universe? That sounds logical to me because then you could choose any arbitrary sphere to define your event horizon but that is of course not valid.
so just for my understanding: if this is true and I would remove all the mass outside a sphere in the universe where the mass inside causes the Schwarzschild radius to be that sphere, do I then get an event horizon at that radius and a black hole?

If this is true and I then think one step further then the whole universe must be more or less having an equal density of matter because otherwise we would get spontanuous black holes in certain regions with a large mass bubble and sufficient empty space around it, because the equetion you refer makes it more likely to form black holes in a very large structure of space.

 

[PeroK]

If this is true and I then think one step further then the whole universe must be more or less having an equal density of matter because otherwise we would get spontanuous black holes in certain regions with a large mass bubble and sufficient empty space around it, because the equetion you refer makes it more likely to form black holes in a very large structure of space.

The density of matter in any structure other than a star or planet is extremely low. The only object that could collapse to a black hole is a large star.

 

[Ibix]

Do you mean that the formation of a black hole in this way is applicable to the situation where the mass density in the region where the black hole forms is much larger than the mass density outside that region?

Not quite. The Schwarzschild solution is an eternal black hole - it just exists forever, it does not form from anything. It's not a realistic solution, but it's probably the simplest known solution to the Einstein field equations. And once a black hole has formed it won't be very different from a Schwarzschild black hole for most purposes.

if this is true and I would remove all the mass outside a sphere in the universe where the mass inside causes the Schwarzschild radius to be that sphere, do I then get an event horizon at that radius and a black hole?

What you are describing is exactly the time-reversal of the Oppenheimer-Snyder black hole model. Oppenheimer-Snyder is a simple model of a stellar collapse into a black hole that is literally a spherical region of collapsing FLRW spacetime with vacuum outside. You've just replaced the collapsing region with an expanding region, so you are describing a white hole, which has an anti-horizon instead of an event horizon.

If this is true and I then think one step further then the whole universe must be more or less having an equal density of matter because otherwise we would get spontanuous black holes in certain regions with a large mass bubble and sufficient empty space around it, because the equetion you refer makes it more likely to form black holes in a very large structure of space.

No, we model the universe as having more or less uniform mass density everywhere because that's what we see. You can add perturbations that tend to evolve into areas of high density that could form structures like galaxies and stars and eventually black holes. Sufficiently large perturbations might form "primordial" black holes directly, and people have looked for signatures of such things. No success, as far as I am aware.

 

[PeterDonis]

As far as I understand is that for any mass a Schwarzschild radius can be calculated and when alle this mass is compressed withing this Schwarzschild radius then the Schwarzschild radius becomes an event horizon.

That is not correct. What is correct is that if the matter is collapsing, and it collapses to within the Schwarzschild radius for its mass, it becomes a black hole and an event horizon forms around it. But our universe is not collapsing; it is expanding.

Also, as @Ibix has already pointed out, the "collapse" process described above assumes an isolated system of finite extent surrounded by empty space. Our universe is no such thing.

So there are two reasons why the heuristic rule of thumb you describe does not apply to our universe as a whole.

 

[PeterDonis]

For me however it is not clear if it is possible to have a region of normal space (which has on average a density of almost nothing) with all its mass within its own Schwarzschild radius.

Yes, it is possible. A black hole has vacuum inside; all of the matter that originally collapsed to form the hole disappears into the singularity. (At least, that's what classical GR says; we don't have a theory of quantum gravity so we don't know how quantum effects might change this.)

One common misconception is that a black hole has to be "made of" matter in order to have mass. It doesn't. The "mass" of a black hole is a property of its spacetime curvature; spacetime curvature is what the hole is "made" of. As noted above, the hole is vacuum inside, i.e., "empty space".

 

[HansH]

What you are describing is exactly the time-reversal of the Oppenheimer-Snyder black hole model. Oppenheimer-Snyder is a simple model of a stellar collapse into a black hole that is literally a spherical region of collapsing FLRW spacetime with vacuum outside. You've just replaced the collapsing region with an expanding region, so you are describing a white hole, which has an anti-horizon instead of an event horizon.

Not sure If I made myself sufficiently clear. I mean all mass is still withing the sphere and no mass outside the sphere. So then I do not understand why you talk about an expanding region instead of a collapring region as there are only attracting forces between mass particles making the particles move towards each other.

I am basically looking for a proper criterion to be able to determine when in such theoretical situation a black hole forms.(for me the goal is not to understand all the details, but to get a general idea of how black holes could form)
In the first post I used a spherical shell of iron particles with 'sufficient' mass to form a black hole. Now we are talking about what is 'sufficient' and is there the relation between radius of the contracting shell, the moment that an event horizon forms and the minimum amount of mass of the shell to be able to form a black hole at all. and does it make a difference if this mass is all concentrated in a thin shell or evenly dense within the whole sphere. And then some posts back I was wandering if we could extend this situation to very large structures because of the equations posted in #37. but the answers on that still confuse me.

 

[PeroK]

I am basically looking for a proper criterion to be able to determine when in such theoretical situation a black hole forms.

See here for details of the minimum mass of star for a stellar black hole to form:

https://en.wikipedia.org/wiki/Stellar_black_hole

Note that if we take the Newtonian gravity for a star of radius $R$ and fixed density $\rho$, then the gravitational acceleration on the surface is:
$$g = \frac{GM}{R^2} = \frac{4\pi G\rho R^3}{3R^2} = \frac{4\pi G\rho R}{3}$$Where I've used the formula for the volume of a sphere to get the mass in terms of the radius and density. From this you can see that the larger a sphere becomes, the greater its surface gravity becomes (approximately a linear function of $R$). Eventually, if it is large enough, the surface gravity will cause the sphere to start to collapse. The final state for such a sphere depends on how big it is: e.g. it could collapse to a neutron star:

https://en.wikipedia.org/wiki/Neutron_star

Or, if it is even larger and the collapse can overcome neutron pressure, then it may collapse to a black hole.

 

[Ibix]

Not sure If I made myself sufficiently clear. I mean all mass is still withing the sphere and no mass outside the sphere.

It depends on your initial conditions, then. If the matter is already expanding rapidly enough (which, on the large scale, matter in the universe is doing) then it will not collapse into a black hole - instead, that's the dying moments of a white hole. If the matter is static then yes, it'll collapse into a black hole.

I am basically looking for a proper criterion to be able to determine when in such theoretical situation a black hole forms.

Gravitational collapse happens as long as the matter isn't moving outwards fast enough. The collapse can be halted by various processes (e.g. forming a planet or a star) but may start up again if that process is time limited (a planet can support itself against its own gravity and does not collapse further, but a star is supported by emissions from its fusion reactions, which eventually burn out). The last process we're aware of that can stop a star collapsing is neutron degeneracy pressure. If the mass is sufficiently great to overcome this, there's nothing we know of that can stop the star collapsing into a black hole.

So generally, you need about 1.4 solar masses of matter and you need it not to be expanding initially. Or you can use less matter and provide some mechanism to slam it together really hard (which is why people got worried about the LHC possibly producing a black hole, although in reality it's far too low energy to do so).

 

[Hornbein]

The last process we're aware of that can stop a star collapsing is neutron degeneracy pressure. If the mass is sufficiently great to overcome this, there's nothing we know of that can stop the star collapsing into a black hole.

So generally, you need about 1.4 solar masses of matter

Actually about 2 solar masses to overcome neutron degeneracy pressure. The minimal mass of a neutron star is 1.4.

 

[PeterDonis]

If the matter is already expanding rapidly enough (which, on the large scale, matter in the universe is doing) then it will not collapse into a black hole - instead, that's the dying moments of a white hole.

This is not quite true. Our universe did not start out as a white hole. A white hole is an isolated region surrounded by empty space, just like a black hole. Our universe is not like that.

 

[PeterDonis]

I mean all mass is still withing the sphere and no mass outside the sphere.

This is not true of our universe; there is no way to draw a sphere around any region occupied by mass in our universe and have all the mass inside and no mass outside. So the reasoning you are trying to use simply cannot be applied to our universe.

 

[PeterDonis]

I am basically looking for a proper criterion to be able to determine when in such theoretical situation a black hole forms.

A black hole forms from the gravitational collapse of an isolated object like a star. Strictly speaking, stars in our universe are not completely isolated, since there is still other matter in the rest of the universe. But if you think of what a star is like, it's a finite region of continuous matter surrounded by empty space, not extending out forever but at least extending out to a distance many times the star's own size (and many more times the Schwarzschild radius corresponding to the star's mass, which will be the size of a black hole formed from the star).

That is the key property that our universe does not have. If you talk about some large region of our universe, hundreds of millions of light-years across, it is not a single finite region of continuous matter surrounded by empty space. It is lots of matter scattered all through the region, with an average density that is roughly constant throughout the region, and with the same kind of distribution matter outside as inside. So it is not correct to think of any region of our universe as a whole as an isolated finite region of matter surrounded by empty space.

 

[PeterDonis]

The minimal mass of a neutron star is 1.4.

This is not correct. It's correct that the maximal mass of a white dwarf is 1.4 solar masses, so anything larger than that must be either a neutron star or a black hole. But that does not mean neutron stars smaller than that mass cannot exist. They can. They're just much less likely to form because the collapse process of a star that small can stop at the white dwarf stage.

 

[Hornbein]

This is not correct. It's correct that the maximal mass of a white dwarf is 1.4 solar masses, so anything larger than that must be either a neutron star or a black hole. But that does not mean neutron stars smaller than that mass cannot exist. They can. They're just much less likely to form because the collapse process of a star that small can stop at the white dwarf stage.

Right you are. The lightest neutron star so far seems to be 0.77. https://www.researchgate.net/publication/359752835_A_strangely_light_neutron_star

 

[HansH]

This is not true of our universe; there is no way to draw a sphere around any region occupied by mass in our universe and have all the mass inside and no mass outside. So the reasoning you are trying to use simply cannot be applied to our universe.

I know. This is pure a theoretical situation I created in order to better understand if a black hole could form based on a large cloud of (on average) low density material if all mass is within its own Schwarzschild radius.
In theory you could move all mass to another place and create this situation I assume?

 

[HansH]

This is not correct. It's correct that the maximal mass of a white dwarf is 1.4 solar masses, so anything larger than that must be either a neutron star or a black hole. But that does not mean neutron stars smaller than that mass cannot exist. They can. They're just much less likely to form because the collapse process of a star that small can stop at the white dwarf stage.

This situation triggered me to think of another theoretical situation. Suppose there would be another mechanism in nature that generates a very large repelling force at a much closer distance than occurs in a neutron star such that for larger mass than the maximum mass of a neutron star the mass further collapses and an event horizon forms, but within this event horizon while furter collapsing at a certain point this repelling force prevent further collapse. Then I assume you still see a black hole from outside but the formation of a singularity is then prevented. Would this be possble? or does the formation of an event horizon per definition mean that we get a singularity even if such a repelling force would be there? or is this the situation we think could occur due to quantum effects but we do not have a theory for yet and cannot define an experiment to test?

 

[PeterDonis]

This is pure a theoretical situation I created in order to better understand if a black hole could form based on a large cloud of (on average) low density material if all mass is within its own Schwarzschild radius.

Yes, that was the original theoretical situation you posed in the OP of this thread. But you have also asked about our universe as a whole, and our universe as a whole is a different situation, to which the theoretical model you posed in the OP of this thread is not applicable. So if you want to limit discussion in this thread to the theoretical situation you posed in the OP of this thread, which is a very good idea, you need to understand that discussion of our universe as a whole is not covered by that and would need to be done in a separate thread.

In theory you could move all mass to another place and create this situation I assume?

You can't just magically "move all mass to another place". You simply have two different theoretical situations: the one you posed in the OP of this thread, an isolated collapse, and the one that applies to the universe as a whole. They're different. That's all there is to it.

Suppose there would be another mechanism in nature that generates a very large repelling force at a much closer distance than occurs in a neutron star such that for larger mass than the maximum mass of a neutron star the mass further collapses and an event horizon forms, but within this event horizon while furter collapsing at a certain point this repelling force prevent further collapse.

This is not possible. Inside the event horizon collapse cannot be prevented by any means whatever; preventing the collapse would mean matter moving faster than light.

There are speculations connected with quantum effects that might generate the kind of "very large repelling force" you describe, but all of these effects would not just prevent a singularity from forming, they would prevent an event horizon from forming.

 

[Ibix]

This is not quite true. Our universe did not start out as a white hole. A white hole is an isolated region surrounded by empty space, just like a black hole. Our universe is not like that.

Yes. The thread of conversation to which I was responding seemed to me to be considering transplanting a large but finite region of our universe into an otherwise empty spacetime, then asking if it would be a black hole. That's a time reverse of an Oppenheimer-Snyder black hole, so a white hole.

I agree that a plain old FLRW universe did not start as a white hole.

 

[HansH]

transplanting a large but finite region of our universe into an otherwise empty spacetime, then asking if it would be a black hole. That's a time reverse of an Oppenheimer-Snyder black hole, so a white hole.

This I cannot follow. I would assume matter particles in this large region would attract each other, while in a white hole things repell each other. so I am lost here.

 

[PeroK]

I would assume matter particles in this large region would attract each other, while in a white hole things repell each other.

That makes no sense.

so I am lost here.

https://en.wikipedia.org/wiki/White_hole

 

[Ibix]

This I cannot follow. I would assume matter particles in this large region would attract each other, while in a white hole things repell each other. so I am lost here.

No, in a white hole gravity is still attractive. Matter slows down as it comes out instead of speeding up as it falls in.

If your next question is "then why can't you enter a white hole" the answer is because of the way spacetime is curved. This is far past the point that you can try to understand general relativistic gravity as if it were just a few tweaks to Newtonian gravity. There are rough analogues to black holes in Newtonian gravity (although a close inspection shows that Newtonian "black stars" are very different from black holes and not entirely consistent), but there's nothing like a white hole.