# B What if the universe reaches a high enough density to become a black hole?

#### black hole 123

suppose the universe started shrinking. because the density vs mass is a 1/M^2 factor, density can be made arbitrarily small given high enough mass. so it's not hard for all the mass in the universe to quickly reach black hole density. when critical density of the universal is reached, what would it look like? i just still don't understand what the flipped space and time signs in the schwarzchild metric mean, what would it look like to us inside the universe-black hole? does everything suddenly change?

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#### Ibix

There seem to be a lot of questions here. If you don't understand the significance of the negative sign in the metric, you really need a textbook on special relativity and then general relativity (I'd recommend Spacetime Physics by Taylor and Wheeler). All relativistic metrics have one opposite sign - that's what makes it spacetime, not space.

Why would the universe start shrinking? This does happen in some variants of FLRW spacetime, but they don't end in a black hole. One way to see this is that the density of the universe is everywhere the same, so there cannot be any special points. A black hole is very different from the rest of the universe. But maybe you had some other reason in mind for why the universe should suddenly start shrinking.

#### PeterDonis

Mentor
i just still don't understand what the flipped space and time signs in the schwarzchild metric mean
The Schwarzschild metric does not describe a universe that is either expanding or contracting. So it is not relevant for the case you are describing.

The relevant spacetime for a contracting universe is FRW spacetime; in FRW spacetime, there is no black hole as the universe shrinks, no matter how high the density gets, because a black hole is a region of spacetime that cannot send light signals to infinity, and in FRW spacetime there is no infinity.

#### pervect

Staff Emeritus
The current indications are that the expansion of the universe is accelerating, which means that it won't reach the critical density to become a black hole.

It's a bit surprising that the universal expansion is accelerating, but that's what the data is currently showing. See for instance <<https://en.wikipedia.org/w/index.php?title=Accelerating_expansion_of_the_universe&oldid=906250109>>. This is explained as being due to the cosmological constant - sometimes it's called other names.

If we imagine a universe obeying the laws of general relativity without a cosmological constant, the expansion would slow down rather than accelerate, and it's possible the expansion would stop and reverse, at which point such a universe would eventually re-collapse to form a black hole.

We'd expect that such a universe would become a black hole in that case. The details are unclear, however. Our cosmological models make assumptions to simply the problem, one key assumption which is called homogoneity, and it's likely that this assumption would break down sometime during the collapse process.

This difficulties also exists in our attempts to understand the realistic collapse on a lesser scale.

The evidence that there is a black hole at the center of our galaxy is pretty convicing, so we are pretty sure collapse is possible. Many of the details of a realistic collapse process are unclear.

#### PeterDonis

Mentor
The current indications are that the expansion of the universe is accelerating, which means that it won't reach the critical density to become a black hole.
The universe as a whole can't become a black hole. It has the wrong spacetime geometry. See my post #3.

Also, there is no critical density to become a black hole. (I'll expand on this in a response to the OP shortly.)

If we imagine a universe obeying the laws of general relativity without a cosmological constant, the expansion would slow down rather than accelerate, and it's possible the expansion would stop and reverse, at which point such a universe would eventually re-collapse to form a black hole.
The universe would recollapse in this case, but not to a black hole.

#### PeterDonis

Mentor
it's not hard for all the mass in the universe to quickly reach black hole density
There is no critical density to become a black hole. The criterion for becoming a black hole is that an event horizon forms, which means a given amount of mass $M$ collapses in such a way that a 2-sphere with area $16 \pi G^2 M^2 / c^4$ can enclose it. The larger the mass $M$ is, the smaller the density of the collapsing matter needs to be when this criterion is met.

In fact, a black hole doesn't even have a well-defined density since it doesn't have a well-defined interior volume. Some pop science sources will say it has the volume of a sphere with radius equal to the Schwarzschild radius, but that's not correct.

#### DaveC426913

Gold Member
In fact, a black hole doesn't even have a well-defined density since it doesn't have a well-defined interior volume.
That's a fascinating thought.
Is this because of extreme space-time curvature?
Is there an theoretical upper limit on what the volume inside a black hole could be?

#### PeterDonis

Mentor
Is this because of extreme space-time curvature?
No, it's because the hole doesn't have a well-defined volume.

Is there an theoretical upper limit on what the volume inside a black hole could be?
The volume inside isn't even well-defined. There are spacelike slices inside the hole that are infinite in volume, and other ones that are finite, with varying values.

#### black hole 123

The Schwarzschild metric does not describe a universe that is either expanding or contracting. So it is not relevant for the case you are describing.

The relevant spacetime for a contracting universe is FRW spacetime; in FRW spacetime, there is no black hole as the universe shrinks, no matter how high the density gets, because a black hole is a region of spacetime that cannot send light signals to infinity, and in FRW spacetime there is no infinity.
this kind of answers my question, what i was thinking is if all the matter in the universe suddenly started gravitating towards some point (i know this won't happen in our universe but lets assume magic happens), and a person outside this sphere of shrinking matter would see a black hole forming, and my question was what it would look like to people inside the shrinking sphere. but this shouldn't be possible? because the distribution of matter determines spacetime so there can't be large regions of empty flat space outside the shrinking sphere?

i should rephrase my question. suppose a very large nebular started shrinking (so it has a huge schwarzschild radius, many light weeks in radius), and there's an unlucky astronaut in it. when the nebular reaches the 2 sphere with critical area, what would the astronaut see? he will have plenty of time to observe the universe around him before all the nebular reach singularity due to extremely large schwarzschild radius. someone outside would see a black hole, but what would he see? would he suddenly see the universe change once the 2 sphere is reached?

i guess my question can be stated as what an interior observer would see once event horizon is formed, will he see a sudden change in everything around him etc. but for normal small black holes this is too far quick for him to have time to observe and think about everything, plus an astronaut cant see past the inside of a shrinking star (assuming he can somehow survive the temperature). that's why i thought about the universe in my OP question, but a very large nebular that's somewhat transparent would do.

#### PeterDonis

Mentor
if all the matter in the universe suddenly started gravitating towards some point (i know this won't happen in our universe but lets assume magic happens), and a person outside this sphere of shrinking matter
There can't be anyone outside the shrinking matter. It fills the entire universe.

#### PeterDonis

Mentor
suppose a very large nebular started shrinking (so it has a huge schwarzschild radius, many light weeks in radius), and there's an unlucky astronaut in it. when the nebular reaches the 2 sphere with critical area, what would the astronaut see?
This is a different question, because the nebula is not the entire universe.

That said, the astronaut inside the nebula would not see anything unusual during the collapse except that the density of matter around him would be increasing (but smoothly, with no sudden discontinuity), and that if he were able to see anything of the universe outside the nebula (which in this case is assumed to not be collapsing), he would see it redshifted.

what an interior observer would see once event horizon is formed, will he see a sudden change in everything around him
No. See above.

#### DaveC426913

Gold Member
No, it's because the hole doesn't have a well-defined volume.
Can you elaborate?, I'm reading circular logic there.

Q: Why doesn't a BH have a well-defined volume?
A: Because the hole doesn't have a well-defined volume.

Are you simply saying 'since the radius of a black hole is ambiguous and arbitrary, so must its derived volume be.'?

If so, does that mean if we choose a radius (say, the EH, or the SR) we can calculate the volume inside that radius. (No, you already disqualified that as a pop-sci error.)

So what is coming into play here?

The volume inside isn't even well-defined. There are spacelike slices inside the hole that are infinite in volume, and other ones that are finite, with varying values.
And you're saying this is not caused by extreme curvature of space-time. So what is it caused by?
(I'm not looking for an exhaustive explanation, just a nudge in the right direction.)

#### PeterDonis

Mentor
Q: Why doesn't a BH have a well-defined volume?
A: Because the hole doesn't have a well-defined volume.
That's not what I said. What I said was:

Q: Why doesn't a BH have a well-defined density?
A: Because it doesn't have a well-defined volume.

Then I said:

Q: Why doesn't a BH have a well-defined volume?
A: Because there are spacelike slices entirely within the event horizon that have different volumes--some of which are even infinite.

And you're saying this is not caused by extreme curvature of space-time.
That's right, because the curvature of spacetime does not have to be "extreme" everywhere inside a black hole. It will increase without bound as you get close to the singularity, but not everywhere inside the hole is that close to the singularity. And there are infinite spacelike slices inside the horizon that are not close to the singularity anywhere.

So what is it caused by?
By the facts about spacelike slices that I have said.

#### DaveC426913

Gold Member
That's not what I said. What I said was:

Q: Why doesn't a BH have a well-defined density?
A: Because it doesn't have a well-defined volume.
We've gotten signals a bit crossed.: My question in post 7 'Is this because of extreme space-time curvature? ' was asking about the volume being poorly-defined. So explaining it's because the volume is poorly defined (post 8) didn't help me much. My bad for an ambiguous question.

But OK, you elaborated:
That's right, because the curvature of spacetime does not have to be "extreme" everywhere inside a black hole. It will increase without bound as you get close to the singularity, but not everywhere inside the hole is that close to the singularity. And there are infinite spacelike slices inside the horizon that are not close to the singularity anywhere.
OK, so we have slices of the interior of a BH that do not have extreme curvature (such as those not near the singularity). These slices - despite not having extreme curvature - can still have infinite volume. (Correct me if that's not properly paraphrased).

I guess I'll have to read up on this a little.

#### Ibix

OK, so we have slices of the interior of a BH that do not have extreme curvature (such as those not near the singularity). These slices - despite not having extreme curvature - can still have infinite volume. (Correct me if that's not properly paraphrased).
I think it's a consequence of the event horizon being a null surface. For things like stars, which have timelike surfaces, any spacelike surface has to cut through it somewhere and, for a static situation, there's a timelike Killing vector field which picks out a time-independent definition of space.

But for the black hole, the Killing vector field that's timelike outside is spacelike inside the hole. So there's no obvious definition of space, and the event horizon is null so you can engineer spacelike slices that are inside the horizon and infinite in extent.

So it's a result of curvature, but it doesn't have to be extreme. It's kind of like how a Euclidean plane is infinite in extent but the surface of the Earth is finite. The curvature is pretty gentle (you have to be building something really big before you need to care), but it is there.

#### DaveC426913

Gold Member
So it's a result of curvature, but it doesn't have to be extreme.
Ah. This is what I was intuiting.
It's not extreme curvature in-and-of-itself that causes infinite spacelike slices - but there's got to be a proximity to it. After all, it doesn't just happen in out here in normal flattish space-time.

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