Does a Black Hole with Lower Density Than Water Float or Sink?

[fluidistic]

Would a black hole whose density is lower than that of water, sink?
The answer is likely over my head. So far, I've found 5 answers by "experts", with quite different opinions.

For instance on this website, one reads

Well, it can't (float), since a Black Hole is not a solid object that has any kind of surface.

. On IRC I was told by another expert that it would sink.

Here are the other opinions:

The black hole would float in water, if you could make a large enough pool to submerge it, and with enough replenishment to replace the water that the black hole will sucks up. The black hole will remove water from its surroundings, but the water below will come into the horizon at higher pressure than the water above, so the velocity inward will not be uniform.

If the black hole is denser than water, it will sink for a while, because the pressure difference is not enough to compensate for the pull of gravity. If the black hole has less density than water, it will float. It's like a balloon that sucks in water and expands, always maintaining a volume which is big enough to keep itself lighter than water.

Often people are hesitant to answer any physical questions about black holes. They're like, well, this is the domain of GR, so the only "allowed" objects of discussion are black holes, point-like planets, and ideal point-like spaceships.

But if you were to actually mix other physics in, which you can, I think what Ron is saying is totally reasonable.

Even if Ron's answer is not ideal, it seems to me to be much better than the existing accepted answer, which basically just says "I refuse to answer because black holes are too weird to think about".

Finally, expert 4 claims:

A black hole so massive that it has such a low density is so large that there is no continuous body of water in existence on which it could float.
Even if you have enough water, I question the idea that, with the immense gravitation of such a hole, you can still meaningfully treat the water being sucked in as a continuous fluid to which ordinary hydrodynamics apply.

In the link I give above, one can read the comments below the answers given. Essentially expert 2 and expert 1 are arguing about their point of view, leaving me (and probably a few billion people) clueless on the final answer.

 

[Dale]

Would a black hole whose density is lower than that of water, sink?

The question doesn’t make sense. A black hole less dense than water would be at least a few billion times as massive as the sun. Any body of water large enough for this to float in would itself be a black hole.

The final answer is that the question itself is incorrect: mu

 

[jbriggs444]

In the link I give above, one can read the comments below the answers given. Essentially expert 2 and expert 1 are arguing about their point of view, leaving me (and probably a few billion people) clueless on the final answer.

It seems to me that the experts are all in agreement that the question carries too many implausible assumptions with it to permit a meaningful answer.

 

[PeterDonis]

a black hole whose density is lower than that of water

In addition to the other issues already raised, the concept of "density" does not apply to a black hole, at least not as it is being used here. A black hole is vacuum, not a substance, and it does not have a well-defined interior volume. The "density" that is often quoted (in pop science sources) for a black hole is obtained by dividing its mass by the volume of a sphere in Euclidean space with the same surface area as the hole's horizon. But that number is meaningless physically, because a black hole is not a Euclidean sphere with "stuff" inside it.

 

[russ_watters]

Would a black hole whose density is lower than that of water, sink?
The answer is likely over my head. So far, I've found 5 answers by "experts", with quite different opinions.

Often when faced with impossible or incomplete scenarios, people will make assumptions that plug holes or overlook the impossible in order to answer it. Different assumptions result in different answers.

 

[fresh_42]

But that number is meaningless physically, because a black hole is not a Euclidean sphere with "stuff" inside it.

I understand this from a geometrical point of view, but given that a black hole resulted from a collapse of a massive star, what does this mean for the star's matter? Shouldn't there still be "stuff"?

 

[jbriggs444]

I understand this from a geometrical point of view, but given that a black hole resulted from a collapse of a massive star, what does this mean for the star's matter? Shouldn't there still be "stuff"?

Any stuff which finds itself within the boundary of the hole will quickly fall to the singularity where it becomes irrelevant. The interior of a black hole is a vacuum. Nothing remains inside.

The Schwarzschild solution to the Einstein field equations is a "vacuum solution"

 

[Ibix]

I understand this from a geometrical point of view, but given that a black hole resulted from a collapse of a massive star, what does this mean for the star's matter? Shouldn't there still be "stuff"?

GR says the stuff falls quickly into the singularity, which is not part of the manifold and concepts like volume can't be defined. You can still measure its mass, angular momentum, and electric charge, but that's it (until the hole evaporates again).

 

[PAllen]

The question doesn’t make sense. A black hole less dense than water would be at least a few billion times as massive as the sun. Any body of water large enough for this to float in would itself be a black hole.

The final answer is that the question itself is incorrect: mu

I think this key point was missed by at least one of the experts.

Now I’ll risk playing the game of making the question more precise I order to answer, in a way that dodges the issue Peter raised.

A BH forming from collapse has a history where there is a horizon but no singularity. Generally, there exists a (possibly nonunique) largest volume hyper slice of its history such that all the initially collapsing matter is inside the horizon and there is no singularity. Somewhat arbitrarily define the initial density of a BH as its externally measured mass via test orbits divided this volume. What happens to its density after later infall, I choose not to try to define as it isn’t needed for the argument.

Ok, if you have a BH whose initial density is less than that of water, then anybody of water with volume larger than the initial BH volume defined above will already be inside its own BH horizon and will be extremely rapidly undergoing catastrophic collapse. Thus, there cannot exist a body of water such that the question can be meaningfully posed.

 

[PeterDonis]

given that a black hole resulted from a collapse of a massive star, what does this mean for the star's matter? Shouldn't there still be "stuff"?

Yes, but the "stuff" rapidly collapses to $r = 0$ and forms the singularity, so for any hole that did not form extremely recently, the "stuff" has already collapsed inside and the interior can be treated as vacuum.

A BH forming from collapse has a history where there is a horizon but no singularity.

This is not correct; there is still a singularity in this spacetime.

Generally, there exists a (possibly nonunique) largest volume hyper slice of its history such that all the initially collapsing matter is inside the horizon and there is no singularity.

By "largest volume" do you mean the volume of the collapsing matter? (The 3-volume of an entire spacelike slice, including the vacuum region, is infinite.)

It is true that one can find such slices that do not include the singularity, yes.

 

[fresh_42]

GR says the stuff falls quickly into the singularity, which is not part of the manifold and concepts like volume can't be defined. You can still measure its mass, angular momentum, and electric charge, but that's it (until the hole evaporates again).

I understand. But it reads more like a lack of mathematics than a lack of physics. Whichever process a single atom or whatever will undergo, at least there has to be a process. I don't want to define volume or any other quantity like density for a point that isn't part of the manifold, and I see it isn't, but do we have an idea about the destiny of a single particle? Does it simply collapse to zero? Probably not, as it still contributes momentum and charge.

 

[Ibix]

Ok, if you have a BH whose initial density is less than that of water, then anybody of water with volume larger than the initial BH volume defined above will already be inside its own BH horizon and will be extremely rapidly undergoing catastrophic collapse. Thus, there cannot exist a body of water such that the question can be meaningfully posed.

That's what I was thinking myself - but is it correct? I agree it is for an isolated mass, but what about as part of a larger mass? FLRW spacetime has an arbitrarily high density in the past and no black hole. Obviously there's no sense of "down" either, so a hole added manually would go nowhere.

I believe Oppenheimer-Snyder collapse is something like an FLRW region surrounded by vacuum. Could that have a black hole added to it that might sink or not? Or would a horizon have to form per your argument before the density worked out?

 

[Ibix]

I understand. But it reads more like a lack of mathematics than a lack of physics. Whichever process a single atom or whatever will undergo, at least there has to be a process. I don't want to define volume or any other quantity like density for a point that isn't part of the manifold, and I see it isn't, but do we have an idea about the destiny of a single particle? Does it simply collapse to zero? Probably not, as it still contributes momentum and charge.

As far as I understand it, we'll get back to you when we have a theory of quantum gravity. GR just says you drop through the horizon (before or after being torn apart by tidal forces - depends on the hole size) and then you run into geodesic incompleteness at the singularity and your particles' worldlines end. This is most likely a failure of our model to adequately describe the reality, but that's what it says and it's the best we can do.

 

[PeterDonis]

I believe Oppenheimer-Snyder collapse is something like an FLRW region surrounded by vacuum

A collapsing FRW region surrounded by Schwarzschild vacuum, yes.

Could that have a black hole added to it

The O-S collapse already has a black hole in it--the model describes the collapse of an ordinary object to form a black hole.

 

[PAllen]

Yes, but the "stuff" rapidly collapses to $r = 0$ and forms the singularity, so for any hole that did not form extremely recently, the "stuff" has already collapsed inside and the interior can be treated as vacuum.
This is not correct; there is still a singularity in this spacetime.

Yes, but not on the slice I was about to define.

By "largest volume" do you mean the volume of the collapsing matter? (The 3-volume of an entire spacelike slice, including the vacuum region, is infinite.)

It is true that one can find such slices that do not include the singularity, yes.

3-volume inside the horizon, irrespective of what part is matter or not. In effect, you find the largest 3-volume possible among all spacelike slices bounded by the horizon that do not include the singularity. If there is no upper bound on this, my construction fails. I would like to add some notion of maximal proper time for free fall geodesics to the singularity, but this would only work for ideally symmetric collapse.

 

[Ibix]

The O-S collapse already has a black hole in it--the model describes the collapse of an ordinary object to form a black hole.

Yes - but can we add a small hole by hand before the O-S horizon forms and see if it sinks? In the sense of in-falling faster than co-moving matter?

I was trying to get around @PAllen's argument that a chunk of matter would be inside its own horizon if it were more dense than a black hole (accepting that such a thing is definable in some sense). You can have arbitrarily high densities in early FLRW spacetime with no horizons, but there's no "down" for a manually-added hole to sink towards. There is a "down" in pre-horizon O-S but I don't know how high the density gets.

 

[PeterDonis]

In effect, you find the largest 3-volume possible among all spacelike slices bounded by the horizon that do not include the singularity. If there is no upper bound on this, my construction fails.

Ah, ok. I think there is an upper bound (because of the "bounded by the horizon" requirement--there are spacelike slices not including the singularity that are infinite in 3-volume but they are entirely inside the horizon, not bounded by it). But I don't have a simple proof.

 

[PeterDonis]

can we add a small hole by hand before the O-S horizon forms

Not without completely changing the model and thereby removing any advantage given by the known properties of the O-S model.

You can have arbitrarily high densities in early FLRW spacetime with no horizons

"Early" here means early in an expanding FRW spacetime. The O-S model includes a contracting FRW region. Not the same thing.

Also, the O-S model only includes a finite portion of a contracting FRW spacetime, matched at its boundary to a Schwarzschild vacuum that extends out to infinity. This model is very different from the FRW spacetime used in cosmology (the one with no horizons), because in the latter model, the FRW region is the entire spacetime and it has no boundary. Also, the FRW spacetime used in cosmology is not asymptotically flat, so you can't even define the concept of "event horizon" in it, since that concept requires the concept of "escape to infinity", and in the FRW spacetime used in cosmology there is no "infinity" to escape to. The O-S model is asymptotically flat and does have an infinity.

 

[PAllen]

Yes - but can we add a small hole by hand before the O-S horizon forms and see if it sinks? In the sense of in-falling faster than co-moving matter?

I was trying to get around @PAllen's argument that a chunk of matter would be inside its own horizon if it were more dense than a black hole (accepting that such a thing is definable in some sense). You can have arbitrarily high densities in early FLRW spacetime with no horizons, but there's no "down" for a manually-added hole to sink towards. There is a "down" in pre-horizon O-S but I don't know how high the density gets.

If the OS collapse (embedded in an exterior SC geometry) reaches the density of water, it will necessarily either be already inside its horizon or it will be smaller than the BH whose density is less than water.

 

[PeterDonis]

I was trying to get around @PAllen's argument that a chunk of matter would be inside its own horizon if it were more dense than a black hole (accepting that such a thing is definable in some sense).

More precisely, more dense than (meaning, with a surface area less than or equal to that of) a black hole of the same mass. You can't get around that argument; it's valid.

 

[Dale]

I’ll risk playing the game of making the question more precise I order to answer, in a way that dodges the issue Peter raised.

A BH forming from collapse has a history where there is a horizon ... Thus, there cannot exist a body of water such that the question can be meaningfully posed.

Yes, that is almost exactly my thought process that led to my response.