What Happens When a Black Hole Evaporates Below its Schwarzschild Radius?

[Ascending One]

Hi,

I have a question about black holes that has to do with their evaporation. What is predicted to happen once a black hole evaporates enough matter that its Schwarzschild Radius is no longer larger than its size (i.e. the point at which it ceases to be a black hole)? Will the mass of the black hole 'snap' back into 'normal' space (from 'quantum space'), for lack of better words.

 

[cesiumfrog]

Just to be perfectly clear, how about you explain what you mean by "its size"?

 

[Ascending One]

I think what I mean is its volume, or the radius of its volume. I'm not really sure actually.

 

[mgb_phys]

As a black hole shrinks it's temperature rises, so it's final mass will be emitted as gamma rays.

 

[Ascending One]

So - are you implying that a black hole will remain a black hole until it evaporates its last bit of matter?

That really doesn't seem to make sense... wouldn't it, at some point lose enough matter that it would stop being a black before it lost all its matter?

 

[genneth]

No. Any amount of mass can be a black hole, as long as its confined to within its Schwarzschild radius.

 

[DarkMatter258]

I think the original question was what would happen once it was no longer confined to within its Schwarzschild radius... no?

 

[mgb_phys]

The Schwarzschild is proportional to mass, so as the mass decreases it's radius decreases until it 'classically' dissapears.
As it gets very small quantum effects start to matter and at some point the final mass energy escapes as a couple of gamma photons.

 

[genneth]

I've always wondered about the formation process more than the end. In particular, from the point of view of an asymptotic observer, in-falling matter will take forever before crossing the event-horizon. This includes the initial collapse. However, it will start to Hawking radiate all the time, as soon as there is curvature. In fact, it seems the black hole will evaporate before in-falling matter gets to the event horizon.

 

[mgb_phys]

The point of view of the infalling matter is irrelavant - the universe doesn't care!
Forming a black hole is easy - you just have to push matter together, closer than the Schwarzschild limit before the nuclear force can push it apart.
You only have to worry about evaporation for very small holes - this is a good thing, it stops any small holes formed by cosmic rays in the atmosphere from growing!

 

[Ascending One]

I think the original question was what would happen once it was no longer confined to within its Schwarzschild radius... no?

Yes - this was the original question.

In response to this:

The Schwarzschild is proportional to mass, so as the mass decreases it's radius decreases until it 'classically' dissapears.
As it gets very small quantum effects start to matter and at some point the final mass energy escapes as a couple of gamma photons.

Yes - I understand that the Schwarzschild radius is proportional to the mass. But I thought that the Schwarzschild radius was more of a theoretical concept - not a physical measure.

EDIT: I guess I should clarify - I know that the Schwarzschild radius can be manifested as a physical measure - namely the event horizon. But I also thought that the schwarzchild radius could be independent of the radius of the volume of the mass contained in an object.

 

[mgb_phys]

It's about the only physical measure of a black hole. It's basically the point at which sensible classical physics stops - inside this radius everything gets a bit silly.

 

[Ascending One]

But - anything has a schwarzchild radius, correct? The sun has a schwarzchild radius which is much smaller than the size of the sun. I'm saying 'size of the sun' because I'm not sure what measurement that actually is. I'm assuming that it is the 'radius of the volume of the sun.'

With a black hole, it's just that the schwarzchild radius is larger than the 'size of the black hole' before it actually turns into a black hole. At that point, what happens?

 

[genneth]

With a black hole, the mass has zero volume, since no know physical process can stop the inwards collapse. So no matter how much mass you lose to evaporation, the matter inside is still smaller than the Schwarzschild radius.

 

[mgb_phys]

Yes everything has a value for the schwarschild radius, ie the size it would have if it were to be squashed down to form a black hole.
The definition of a black hole is that it has mass 'm' inside it's schwarschild radius.
For something where this has happened the schwarschild radius = the event horizon, has a definite physical meaning.

The schwarschild radius is the boundary at which the escape volocity is greater than c, so inside this radius classical physics doesn't work, outside this radius you can still do the sums.
In practical terms not a lot actually happens as a the size goes below the radius and it becomes a black hole, the gravitiational effects of a star squeezed into a size just slightly larger than it's black hole radius are still pretty extreme.

 

[dilletante]

In practical terms not a lot actually happens as a the size goes below the radius and it becomes a black hole, the gravitiational effects of a star squeezed into a size just slightly larger than it's black hole radius are still pretty extreme.

I take this to mean then, that the radius of a star can be one foot (or one mile, doesn't matter) larger than the Schwarzschild radius. Does anyone theorize that it can be one foot less than the Schwarzschild radius, or do all theories assume zero volume of any mass that is within an event horizon? In other words, does current physics predict a sudden catastrophic collapse the moment mass is squeezed smaller than the BH radius, or is this a mathematical artifact of the "singularity" (infinity) problem of GR?

 

[genneth]

I take this to mean then, that the radius of a star can be one foot (or one mile, doesn't matter) larger than the Schwarzschild radius. Does anyone theorize that it can be one foot less than the Schwarzschild radius, or do all theories assume zero volume of any mass that is within an event horizon? In other words, does current physics predict a sudden catastrophic collapse the moment mass is squeezed smaller than the BH radius, or is this a mathematical artifact of the "singularity" (infinity) problem of GR?

All this stuff is in any textbook on the subject. You cannot expect PF members to recite what books take chapters to explain.

You seem to still be in a state confusion over the different times experienced by different points of view. For someone standing on the surface of a collapsing star, nothing extraordinary (as far as gravity goes, there's all sorts of stellar stuff flying about) happens. They can't even tell when they've passed the event horizon. However, they will measure a finite amount of time (as they would by looking at their watch), before they reach the singularity. At which point, our current physics doesn't say what happens.

From an observer far away, the surface of the star red-shifts out to black. As it happens, this is exponentially fast -- for a typical star, the time taken to go from 1.5x the event-horizon radius to the point where the last photon (taking into account the discreteness of energy) is likely to be emitted is about 10^-4 seconds. Furthermore, it becomes impossible to interact with the in-falling stellar surface. A photon launched at the stellar surface won't reach it before the stellar surface crosses the event-horizon.

Now, it's certainly possible for a star to have a size just above its Schwarzschild radius. It is even possible for it to be below -- though only for a finite amount of time as seen by the star. For everyone else, the star collapses to the size of the Schwarzschild radius, and then cannot be interacted with any more, so we call it a black hole. For external observers, there is no point in talking about stars smaller than their Schwarzschild radius, because there's no observational way to distinguish them -- they're all the same hairless hole.

 

[dilletante]

All this stuff is in any textbook on the subject. You cannot expect PF members to recite what books take chapters to explain.

Sorry for the inconvenience, I seem to have misplaced my textbooks 40 years ago.

 

[genneth]

Sorry for the inconvenience, I seem to have misplaced my textbooks 40 years ago.

I tend to get mine out from a library when I need them...

 

[dilletante]

I tend to get mine out from a library when I need them...

First, I wish to say that I appreciate your taking the time to answer my question. Neither you nor anyone else on the forum is obligated to answer anything, if you feel a question is stupid, or obvious, or too much trouble. So my appreciation for the time you took is genuine.

I find great value in being able to ask questions and have experts such as you and others share their knowledge. In this particular instance however, I felt berated for having asked a question, and am less likely to ask one in the future as a result, which saddens me.

If, as you put it, no one on this forum is going to take the time to answer questions that physics books devote chapters to, or that can be answered by researching textbooks in the library, then I suppose I have misunderstood the purpose of these forums. It was my understanding that discussions should be precisely about those things, and not about theories not found in textbooks.

You certainly have the option of ignoring a question if you find it offensive or beneath discussion, and I believe most laymen asking questions here would prefer that to condescension.

 

[SpaceTiger]

I take this to mean then, that the radius of a star can be one foot (or one mile, doesn't matter) larger than the Schwarzschild radius. Does anyone theorize that it can be one foot less than the Schwarzschild radius, or do all theories assume zero volume of any mass that is within an event horizon? In other words, does current physics predict a sudden catastrophic collapse the moment mass is squeezed smaller than the BH radius, or is this a mathematical artifact of the "singularity" (infinity) problem of GR?

The collapse of a star into a black hole is a highly non-trivial problem, particularly if the star is rotating. With a number of questionable simplifying assumptions, some textbooks will give models that are meant to simulate a collapsing star, but in reality the problem hasn't even been solved. For example, some theories suggest that a Type II supernova should accompany the collapse into a black hole, while others suggest that the collapse would occur with minimal radiation output.

Also note that the collapse is a result of the loss of a restoring force, not compression beyond the Schwarzschild limit. Extremely compact stars, like neutron stars and the cores of massive stars, are held together by degeneracy pressure, a quantum effect that results from the limited number of states available to particles within the star. This degeneracy pressure fails when the objects exceeds the Chandrasekhar mass, resulting in rapid gravitational collapse.

 

[dilletante]

The collapse of a star into a black hole is a highly non-trivial problem, particularly if the star is rotating. With a number of questionable simplifying assumptions, some textbooks will give models that are meant to simulate a collapsing star, but in reality the problem hasn't even been solved. For example, some theories suggest that a Type II supernova should accompany the collapse into a black hole, while others suggest that the collapse would occur with minimal radiation output.

Also note that the collapse is a result of the loss of a restoring force, not compression beyond the Schwarzschild limit. Extremely compact stars, like neutron stars and the cores of massive stars, are held together by degeneracy pressure, a quantum effect that results from the limited number of states available to particles within the star. This degeneracy pressure fails when the objects exceeds the Chandrasekhar mass, resulting in rapid gravitational collapse.

That is quite helpful information about the degeneracy pressure. It seems then that collapsed stars can be in a limited number of states of collapse, depending on mass. If I understand correctly, white dwarfs are supported by electron degeneracy pressure, neutron stars by neutron degeneracy, quark stars (if they exist) by quark degeneracy, and finally black holes lack the pressure to maintain a radius larger than the Schwarzschild radius.

 

[SpaceTiger]

That is quite helpful information about the degeneracy pressure. It seems then that collapsed stars can be in a limited number of states of collapse, depending on mass. If I understand correctly, white dwarfs are supported by electron degeneracy pressure, neutron stars by neutron degeneracy, quark stars (if they exist) by quark degeneracy, and finally black holes lack the pressure to maintain a radius larger than the Schwarzschild radius.

Looks about right. In reality, these things are always more complex than the textbooks and internet resources lead one to believe, but those are the basic models we work with. At the moment, we have direct observational evidence for the existence of white dwarfs (very convincing) and neutron stars (fairly convincing). Black holes have not been directly observed, but there is a great deal of indirect evidence for their existence. This last issue has been discussed at great length on PF, but you'd have to do an archive search to find it.

 

[Chris Hillman]

Ditto ST, except:

black holes lack the pressure to maintain a radius larger than the Schwarzschild radius.

Bad way of thinking about it: the whole point is that no pressure can sustain a body which has been compacted enough that it lies within its Schwarzschild radius (or rather a somewhat larger radius, given some reasonable assumptions). Be aware that extreme pressure is associated with a substantial contribution to the stress-energy tensor which acts (in gtr) as the source for the gravitational field.

Your discarded textbooks from 40 years ago are out of date anyway, so I'd urge you to get a more recent one. There are many excellent choices, including some inexpensive books.

 

[dilletante]

Ditto ST, except:



Bad way of thinking about it: the whole point is that no pressure can sustain a body which has been compacted enough that it lies within its Schwarzschild radius (or rather a somewhat larger radius, given some reasonable assumptions). Be aware that extreme pressure is associated with a substantial contribution to the stress-energy tensor which acts (in gtr) as the source for the gravitational field.

Your discarded textbooks from 40 years ago are out of date anyway, so I'd urge you to get a more recent one. There are many excellent choices, including some inexpensive books.

Thanks, I phrased that poorly. I have Kip Thorne's "Black Holes and Time Warps" on order -- don't know if that is my best choice but I suspect I would get lost in the math of a pure textbook.

 

[randa177]

Assuming that the black hole at the center of the Milky Way Galaxy (MWG) is Schwarzschild (non-rotating) black hole, how large is its Schwarzschild radius, in AU?

 

[Chris Hillman]

Formation of a black hole?

I've always wondered about the formation process more than the end. In particular, from the point of view of an asymptotic observer, in-falling matter will take forever before crossing the event-horizon. This includes the initial collapse.

Well, as you appear to know, this is misleading. I could try to explain the mathematical details of the Oppenheimer-Snyder collapsing dust ball model or the Vaidya collapsing shell of null dust model for the formation of a black hole, but unless you have a good math/physics background it probably makes more sense to point you at the same excellent nontechnical book I just mentioned in another thread, General Relativity from A to B by Robert Geroch, which aims to explain the geometrical picture of the formation of a black hole via the OS model.

 

[mgb_phys]

Assuming that the black hole at the center of the Milky Way Galaxy (MWG) is Schwarzschild (non-rotating) black hole, how large is its Schwarzschild radius, in AU?

R = 2GM/c^2

Assuming the black hole is around 2.5 million solar masses
G = 6 E-11 m/kg/s^2, Msun = 2E30kg C=3E8 m/s

R = 1.5E-27 * 5E36 = 7.5E9 m = 0.05 AU ( seems pretty small ? )

 

[Chris Hillman]

seems pretty small ?

That's what [post=1497780]I got too[/post] (except that I used a smaller figure for the estimated mass of Sag A*).

 

[DaveC426913]

Yes everything has a value for the schwarschild radius,

Minor poiint: Everything potentially has a S.radius - if it were shrunk. While a compactified object of the sun's mass could have a S.radius, the sun itself does not.

Maximum gravitational curvature is acquired at the sun's surface. Any deeper points have less curvature, just like any other solid spheroidal object.