Black hole help needed

[Professor Yaffle]

This Stephen Hawking lecture might help explain things:

 

[RecoveringYuppy]

@Sol,

Yes, there is no gravity at the center. But there is tremendous pressure and it's fluid. I don't think there is any plausible way a shell can form. Except for some local anomolies I'm pretty sure most all of the surface is supported from below rather than the side.

I'd also have to wonder if cooling is really what would happen. For even a small black hole wouldn't it still be true that the fricitional heating and accelaration occuring just outside the horizon is going to radiate away a substantial fraction of the mass falling in as energy?

I think I agree about the necessary black hole being too massive to be plausibly maneuvered.

 

[Ocelot]

But there is tremendous pressure and it's fluid.

tremendous pressure yes, but the inner core is believed to be solid crystaline iron.

http://www.seismo.unr.edu/ftp/pub/louie/class/100/interior.html

http://en.wikipedia.org/wiki/Structure_of_the_Earth

http://www.crystalinks.com/corecrystal.html

etc.

 

[sol invictus]

tremendous pressure yes, but the inner core is believed to be solid crystaline iron.

Interesting! Thanks, I didn't know that. Looks like it's only the very center that's solid, though - a solid ball surrounded by a thick layer of liquid.

Yes, there is no gravity at the center. But there is tremendous pressure and it's fluid. I don't think there is any plausible way a shell can form. Except for some local anomolies I'm pretty sure most all of the surface is supported from below rather than the side.

As the hole swallows more and more of the center, the pressure would decrease. Come to think of it, if we treated everything as a liquid this is essentially like draining a bathtub through an extremely small hole. The black hole can only eat the layer of fluid immediately surrounding it, and the fluid can only flow inward so fast. One could probably use that model to get a basic estimate of the time.

For a solid it could be much slower since the hole will need to drift around by itself to eat anything.

I'd also have to wonder if cooling is really what would happen. For even a small black hole wouldn't it still be true that the fricitional heating and accelaration occuring just outside the horizon is going to radiate away a substantial fraction of the mass falling in as energy?

Maybe, yes. But that might still result in a net cooling, especially once you take into account the temperature gradients that must exist in the core.

 

[Ziggurat]

But I realized I had forgotten something else important - the gravitational field inside a hollow earth is essentially zero. That means if the hole managed to eat out a spherical cavity, it would drift around inside it at constant velocity.

You're forgetting the moon again. The gravitational field of the shell itself might be zero inside, but the gravitational pull from the moon is not. If it drifts off center at all, tidal forces from the moon will pull the shell and the black hole in opposite directions, and they'll collide.

 

[Ziggurat]

As the hole swallows more and more of the center, the pressure would decrease. Come to think of it, if we treated everything as a liquid this is essentially like draining a bathtub through an extremely small hole. The black hole can only eat the layer of fluid immediately surrounding it, and the fluid can only flow inward so fast. One could probably use that model to get a basic estimate of the time.

The only drop in pressure would be due to Bernoili's principle (the velocity change as liquid flows inwards) - in other words, negligible.

But actually, the black hole wouldn't eat the entire earth. As the earth shrinks inwards, Coriolis forces will speed up its rotation. At some point, the remains will spin fast enough to form a donut of mater around the black hole.

For a solid it could be much slower since the hole will need to drift around by itself to eat anything.

The core of the earth is solid under hydrostatic pressure, but if you eat out the inside, the inner layer is no longer under hydrostatic pressure. The pressure differentials are going to be far larger than iron can possibly withstand, and it will flow like playdough does if you squish it.

 

[sol invictus]

You're forgetting the moon again. The gravitational field of the shell itself might be zero inside, but the gravitational pull from the moon is not. If it drifts off center at all, tidal forces from the moon will pull the shell and the black hole in opposite directions, and they'll collide.

Again?

Anyway, if what you meant to say was that it will pull them in more or less the same direction but with different accelerations, then yes, I agree. But tidal forces are very weak compared to everything else that's going to act on this thing, like the force it will experience when it hits the wall and starts eating stuff.

Anyway, here's a simpler but related question - how long does it take a bathtub of given size and volume to drain through a hole of given size? You can assume the hole is connected to a vertical pipe that does down forever.

 

[RecoveringYuppy]

The black hole can only eat the layer of fluid immediately surrounding it, and the fluid can only flow inward so fast. One could probably use that model to get a basic estimate of the time.

The pressure differentials would transmit at the speed of sound in whatever material they are travelling through. That's what caused me to extend my guesstimate to 25 minutes or so.

Maybe, yes. But that might still result in a net cooling, especially once you take into account the temperature gradients that must exist in the core.

Not sure what you're getting at there but the surface of the Earth is 300 kelvin and the center is at about 7,000 kelvin. That constrains the gradients that might exist.

Anyway, here's a simpler but related question - how long does it take a bathtub of given size and volume to drain through a hole of given size? You can assume the hole is connected to a vertical pipe that does down forever.

I think the pressures involved make that a misleading analogy.

 

[RecoveringYuppy]

The core of the earth is solid under hydrostatic pressure, but if you eat out the inside, the inner layer is no longer under hydrostatic pressure. The pressure differentials are going to be far larger than iron can possibly withstand, and it will flow like playdough does if you squish it.

That understates the case I believe. The pressures at the center of the Earth are 4 million atmospheres or 400 Gigapascals. Tensile strengths rarely exceed 1 Gigapascal (and I think it would be tensile strength that would resist flow).

 

[sol invictus]

The pressure differentials would transmit at the speed of sound in whatever material they are travelling through. That's what caused me to extend my guesstimate to 25 minutes or so.

But that's clearly the wrong estimate. A bathtub doesn't drain in the time it takes sound to propagate across it. Increase the pressure on the water in the tub all you want; it's still not the speed of sound that matters (and the speed depends only slightly on the pressure).

Not sure what you're getting at there but the surface of the Earth is 300 kelvin and the center is at about 7,000 kelvin. That constrains the gradients that might exist.

My point is that if the hole eats the hottest stuff at the center first, it may well decrease the average temperature .

I think the pressures involved make that a misleading analogy.

Make it a high-pressure bathtub!

Look - the rate of bathtub draining obviously depends on the size of the hole, right? It's equally obvious that our rate depends on the size of the black hole. Even if we assume the core is liquid (wrong) and the liquid can move at the speed of sound (also wrong), the total flux of liquid into the hole is the speed of inflow times the area of the horizon - which is on the order of square millimeters even for an earth-mass hole. So the total volume per second swallowed is only of order .001 m^3/s if I assume the speed of sound in molten metal is 1000 m/s (wild guess). The earth contains of order 10^19 m^3 of stuff. So that would give 10^22 seconds, which is longer than the current age of the universe!

Now maybe one should use a larger number for the horizon, since gravitational effects on the fluid will become very important some distance from it, but I doubt that will matter much. I guess on the speed of sound, and took guestimates at all the other numbers. But I doubt you could possibly get more than a few orders of magnitude out of my sloppyness.

Can someone check that estimate? It seems a little too slow...

 

[RecoveringYuppy]

But that's clearly the wrong estimate. A bathtub doesn't drain in the time it takes sound to propagate across it.

I didn't estimate that the Earth would collapse in that time. I estimated that was the time it would take the pressure differentials to propagate. And pressure differentials do propagate across a bath tub at the speed of sound. So I used that number to guesstimate when each piece of the Earth would start it's motion downward.

... the total flux of liquid into the hole is the speed of inflow times the area of the horizon -

Sure, but at the horizon the speed of inflow is going to be the speed of light and the "liquid" is likely to be very highly compressed near the horizon.

 

[RecoveringYuppy]

@Sol,

Hmmm. I'm thinking of your "bathtub" approach to evaluating how fast stuff would pass the horizon.

I think that trying to estimate how much can cross the event horizon is beyond my capabilities. If my calculations are correct the surface gravity of a 4.4 mm Earth mass black hole would be about 10¹⁸ gravities. I think that raises severe questions about the density of the material in that neighborhood. So I don't think I can evaluate how much material can cross the horizon.

So I took a different approach and thought about what would be going on at the surface of a sphere 100 meters in radius surrounding the Earth massed black hole. I come up with that sphere having a surface gravity of about 1/2 million gravities. I also come up with that sphere having an escape velocity of about 2.8 million meters per second. And the surface area is about 125,000 square meters.

So assuming that stuff falling in from a significant distance would reach a significant fraction of that escape velocity and ignoring the possibility that 1/2 million gravities would still lead to some outlandish densities, I come up with something in the neighbordhood of 350 billion cubic meters per second crossing that surface per second.

From your number of 10¹⁹ (low, I think) cubic meters that leads to the entire Earth crossing that surface in 3,256 days.

So maybe I've made a mistake in my numbers, but this looks contradictory to me. The contradicton is, if there is stuff in the center of the Earth falling at many times the surface free fall why does the collapse of the entire Earth wind up taking so much longer than a simple free fall? (Sounds like a 9/11 conspiracy theory brewing).

BTW I haven't made any attempt here to try to figure out if radiation pressure would slow things down.

 

[sol invictus]

@Sol,

Hmmm. I'm thinking of your "bathtub" approach to evaluating how fast stuff would pass the horizon.

I think that trying to estimate how much can cross the event horizon is beyond my capabilities. If my calculations are correct the surface gravity of a 4.4 mm Earth mass black hole would be about 10¹⁸ gravities. I think that raises severe questions about the density of the material in that neighborhood. So I don't think I can evaluate how much material can cross the horizon.

Yes, agreed. There should be some larger "effective horizon". Since the time goes down like the horizon radius squared, if the effective horizon is significantly larger than the actual horizon the rate would be much faster.

From your number of 10¹⁹ (low, I think) cubic meters that leads to the entire Earth crossing that surface in 3,256 days.

So maybe I've made a mistake in my numbers, but this looks contradictory to me. The contradicton is, if there is stuff in the center of the Earth falling at many times the surface free fall why does the collapse of the entire Earth wind up taking so much longer than a simple free fall? (Sounds like a 9/11 conspiracy theory brewing).

The discrepancy is due to the fact that your treatment (and mine) assumed the density didn't change. If stuff fell in at its free-fall rate, the density would be enormous near the horizon. But since you used volume rather than mass, you were implicitly assuming it didn't, and that's why you got a slow rate.

I'm starting to think your first estimate was close to correct...

 

[Ziggurat]

Make it a high-pressure bathtub!

Look - the rate of bathtub draining obviously depends on the size of the hole, right? It's equally obvious that our rate depends on the size of the black hole.

True.

Even if we assume the core is liquid (wrong)

It will act like a viscous liquid under non-hydrostatic pressures far in excess of its material limits. Which is the case at the center of the earth of you make a hollow spot, especially if you stick a very massive body in there to exert significant extra force.

and the liquid can move at the speed of sound (also wrong),

Of course liquid can move at the speed of sound. Signals cannot, waves cannot, but if you start accelerating a liquid and keep accelerating it, at some point it'll start moving faster than the speed of sound. Why is that a problem? It isn't. There's nothing about fluid dynamics that prohibits super-accoustic flows.

 

[dahduh]

Suppose we choose a mass that would survive 1 year against evaporation; that would make the hole mass 72,000 tonnes, a bit less than the mass of the USS Enterprise aircraft carrier.

If you dropped a black hole from infinity onto the earth, it would arrive at the earth with escape velocity of about 11km/s.

As the hole approaches Earth, it will be radiating 6.8 x 10^16 W of energy. At 25,000km, this will be as much energy per square meter as the sun deposits on earth. At 1,000 km, it will be 16,000 times more. So if you are out in the open, even with your sunglasses on, you are toast, because at 11km/s it is going to take about 15 minutes to cover the last 10,000km. Most of the people on the side of the planet the hole approaches from are toast, even before the hole hits.

The hole is only 10^-19m big - that's about 10^9 times smaller than an atom, or 10^4 smaller than an atomic nucleus. Perhaps more relevant, at a distance of about 2cm the gravitational acceleration of the black hole will be the same as the gravitational acceleration of the earth. This means that any unattached object approaching within about 2cm will tend to be drawn into the hole. Solid material will only be drawn in if it approaches to a distance close enough so that the force exceeds its tensile strength - for rock this is not terribly high and very variable, but at a thumb-suck lets take 10MPa. At a density of 5g/cm^3, this means rock will be torn apart at about 200g, or when it approaches to within 1.4mm of the hole.

In a liquid, which is what we can take the center of the earth to be (it is molten and plastic), material will just flow to wherever there is lower pressure. The rate at which material is sucked up is limited by the rate at which it can fall into the hole. The escape velocity at a radius of 2cm is only 0.69m/s, so you have material falling through at a rate of about 11kg/s.

But this is assuming the hole is sitting still. Really, it's zipping along at 11km/s, which means it can only pick up matter that approaches to a radius at which the escape velocity is 11km/s - that's a distance of about 1 Angstrom, the radius of an atom. So as our hole zips through the earth at this high speed, it will pick up just four lines of atoms as it goes, or only 3.5 x 10^-9kg/s. As it passes through earth, it will pick up almost nothing at all, and will head back out into space with its speed only very slightly retarded by the matter it's picked up.

BUT, this is all rather academic. As noted, this hole is radiating 6.8 x 10^16W of energy - that's the equivalent of a 16 megaton nuke going off every second. This energy is going to blow a rather nasty hole hole in the earth - not catastrophic, since the hole is moving so fast, think of a line of 16 megaton nukes going off each second spaced 11km apart. I think we would get at least a new volcano and lost of dust in the atmosphere, but not much worse than that. But the point is that even if the hole is stationary, no matter is going to approach it anyway - it will all be blown away by the radiation.

I haven't exactly answered your questions but it was fun to think about it with all the hoopla with the LHC; and I haven't checked by sums! Someone else will have to work out a scenario with a hole perhaps 10^6 times bigger, which is what you would need to avoid the radiation problem - but then you have the problem of about 10% of the mass of the matter falling into the hole being converted into radiant energy, so it's even worse - I think I'd better stop now...

 

[sol invictus]

Of course liquid can move at the speed of sound. Signals cannot, waves cannot, but if you start accelerating a liquid and keep accelerating it, at some point it'll start moving faster than the speed of sound. Why is that a problem? It isn't. There's nothing about fluid dynamics that prohibits super-accoustic flows.

I wasn't saying it was impossible under any circumstances for a liquid to move faster than the speed of sound, I was saying it wouldn't in this situation, and so using that speed was an overestimate. However I now think that was wrong.