Black hole help needed

[NobbyNobbs]

A friend of mine is running a space-based RPG, and turned to me for some scientific help. I, therefore, turn to you.

He wants his villians to have a weapon that can use a black hole to destroy a planet. Now, I was able to tell him that such a black hole wouldn't need to be very big, and that shot correctly, it could be captured by the Earth's gravity, zigzagging back and forth, turning the Earth into swiss cheese. What I wasn't able to tell him was:

1) How big a black hole would it need to be? Bullet-sized? Smaller?
2) How big would it be after consuming the earth?
3) If you shot it at the earth from orbit (low earth, moon's orbit, I don't really care which) how fast would you have to shoot it so that it stays zigzagging under the surface, eating the planet, instead of passing through once and shooting off into space?
4) How long before things are affected on a macro scale, for example, earthquakes, volcanos, etc?
5) Once started, would there be any way (short of Star Trek technobabble) to stop it?

Thanks in advance for all your suggestions.

 

[Hellbound]

I can help some

1) Depends on how big you want your holes to be. It'd have to be fairly big to begin with to last long enough to do damage. IIRC, current theories predict black holes evaporate, and they do this faster as they get smaller. You'd probably need something at least Everest-massed to be stable, but that won't be very big. Of course, it will be spewing copious amounts of hard radiation the whole time, which just adds to the fun and carnage.
2) I think the entire Earth squeezed to a black hole would be roughly pea-sized. That's a close enough approximation.
3) As long as it's final velocity is less than Earth's escape velocity, you should be fine. But remember, firing at Earth it's going to be accelerated by gravity, as well. You could likely just drop it...depending on how far above planet you are.
4) Not a clue.
5) Maybe. Technically. Firs,t there has to be some way to actually fire the hole. You can't touch it directly. This means either:
a) the black hole has an electric charge and so can be moved with magnetic fields or
b) your villians have some sort of tractor-beam like technology.
Either of these processes could be used by Earth-ites to either slow the hole down and capture it, or speed it up to an escape velocity. It could also slow down on it's own as it encoutners and absorbs other matter/energy, depending on if it's absorbing or emitting faster.

Consideirng that black holes evaporate, a much better idea might be using highly unstable micro black holes, timed to explode as they enter Earth's atmosphere or under the surface. Since black hole evaporation is, theoretically, predictible based on their size, this seems a possibility. The explosions are immense; I don't have any references handy but I believe it's a huge number of megatons. This also has the advantages (for a weapon system) of:
1. Requiring much less mass and/or energy to create
2. Requiring much less energy to fire (you still have to move all that mass).
3. Larger micro-black holes, if pre-created, could be used as a very compact, very powerful energy source.
4. Less time for the enemy to interupt or disrupt your plans.
5. Big boom

I'm sure there are physics people on here that will correct the errors I no doubt made above, but anyway, figured I'd give it a shot

 

[Ziggurat]

1) How big a black hole would it need to be? Bullet-sized? Smaller?

The Schwarzschild radius for the earth itself would be 2MG/c², or about 4.4 mm. Since presumably you're going to use a black hole with less mass than the earth itself, it would be smaller than a bullet - the real question becomes not what its radius would be, but what its mass would be. I don't have a good answer for that. What you want is something big enough that it won't evaporate over the time scale we're interested in, but it may take some work to figure that out.

2) How big would it be after consuming the earth?

Well, the Schwarzschild radius scales linearly with mass, so you'd add 4.4 mm to that radius after swallowing the earth.

3) If you shot it at the earth from orbit (low earth, moon's orbit, I don't really care which) how fast would you have to shoot it so that it stays zigzagging under the surface, eating the planet, instead of passing through once and shooting off into space?

Well, if you dropped it from orbit, it wouldn't shoot into space on the far side, but it would exit the surface on the other side. No way to prevent that.

4) How long before things are affected on a macro scale, for example, earthquakes, volcanos, etc?

If it's massive enough (which it might need to be in order to avoid evaporating), immediately.

5) Once started, would there be any way (short of Star Trek technobabble) to stop it?

Nope - provided, of course, that it's massive enough that it doesn't evaporate on its own.

 

[NobbyNobbs]

Thanks for all the info so far.

There's one other question I forgot to ask:

6) How long woudl it take to eat the earth?

 

[Fredrik]

One problem that occurred to me is that if the mass is small enough, the pressure from the Hawking radiation might prevent other matter from "falling" into the black hole.

I think I read somewhere that a black hole with the appropriate mass (don't remember exactly, maybe the mass of a mountain) would radiate as much per unit of time as a hydrogen bomb at the peak of its explosion (which only lasts something like a millionth of a second, but I don't remember that number either, so I may be off by many orders of magnitude).

I'm sure that to have something like that going through the Earth over and over again would kill us soon enough, but I'm not convinced that the planet will get sucked into the black hole. Maybe the black hole has to be much more massive for that. I wouldn't be surprised if it needs to be as massive as the Sun.

 

[Ziggurat]

I'm sure that to have something like that going through the Earth over and over again would kill us soon enough, but I'm not convinced that the planet will get sucked into the black hole. Maybe the black hole has to be much more massive for that. I wouldn't be surprised if it needs to be as massive as the Sun.

According to this, a black hole has to have a mass less than the moon in order to evaporate at all (above that and the CMB is enough to counter its own losses) - put it in contact with a food source and the minimum mass will drop further. How much I don't know, but we definitely don't need solar-mass black holes for this.

 

[MattusMaximus]

My advice on black holes:

Don't go near them

 

[RecoveringYuppy]

Consideirng that black holes evaporate, a much better idea might be using highly unstable micro black holes, timed to explode as they enter Earth's atmosphere or under the surface. Since black hole evaporation is, theoretically, predictible based on their size, this seems a possibility.

In order to evaporate there has to be more radiation coming out of the black hole than going in. Most approaches towards Earth are going to have the black hole approaching a warmer region of space than where it started and that's going to act against the evaporation.

 

[RecoveringYuppy]

6) How long woudl it take to eat the earth?

Less than five minutes I think.

I think the black hole would quickly come to rest near the center of the Earth because it would be consuming material that isn't moving and it would then cease moving itself. At that time it would then be a matter of figuring out how long it would take the rest of the Earth to fall in to the hole.

I suspect a lot of the Earth would radiate away as X-rays in those five minutes.

ETA: On second thought I think there might be two aspects to Earth's collapse. One would be that the surface wouldn't start collapsing until the stuff underneath it had fallen away. I think the collapse would propagate outward at the speed of sound through the Earth and then you add to that the time it takes to fall to the center of the Earth. Sound right? So 25 minutes or so would be my new guess at an upper bound.

 

[RecoveringYuppy]

Well, if you dropped it from orbit, it wouldn't shoot into space on the far side, but it would exit the surface on the other side. No way to prevent that.

Would it necessarily exit the other side or would it have consumed enough that it's velocity would have to slow down before it reached the other side?

 

[sol invictus]

Black holes have an evaporation time that scales as their mass cubed. The formula for their lifetime, with all the constants in there, is

[latex]$ t={5120 \,\pi \, G^2M_0^{3}\over\hbar \, c^4} \sim 10^{-16} (M_0/{\rm kg})^3 s$[/latex]

So very roughly, a black hole with mass 10^5 kg has a lifetime of one second, and if you increase the mass the lifetime goes up by the cube.

As for the time it would take to eat the earth, a black hole's gravitational field some distance away is identical in strength to the gravitational field of any other object of the same mass. Suppose it had the mass of a small mountain - then a kilometer away its pull would be identical to that of a mountain a kilometer away (e.g. essentially zero). Only the stuff very close to the hole would be pulled into it.

I'm not sure it would actually eat the earth at all. Suppose it fell in, absorbing stuff along the way, and came to rest in the core. The field of the hole would be so weak that we can neglect it outside some radius, and just assume that only fluid that comes within that radius will be absorbed. I could estimate that radius; my offhand guess is it's a few meters.

Now the liquid core is probably rotating (?). If so it will be "held up" by angular momentum. So the hole might eat a little liquid right near the center, leaving it empty, while the rest of the fluid orbits it inside the cavity. If so the whole process could be very slow.

Even if all the liquid is swallowed, would the crust collapse? I'm not sure... the gravitational force on it would be identical to what it is normally, but there'd be no magma underneath to apply a pressure. So if you empty all the molten stuff from the center of the earth, could the crust hold itself up?

Anyway, I think it would be a pretty slow process.

 

[Ziggurat]

Even if all the liquid is swallowed, would the crust collapse?

Definitely. On a global scale it's not rigid (think plate tectonics), and don't forget that tides from the moon constantly perturb it, so it would buckle and collapse.

 

[RecoveringYuppy]

As for the time it would take to eat the earth, a black hole's gravitational field some distance away is identical in strength to the gravitational field of any other object of the same mass. Suppose it had the mass of a small mountain - then a kilometer away its pull would be identical to that of a mountain a kilometer away (e.g. essentially zero). Only the stuff very close to the hole would be pulled into it.

I would think that the existing pressures and gravity field of the Earth would be sufficient to drive a collapse if something at the center started creating an empty volume.

I'm not sure it would actually eat the earth at all.

Well, the presumption of the thread is a sci-fi way to destroy the Earth, so if whatever black hole you're assuming won't do it, you've assumed the wrong one. I think I agree that if the black hole were significantly smaller than atoms then it's own gravitational field would not be sufficient to move matter toward it and it would have to wait (possibly a long time) for random movements to bring material toward it. So I think we have to presume the black hole is large enough to interact with matter. The 4.4 mm cited earlier for an Earth massed black hole would seem more than sufficient to me. Not sure what a lower bound would be.

Now the liquid core is probably rotating (?). If so it will be "held up" by angular momentum. So the hole might eat a little liquid right near the center, leaving it empty, while the rest of the fluid orbits it inside the cavity. If so the whole process could be very slow.

Earth's core rotates fast enough to hold itself by it's angular momentum? That doesn't sound right.

 

[Ocelot]

I'm imagining a singularity the mass of the earth appearing aout of "hyperspace" above the north pole and slicing through the polar axis. It starts accelerating at 9.8 ms-2. It will pick up the mass of the column it passes through. We have best guesses of the density of the earth at various depths but being only 4.4mm wide that not too much mass.

At the equator the direction of gravity will have changed by around 27 degrees and increased to just 1.6 of earth normal. Over the course of the next ten minutes. The black hole travels south and the earth travels north the same distance and accelerates at the same rate. After ten minutes thorth seconds it's at the centre of the earth (which has moved half an earth radius north) and gravity is straight down once more but double normal strength. That said the earth is moving north at 71km per second and the black hole is moving south at the same speed. The acceleration are enormous. When the black hole breaks the surface on the other side a dozen or so seconds later it's picked up so much speed that it'll never be seen again and the earth is hurtling away from the ecliptic. Everything not strenuously nailed down will have been shaken loose. That probably includes much of the planet's crust. I must confess I was expecting a periodic motion rather than a sling shot when I set up excel but when I saw what was happening it made sense. No we don't get swallowed up but the planet is shattered and in a very dodgy orbit.

 

[sol invictus]

Definitely. On a global scale it's not rigid (think plate tectonics), and don't forget that tides from the moon constantly perturb it, so it would buckle and collapse.

Yes, I suppose.... but if the interior were to gradually disappear from the inside out there would probably be significant cooling, which might eventually harden the top layer of magma into solid rock. You wouldn't need that thick a layer to hold up the crust, I don't think.

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. Once it arrived at the wall, it would probably get stuck at least for a while. It's not very clear to me what would happen then... one possibility is that it would create a new hollow region. Then gravity would pull it back towards the central cavity, so it might drift across and hit the wall on the other side.

Clearly it would eventually eat the earth, but estimating the time seems rather non-trivial.

 

[Rob Lister]

what holds a small black hole together? Certainly not gravity. If it can't hold together, wouldn't it evaporate within nano-seconds?

 

[sol invictus]

I would think that the existing pressures and gravity field of the Earth would be sufficient to drive a collapse if something at the center started creating an empty volume.

Don't forget that there is essentially no gravity at the center of the earth.

Well, the presumption of the thread is a sci-fi way to destroy the Earth, so if whatever black hole you're assuming won't do it, you've assumed the wrong one. I think I agree that if the black hole were significantly smaller than atoms then it's own gravitational field would not be sufficient to move matter toward it and it would have to wait (possibly a long time) for random movements to bring material toward it. So I think we have to presume the black hole is large enough to interact with matter. The 4.4 mm cited earlier for an Earth massed black hole would seem more than sufficient to me. Not sure what a lower bound would be.

I had in mind something like a mountain-mass black hole. Bringing something the mass of the earth from far away to destroy the earth is not a very efficient way to do things. There are much easier ways to do it than that. So at least to me, the premise only makes sense if the black hole is of much smaller than earth mass.

Earth's core rotates fast enough to hold itself by it's angular momentum? That doesn't sound right.

Again, remember that there is no gravity at the center.

 

[sol invictus]

what holds a small black hole together? Certainly not gravity. If it can't hold together, wouldn't it evaporate within nano-seconds?

Gravity holds it together.

What makes a black hole different from another object with similar mass is its extremely small size. First, think of it as a point with mass (so a point with infinity density). Since the gravitational field gets stronger and stronger the closer you get to the source, and since you can get arbitrarily close to a point, the field gets arbitrarily strong. At some distance its so strong nothing can escape it. Anything inside that radius should be though of as part of the object (since it can never escape it).

So once any collection of stuff forms which is confined to a region smaller than that distance (which is called the Schwarzschild radius), nothing can ever escape... (unless you include quantum tunneling, which does allow particles to escape even then and the black hole to evaporate).

 

[OneDollarWilliam]

Even if all the liquid is swallowed, would the crust collapse? I'm not sure... the gravitational force on it would be identical to what it is normally, but there'd be no magma underneath to apply a pressure. So if you empty all the molten stuff from the center of the earth, could the crust hold itself up?

Even if it doesn't work on this fictional Earth (at least I HOPE it's fictional, or Nobby has some explaining to do ) That is an AWESOME idea for a fictional planet. A crust held together by the force of a small black hole, but with no plate tectonics and existing at a sufficient distance around the hole that it doesn't fall in. Maybe have two societies, one living on the surface, and the other hanging underneath the crust... (wanders off to write)

 

[sol invictus]

Even if it doesn't work on this fictional Earth (at least I HOPE it's fictional, or Nobby has some explaining to do ) That is an AWESOME idea for a fictional planet. A crust held together by the force of a small black hole, but with no plate tectonics and existing at a sufficient distance around the hole that it doesn't fall in. Maybe have two societies, one living on the surface, and the other hanging underneath the crust... (wanders off to write)

One problem - there's no gravity at all inside a spherical shell of mass. That means there's nothing to keep the black hole centered. If you bumped the shell, it would drift to one side until some part of it collided with the hole.

So your people will need to do something to keep the hole away from the crust...