Mini Black Hole Loose on Earth, eek

[Phrak]

Would a miniature black hole sink through the center of the Earth, or be supsended within some sort of bubble consisting of near-vacuum and radiation?

 

[Phrak]

Hard question?

 

[JesseM]

How small is "mini"? The smallest possible black would have a mass around the Planck mass, around 20 micrograms, and they would evaporate via Hawking radiation very quickly. http://www.kressworks.com/Science/A_black_hole_ate_my_planet.htm says:

To last long enough even to begin sucking in matter rather than going off pop, a black hole would have to be many orders of magnitude bigger. According to Cliff Pickover, author of Black Holes: A Traveler's Guide, "Even a black hole with the mass of Mount Everest would have a radius of only about 10^-15 metres, roughly the size of an atomic nucleus. Current thinking is that it would be hard for such a black hole to swallow anything at all--even consuming a proton or neutron would be difficult."

There might be primordial black holes left over from the Big Bang that would be big enough to swallow atoms but still much less massive than the Earth--here is a paper speculating on what would happen if one fell into a star or large planet, and here is an article giving a simplified summary of the results in that paper. The same website also has this article talking about what would happen if a small primordial black hole punched through the Earth at high speed (the effects wouldn't be very noticeable).

 

[Phrak]

You're right. The problem becomes non-interesting. A small black hole doesn't last long enough to consider it's dynamic behavior with terrestrial matter at low speeds --say, less than sound in the medium.

According to this link

http://library.thinkquest.org/C007571/english/advance/core8.htm"

a 230 tonne black hole has a lifetime of about 1 second-- just a really big bomb.

On top of that, it's necessarily predominated by Hawking rather than accretion radiation.

 

[yuiop]

This film http://www.thekroneexperiment.com/videos.html has a black hole that is let loose inside the Earth and is oscillating through the core popping up in different regions of the world about every 81 minutes causing localised havoc. From the film trailer the black hole makes neat holes in the ground that appears to about 0.25 meters across. Assuming the diameter of the tunnel left by the black hole is aproximately the diameter of the black hole's event horizon (is that reasonable?) then I guess we could estimate the mass of the black hole in the Krone experiment. My guess is that the mass would be less than is required for the black hole to survive more than a few seconds.

hmmmm... I just realized that if the black hole keeps tunneling through fresh material as it oscilates, you would have to take into account the amount of Earth core material it consumes per second as it continually tunnels and offset that against the energy it loses due to hawking radiation. I suppose you would also have to take into account how reflective the tunnel is and how much Hawking radiation is reflected right back at the black hole, slowing down its evaporation rate.

...annnd... would I be right in assuming that the passage of a football sized black hole through the Earth would not be perfectly frictionless and it would rapidly slow down? If it is continually accreting essentially stationary material, then conservation of momentum suggests it would slow down.

 

[JesseM]

This film http://www.thekroneexperiment.com/videos.html has a black hole that is let loose inside the Earth and is oscillating through the core popping up in different regions of the world about every 81 minutes causing localised havoc. From the film trailer the black hole makes neat holes in the ground that appears to about 0.25 meters across. Assuming the diameter of the tunnel left by the black hole is aproximately the diameter of the black hole's event horizon (is that reasonable?) then I guess we could estimate the mass of the black hole in the Krone experiment. My guess is that the mass would be less than is required for the black hole to survive more than a few seconds.

A 0.25 meter diameter BH would actually be quite massive, such a black hole would be very long-lived. The Schwarzschild formula is r = 2Gm/c^2, so m = rc^2/2G, and with r = 0.125 meters, c = 299792458 m/s, and G = 6.673 * 10^-11 m^3 / (kg * s^2) that gives m = 8.42 * 10^25 kg, which is larger than the mass of the Earth (5.9742 * 10^24 kg). So, probably the main inaccuracy of the movie is that it doesn't show the huge gravitational effects such a black hole would have on the Earth as a whole, and the major tidal effects on everything near the point of impact.

As for evaporation, this section of the wikipedia article on Hawking radiation gives the evaporation time as 5120*pi*G^2*M^3 / (hbar*c^4). With hbar as 1.0546 * 10^-34 kg * m^2 / s, that means the evaporation time works out to M^3 * 8.4079768 * 10^-17 s/kg^3, so the black hole with a mass of 8.42 * 10^25 kg would evaporate in about 5 * 10^61 seconds, or about 1.6 * 10^54 years, much longer than the age of the universe.

 

[yuiop]

An 0.25 meter diameter would actually be quite massive, such a black hole would be very long-lived. The Schwarzschild formula is r = 2Gm/c^2, so m = rc^2/2G, and with r = 0.125 meters, c = 299792458 m/s, and G = 6.673 * 10^-11 m^3 / (kg * s^2) that gives m = 8.42 * 10^25 kg, which is larger than the mass of the Earth (5.9742 * 10^24 kg). So, probably the main inaccuracy of the movie is that it doesn't show the huge gravitational effects such a black hole would have on the Earth as a whole, and the major tidal effects on everything near the point of impact.

As for evaporation, this section of the wikipedia article on Hawking radiation gives the evaporation time as 5120*pi*G^2*M^3 / (hbar*c^4). With hbar as 1.0546 * 10^-34 kg * m^2 / s, that means the evaporation time works out to M^3 * 8.4079768 * 10^-17 s/kg^3, so the black hole with a mass of 8.42 * 10^25 kg would evaporate in about 5 * 10^61 seconds, or about 1.6 * 10^54 years, much longer than the age of the universe.

This link http://www.moviemistakes.com/film5407 on movie mistakes suggests

"In the course of the story it's said that at the time the black hole went out of control, it had a mass of "half a mountaintop of granite." But in a flashback scene Dr. Krone is shown growing the mass of the hole by feeding it large numbers of lead bricks in the laboratory. It's a little hard to believe his lab could store, or even afford, enough lead bricks to equal that much mass."

so I guess I might have over estimated the diameter of the exit wounds

The same link suggests the krone black hole has a mass of about 10^9 Kgs (That is a lot of lead bricks)

"In the scene in which the scientists observe the emergence of the black hole out in the Lechuguilla desert, as it pokes out of the ground unexpectedly near to their base everything in the tent is shown powerfully accelerating roughly horizontally, toward the object as it whizzes skyward. But, given all the data in the movie(mass = 10^9 kg; it reaches max height of about 1,400 meters above the desert floor before coming back down; it emerges not less than 2 meters outside their tent), it ought to impart a gravitational acceleration of not more than 0.0167 m/s^2, which is only less than two thousandths the force provided by Earth. Given that the hole was also traveling away at about 117 m/s, the relative velocity would've been in excess of the hole's escape velocity(only about 0.25 m/s), so there would be no such dramatic effect."

 

[JesseM]

This link http://www.moviemistakes.com/film5407 on movie mistakes suggests

"In the course of the story it's said that at the time the black hole went out of control, it had a mass of "half a mountaintop of granite." But in a flashback scene Dr. Krone is shown growing the mass of the hole by feeding it large numbers of lead bricks in the laboratory. It's a little hard to believe his lab could store, or even afford, enough lead bricks to equal that much mass."

so I guess I might have over estimated the diameter of the exit wounds

The same link suggests the krone black hole has a mass of about 10^9 Kgs (That is a lot of lead bricks)

Also, how would he "feed" granite bricks to it if the black hole had such a tiny diameter? Even at its maximum mass of 10^9 kg, that would make the radius around 1.5 * 10^-18 meters, about 1000 times smaller than the radius of a proton! In the quote I posted earlier, Cliff Pickover said that even for a larger black hole with the radius of an atomic nucleus, "Current thinking is that it would be hard for such a black hole to swallow anything at all--even consuming a proton or neutron would be difficult."

On the other hand, even such a small BH would not evaporate too quickly--using the same formula I gave before, it would take about 8.4 * 10^10 seconds to evaporate, or around 2700 years.

 

[yuiop]

... "about 1000 times smaller than the radius of a proton!"


so the chances of creating an Earth threatening black hole in the LHC are pretty slim...huh.

 

[JesseM]

... "about 1000 times smaller than the radius of a proton!"


so the chances of creating an Earth threatening black hole in the LHC are pretty slim...huh.

No, that was never a danger that any non-crackpots were worrying about. See this post from the blog of two theoretical physicists for more on why this is a ridiculous idea.

 

[Phrak]

The radius of the black hole is not important in terms of the capture cross-section. It would have everything to do with the accretion disk--something I know nothing about.

Matter falling inward radiates energy colliding with other matter. It looses momentum. It's captured.

 

[JesseM]

The radius of the black hole is not important in terms of the capture cross-section.

What do you mean by "capture cross-section" here? The probability of the black hole capturing some other particle in a gravitational orbit, the probability of it actually capturing some other particle in its event horizon, or something else?

By the way, I also recommend reading this post by the same physicist-bloggers. According to them, the cross-section does depend on the radius:

First of all, mini black holes at the LHC are an option only if one of the theories of "large extra dimensions" was in fact true. But of course, these theories are only speculations so far. Second, should mini black holes be created in high-energy particle collisions, they would evaporate very fast, due to Hawking radiation. Though Hawking radiation has not been experimentally verified so far, its existence is expected in almost all theoretical scenarios investigated (no matter where you go, you will always find somebody who disagrees on something).

But what would happen in the (quite unrealistic) case that tiny black holes were created at the LHC, and that they did not decay by the emission of Hawking radiation?

It's important to keep in mind that black holes do not have some special "vacuum cleaner" property - they just attract other stuff by the force of gravity.

Now, the tiny black holes that could be created at the LHC if theories of large extra dimensions were indeed correct would have masses in the range of a few TeV. 1 TeV corresponds to about 1000 times the mass of a proton, which is 0.94 GeV, or 1.7×10^-27 kg. The corresponding Schwarzschild radius is about 1/1000 fm, or 10^-18 m.

Because gravity is such a weak force, it's safe to assume that nothing happens to matter that encounters the black hole at a larger radial distance than one Schwarzschild radius. Assuming for simplicity that all stuff hitting with a smaller distance gets sucked in, the black hole has a cross section of about 10^-36 m^2, or 10 nanobarn (that's more than typical neutrino cross sections).

 

[Phrak]

What do you mean by "capture cross-section" here? The probability of the black hole capturing some other particle in a gravitational orbit, the probability of it actually capturing some other particle in its event horizon, or something else?

By the way, I also recommend reading this post by the same physicist-bloggers. According to them, the cross-section does depend on the radius:

I read it.

The interesting question, in my mind is, "what diameter hole will a black hole cut-out traveling at a particular speed?" That which it keeps and eats is what I'm calling the capture cross-section.

Given a few mild considerations, the answer can't be simple. It's complicated by the amount and location of entrained matter already circulating the black hole. It depends upon the pressure due to Hawking radiation driving matter away from the black hole. Enough Hawking radiation and the cross-section drops to nil but for dark matter and neutrinos.

There is a photosphere, of a given radius, less than which light cannot escape. For ponderable matter, this radius must be greater. The radius would depend upon kinetic energy.

I don't know enough of the physics to say if unimpeded orbits are circular or spiral. but in my rough analysis it's moot.

Matter colliding with matter already in orbit will radiate emf. The net effect is to reduce kinetic energy and angular momentum. In a sense the captured matter is supercooled--it would take a lot of energy to raise it to infinity.

Matter falling inward and interacting is conduction and radiation cooled. It would be captured in interaction as it looses kinetic energy.

So the cross-section is dependent upon how much and how cold the previously captured matter has become.

 

[JesseM]

There is a photosphere, of a given radius, less than which light cannot escape. For ponderable matter, this radius must be greater. The radius would depend upon kinetic energy.

I think you mean photon sphere rather than photosphere, and the photon sphere does not mark the boundary beyond which light can't escape (that's the event horizon), it just represents a sphere such that photons moving tangentially to the sphere will be in (unstable) circular orbits around the black hole. Its radius is only 3/2 larger than the radius of the event horizon (for a nonrotating BH anyway), so even if we were to assume things within the photon sphere would be unlikely to have the velocity needed to escape, it would make little difference in terms of an order-of-magnitude calculation.

Anyway, for black holes smaller than an atom I assume you can't do a purely classical analysis of nearby particles--quantum tunneling for particles in the BH's potential well might be significant, for example. Certainly you couldn't treat particles as moving in nice well-defined orbits of the kind seen in large classical BHs. As for quantum orbits, you have to consider that the gravitational attractive force between some small particle and a black hole just 1000 times more massive than a proton would probably be far smaller than the electromagnetic attractive force between a proton and an electron, simply because the electromagnetic force is so much stronger than the gravitational force; so, if you don't see accretion discs around protons, you shouldn't expect to see them around black holes with masses about 1000 times the mass of a proton.

To get a rough idea of the differences in forces, law[/url] says that the electromagnetic force between two objects with charges q1 and q2 and separation r will be about (9*10^9 Newtons*meters^2/Coulombs^2)*(q1 * q2)/r^2, so if the particles are like protons or electrons and have the elementary charge, that means the force as a function of r will be about (2.3 * 10^-28 Newtons*meters^2)/r^2. On the other hand, for a black hole with a mass of about 1000 times the mass of a proton acting on an actual proton near it (mass of proton = 1.7 * 10^-27 kg), Newton's gravitational formula says the force will be GMm/r^2 which gives the force as a function of r as (1.9 * 10^-61 Newtons*meters^2)/r^2, so the proton near a black hole would feel a gravitational force about 8 * 10^-34 times weaker than the electromagnetic force if it was at the same distance from another proton. Of course a less massive particle like an electron would feel the same electromagnetic force but an even weaker gravitational force at a given radius (about 1836 times weaker than the gravitational force above, since that's the proton/electron mass ratio).

I assume it's these sorts of considerations that led the physicists who wrote that blog post to say "Because gravity is such a weak force, it's safe to assume that nothing happens to matter that encounters the black hole at a larger radial distance than one Schwarzschild radius", and to treat the cross section as proportional to the radius of the event horizon squared. As they are actual physicists, I trust that they have good physical arguments as to why the cross section wouldn't be significantly larger.

 

[peter0302]

That film looks like garbage.

 

[nitsuj]

In any case, I hope no one will pre-judge the film without seeing it.

Well I never heard of it before reading this. I want to see it now.