Energy needed to create a black hole vs energy needed to run the Large Hadron Collider?

Question

I was disappointed to hear Brian Cox's flippant reply to whether or not the LHC could create a black hole big enough to swallow the earth in this video:

https://www.youtube.com/watch?v=QFy87tFTZwY

I'm guessing that the power used by the LHC is nothing like that used when gravity crushes a star to a singularity, but ...

... how big is "nothing like"?

Answer 1

Your question touches on a few points.

First, yes, he was flippant, but the risk was super-low. The simplest way to explain this is that nothing happens in CERN that doesn't happen all over the universe and in the upper atmosphere of the Earth or on the surface of all the moons, planets and stars every day. For example, the Oh My God particle was traveling with several million times the energy of the particle collisions inside CERN. At those speeds, accelerating the particle further doesn't add much velocity, what they're basically doing is adding energy or mass to the particle and only slightly increasing it's velocity as it's speed gradually moves closer to the speed of light.

What makes CERN useful, isn't that the high speed collisions don't happen anywhere else (they happen all the time all over the Universe). What makes it useful is the million dollar equipment that zooms in on those collisions, recording what happens with the best cameras in the world. We can't predict when and where a very fast moving cosmic particle will hit the Earth, but they know when and where the CERN particles are going to collide, because that's planned to happen in a specific place and time.

So, if those kinds of collisions happen all the time and no planet or star has ever been observed being eaten by a particle collision created black hole, then such events are, at the very least, extremely rare.

Add to that, Hawking radiation dictates that black holes that small should evaporate before they can eat anything. So, when Brian Cox laughs at that question, he has good reason to laugh, because it was never a real risk and it's more of a joke question than a genuine risk.

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Now, when you say "pressure" to create black holes. The pressure inside a black hole, or, I should say, pressure inside a Neutron Star about to turn into a black hole, or the pressure inside a collapsing core of a very large star about to create a black hole - those pressures are crazy high, but if we could re-create that pressure on earth (We can't), but if we could, it wouldn't be enough to make a black hole. Pressure alone doesn't do it, you need mass (at last about 3 solar masses) to fit within a certain size and the less mass you have, the greater the required density (and greater pressure) is needed to achieve the necessary mass to schwarzchild radius ratio. The pressure required inside the forming black hole actually decreases as the mass of the black hole increases. (based on simple pressure equations and schwarzchild radii). Equal pressure in a lab on Earth wouldn't accomplish it.

What makes black holes theoretically possible is the small quantum size and possibility of extra dimensions and quantum rules that only apply to the very small, so it's a very different set of circumstances.

Collisions in CERN are measured in Electron-volts. It was theorized that if there were extra dimensions on the quantum level, then micro-black holes might form at the energy of CERN's collisions (this would imply that black holes of this type form in Earth's upper atmosphere all the time, implying they're not dangerous).

But so far, no luck. No micro-black-holes have been observed. They should dissipate very rapidly but their decay streams should be recognizable so they should get noticed, but so far, none have been spotted and as a result, there's no evidence of extra dimensions. That doesn't mean they're not there, and that micro-black-holes can't be created, but so far there's no evidence that they've been created up to about 13 trillion electron volts, the current peak of CERN's collision energy. See here

Hope that makes sense. Corrections welcome.

Answer 2

According to our current understanding, there is no lower bound for the energy needed to make a black hole. Any object, no matter how small, if compressed enough, could in theory form a black hole. Make it small enough, and it would not require much energy at all.

But here's the main issue. Black holes have this thing called Hawking radiation: they constantly lose energy (and therefore mass) via radiation, and the smaller the BH is, the faster it loses energy. Super-tiny BHs basically pop out of existence the very moment they are created, that's how quick the process is.

For a BH to be a threat, it would have to be big enough to be stable long enough to actually start eating surrounding matter, to compensate for loss via radiation. I believe calculations have shown that the BH would then have to be as massive as a big mountain. That's A LOT of energy. The LHC is nowhere near that level.

Answer 3

While it's not recommended, your comments cover enough ground that rather than make a long answer, I'll address them here. If my answer is unsatisfactory, you might want to create a new question, clarifying the specifics.

A star 3 or 5 times the size of our sun will gradually burn up its hydrogen and collapse to form a black hole big enough to swallow the earth

Stars don't operate that way. During the final stages, (red giant, helium flash, etc), stars lose a significant percentage of their matter into space. Our sun is expected to lose about half it's mass by the time it becomes a white dwarf. larger stars lose an even higher percentage so you need a much more massive star to end up with a black hole. About 8 solar masses to end up a neutron star and about 25 solar masses (same link) for a black hole, though a neutron star can accrue mass and become a black hole that way too if it's in a binary system.

I realize that's not precisely what you asked, but one of the problems with compression on this scale is you have to ask what you are compressing. Compressing fissionable materials can lead to explosive events that would resist the compression.

If you use iron, which won't create any new energy upon being very tightly compacted, then you have a relatively straightforward mass to size problem where you can estimate the pressure. 3 solar masses of iron should be sufficient to form a black hole by its own mass and gravity. It might even be a bit less than that, as it's never been observed and would vary some with speed of rotation and temperature. But 3.0 solar masses is close enough.

The force of gravity needed to crush the star to a singularity is $10^x$ joules

This isn't straightforward because it would vary with the initial mass you wanted to force into a black hole and the math gets very complex, involving equations based on the Pauli exclusion principle and neutron degeneracy pressure.

But, in general, with enough mass, no force is needed, because the gravity does all the work itself. There is no energy expenditure with 3 solar masses of iron. There's a huge (enormous) gravitational force, but no energy.

With less mass, the force increases because a smaller Schwarzschild radius requires the Neutrons and quarks to get even closer, and the energy of such compression is enormous and hard to calculate.

Fortunately, there is a way to cheat and get a very simplified answer. The degenerate pressure that keeps the neutron star from becoming a black hole is in balance with the gravitational binding energy and gravitational binding energy can be estimated, assuming uniform density,

$$U=\frac{3GM^2}{5R}$$

(Source).

So, let's take a millionth of a gram of stuff, something heavy, like a piece of iron dust to avoid any fission energy that would work against black hole formation, and use our handy Hawking radiation calculator.

A one millionth of a gram black hole would have a radius of about $1.5 \times 10^{-36}$ meters (calculator above), so the gravitational binding energy, roughly equivalent to the neutron degeneracy pressure at Schwarzchild radius:

$$\frac{3 \times 6.67 \times 10^{-11} \times 10^{-9}}{5 \times 1.5 \times 10^{-36}} = 2.67 \times 10^{15}$$

or 26.7 million billion joules, crammed into a space smaller than an atom, just to make a 1 millionth of a gram black hole that should evaporate in a fraction of a fraction of a second. Now, there's probably a thousand reasons why my estimate is wrong, but it's ballpark (a very big, Texas-sized ballpark), but still in the ballpark.

Your estimate for CERN is that it uses 800 million watts. Now, remember, CERN accelerates beams of protons, thousands if not millions, and at current peak energy, each individual proton carries a maximum of about 13 trillion electron volts, and there are about $6.24 \times 10^{18}$ electron volts in a joule, so each individual CERN collision, proton onto proton has about 1/48,000 joules, at maximum power.

Now, even with my Texas-sized ballpark estimate, 1/48,000 is a much smaller number than 26.7 million billion. When the numbers are that far off, "bollocks" kinda covers it.

And a millionth of a gram black hole would evaporate anyway. It would be much smaller than an atom and would have a hard time absorbing much of anything. Not to mention the creation of such an object would probably have an initial velocity well above escape velocity and it would simply fly away from the earth (or right through it and away - like a heavy neutrino).

There's no estimate that creates a dangerous scenario using CERN's 1/48,000 joules per collision energy, and for an individual collision, that's enormous, but it's nowhere close to dangerous, unless you make the mistake of standing in the beam (that happened to somebody once, but...I digress). He lived by the way.

CERN can't put anywhere close to all 800 megawatts of energy into the acceleration of one proton. If they could, we might discover some very interesting things, but they can't even come close. All that energy powers the magnets and the vacuum and keeps everything cool, but the individual proton energy is a tiny fraction of that.

So, depending on what size black hole you want to create, CERN would need to be, oh, maybe a billion billion billion times more powerful than it currently is, and even so, the black hole created would probably evaporate before it ate anything, or, if Hawking radiation is wrong and it doesn't evaporate it would most likely fly harmlessly through the Earth, or, mostly harmlessly as it might do some harm if it flew through a person.

To create a black hole of low enough temperature to actually be stable, (and theoretically eat the Earth), you'd need, by the Hawking radiation calculator, millions of tons. And clearly we're deep into the impossible if you think CERN can crush millions of tons of iron into a speck the size of an atom. The orders of magnitude of improbability are higher than I can calculate.

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Now, as to the flip side of this question and why it was thought that CERN might be able to create a black hole (so far they haven't), is because quantum black holes (theoretically) obey different rules over very short distances. They wouldn't last long and they wouldn't be dangerous. If quantum black holes can be created in CERN, then they are being created all the time by cosmic rays, on the surface of the moon, in the upper atmosphere of Earth. CERN does nothing that doesn't happen in the upper atmosphere all the time.

But to actually turn a speck of matter like a grain of sand into a black hole, the energy required would be billions and billions of times over what CERN is capable of. I can't give you an exact answer because I don't have the training in quantum calculations, and I'm not sure anyone can, or if they can, it will be very complicated.

In my opinion, "bollocks", covers it. But he could have said "yes, that could happen, if CERN were a billion billion billion billion times more powerful.

There's much easier ways to kill everyone on Earth than making a black hole that eats the Earth. In fact, that might be the hardest way to kill everyone ever thought up.

Hope you don't mind my narrative, but I think the gravitational binding energy approach isn't a bad way to answer your specific questions on energy required.