To start with, the main reason no one working on the LHC is too concerned about black holes is because of Hawking radiation. While we usually think of black holes as objects that nothing can escape from, Stephen Hawking predicted that black holes actually do emit some light, losing energy (and mass) in the process. In the case of the little bitty black holes that the LHC could produce, they should just evaporate in a shower of Hawking radiation.

That's great you say, but Hawking radiation has never actually been observed. What if Hawking is wrong and the black holes won't evaporate? Well, the usual next argument is that cosmic rays from space bombard the earth all the time, producing collisions many times more energetic than what we'll be able to produce at the LHC. To me, this is a fairly convincing argument. However, let's pretend we don't know about these cosmic rays and that there's no Hawking radiation. We can calculate what effect black holes produced by the LHC would have on the earth if they do stick around.

To start out with, the most massive black hole the LHC could produce would be around 10 Tera-electron-volts, or TeV. We're probably overestimating here. The eventual goal is for the LHC collisions to be 14 TeV, but producing a particle with the entire collision energy is incredibly unlikely (see Tomasso Dorigo's post for more details on why, along with more details than you probably wanted to know about hadron colliders). However, we want to think about the worst case scenario here, and we're just going to do an order of magnitude calculation, so 10 TeV is a good number. Note that I'm using a particle physics convention here of giving masses in terms of energies using E=mc^2. For reference, 10 TeV is about 1000 times smaller than a small virus.

Now from the mass of our black hole, we can get its size by calculating something called the Schwarzschild radius. The Schwarzchild radius for a black hole of mass m is given by

\[r = \frac{2Gm}{c^2}\text{.}\]

Here G is Newton's gravitational constant and c is the speed of light. Plugging our mass in gives us

\[r = 10^{-50} \text{meters.}\]

This is incredibly small! In fact as I was writing this, I realized that it's actually smaller than the Planck length, which means our equation for the Schwarzschild radius may be somewhat suspect. Nonetheless, let's hope that if we ever figure out quantum gravity, it gives us a correction of order one and proceed with our calculation, which is just an order-of-magnitude affair anyway.

Now, anything that enters the Schwarzschild radius of the black hole is absorbed by it. The lightest thing that we could imagine the black hole swallowing is an electron. Let's figure out how long on average a black hole would have to travel through material with the density of the earth before it absorbs an electron. In the spirit of considering the worst case scenario, we'll have the black hole travel at the speed of light, and consider the earth to be the density of lead.

We could do a complicated cross-section calculation to find the rate at which the black hole accumulates mass, but we can also get it right up to factors of pi through unit analysis. We know that the answer should involve the area of the black hole, the density of the earth, and the speed of the black hole. We want our answer to have units of mass per time to represent the mass accumulation rate of the black hole. The only combination that gives the right units is

\[a=\frac{\text{mass}}{\text{time}} = \rho c r^2=\frac{10,000\text{kg}}{\text{m}^3}\frac{3\times 10^8 \text{m}}{\text{s}}(10^{-50}\text{m})^2} = 10^{-88}\text{kg/s}\text{.}\]

Alright, now that we know how fast our black hole accumulates mass, let's figure out how long it takes it to accumulate an electron. The electron mass is

\[m_e = 10^{-30}\text{kg,}\]

so the time to accumulate an electron is

\[t = \frac{m_e}{a} = 10^{50}\text{s.}\]

Now, the current age of the universe is 10^17 seconds. The time it takes our black hole to accumulate an electron is longer than the age of the universe by many orders of magnitude! So, if the LHC produces black holes, and if Hawking is wrong, the black holes will just fly straight through the earth without interacting with anything. Even if we take the size of the black hole to be the Planck length, our black hole accumulates an electron in 10^25 seconds, which is still much longer than the age of the universe.

So the moral of the story is that you should be excited about the new discoveries that the LHC might produce, and you don't need to worry about black holes.

Cool bit of back-of-an-envelope physics with high school maths, nice :)

ReplyDeleteCouple of questions:

If black holes were created, would they still be moving at relativistic speeds, and would that effect the accumulation time?

What about reversing the calculations, what collision energy is needed to create a Planck length horizon, presumably the smallest size it can possibly be?

Well, if they were going slower, they'd accumulate mass more slowly, since instead of c in my equation for the mass accumulation, we'd have some lower velocity. I really did try to consider the worst case scenario.

ReplyDeleteIf you put r = the plack length (10^-35 m) in the equation for the Schwarzschild radius and solve for the mass, you get 6*10^24 eV, or 6*10^12 TeV. The LHC runs at 14 TeV, so we're not getting to those energies anytime soon.

So in ST (Spinal Tap) units, you'd need an LHC that goes up to six trillion (rather than just fourteen as it does at present) if you wanted to (maybe) destroy the planet?

ReplyDeleteObligatory XKCD reference: http://imgs.xkcd.com/comics/spinal_tap_amps.png