Ok, being sick on my Saturday night, I figured I'd finally respond to this. I've been watching it for quite a while without saying much, but I kinda' feel like I should at this point. It may also be because I'm bored as hell.
First off, I'm going to show from firmly established (by that I mean experimentally verified) evidence that all this black hole creation stuff at the LHC cannot happen. Just in case you guys don't trust me, I'm an astrophysics student at U of I in Champaign-Urbana, I do problems like this all the time. Now that I've gotten that out of the way, I'll describe the thought experiment.
So what we want to know is the following. Could we, given optimal black hole creation conditions actually create one in such a way that it could destroy Earth? And if it were to happen, what would the experience entail?
The calculations follow primarily from relativistic mass-energy equivalence, and gravitational laws. There's no calculus here, and should be very easy to follow. Feel free to not look at the math and just look at the consequences of the result in the text if you don't want to sludge through the grunt work.
Facts:
The Large Hadron Collider (LHC) has an energy capability of roughly 14TeV, that 14E12 eV
The LHC accelerates protons to an appreciable portion of the speed of light, as indicated by the energies above. This process is essentially the same as particles that are ejected from the sun on a constant basis. These particles are known as cosmic rays. 90% of the time, these particles are protons. These particles bombard Earth by the billions per second.
The average power of a cosmic ray particle is 1E20 eV, this is more than 6 orders of magnitude more energy than the LHC can produce. This means that cosmic ray particles are far more likely to produce blackholes due to collisions in the atmosphere of Earth. So, for the benefit of the doom and gloomers, let's assume a proton with cosmic ray level energy will be collided in the LHC.
Math:
Using relativistic (these particles are basically moving at the speed of light) mass-energy equivalence, we can find the actual speed of one of these particles, and then compare it to the escape velocity of Earth, to see if it could be caught in the gravitational well.
E: energy, p: momentum, m: mass (in this case, the mass of a proton), c: speed of light, r: radius, G: Gravitational Constant, v: velocity, Θ: angle theta, γ: gamma (relativistic relation), t: time
I will include units most of the time for checking purposes
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energy relation: E^2 = (pc)^2 + (mc^2)^2
momentum: p = γmv = mv/(SQRT[1 - v^2/c^2])
escape velocity: v = SQRT[2Gm/r]
Starting with energy and rearranging to isolate momentum we get:
p^2 = (E^2 - m^2c^4)/(c^2)
p^2 = [(1E40 eV^2)((1.602E-19 j/eV)^2) - ((1.673E-27 kg)^2)((3E8 m/s)^4)]/((3E8 m/s)^2)
p^2 = [(2566.404 j^2)/(9E16 m^2/s^2)
p^2 = 2.85E-14 kg^2 m^2 s^-2
p = γmv = 1.69E-7 kg m s^-1 -> γv = 1.01E20 m s^-1
γv = v/(SQRT[1 - v^2/c^2]) = 1.01E20 m s^-1 -> v^2/(1 - v^2/c^2) = 1.02E40 m^2 s^-2
v^2 = [(1.02E40 m^2 s^-2) - (v^2)(1.02E40 m^2 s^-2)]/(9E16 m^2 s^-2)
v^2 = (1.02E40 M^2 s^-2) - (1.13E23)(v^2) -> v^2 = (1.02E40 m^2 s^-2)/(1.13E23) ~ 9E16 m^2 s^-2
v = 3E8 m s^-1 = c
Now we have shown a cosmic ray essentially moves at the speed of light (for our purposes this is close enough, though not entirely true). We want to see that if a collision at these speeds happened, would the particle be caught by Earth's gravity? I know virtually all of you know it couldn't, but this builds on later ideas.
v_escape = SQRT[2Gm/r] = SQRT[2(6.673E-11)(5.97E24)/(6.38E6)] = 1.12E4 m s^-1
So clearly most collisions couldn't be caught by gravity, but, how close to a perfect head-on Θ = 0 collision do the particles have to be to actually be caught? The following is an oversimplification, but can be made for our purposes. The idea is that a head-on collision has an angle of 0 degrees between the particles, and is inelastic (the particles stick together). A complete deflection, or glancing blow, has an angle of 90 degrees, and results in a complete retention of initial velocity (the particles still stick, but the angle in the frame of Earth is such that the velocity with respect to Earth is the same as the initial collisional velocity.
(v_particle)(sin Θ) = (v_Earth frame)
Rearranging for Θ
Θ = sin^-1(v_Earth frame/v_particle) = sin^-1(1.12E4/3E8) = .002 degrees
So the two particles would have to be on the same but opposite direction trajectory to within .002 degrees of each other to have a collision such that the resulting Earth frame velocity would be less than or equal to the escape velocity. The chances of this are ridiculously, obscenely low. However, let's assume it did happen, and it floated on down to the core of Earth. What would it's gravitational reach be? In other words, what is the distance at which it would be guaranteed to catch a particle that passed near it, and hence be able to grow in size and eventually consume Earth? This radius is called the Schwarzschild radius.
r = (2Gm)/(c^2) = 2(6.673E-11)(1.673E-27)/(9E16) = 4.96E-54 m
This length says that 2 proton mass black hole essentially has no gravitational radius at which it can trap anything. just for comparison, the length scale at which gravity and quantum mechanics interface is 1.62E-35 m. That is 20 orders of magnitude smaller. TWENTY, it would be impossible for a black hole of this size to do any damager whatsoever. However, let's just say it might possibly be able to get lucky and start a snowball effect. How much time would a black hole of this size have to grab something before it evaporated away? Well, to be honest, this part is theoretical but it is showing promise through other experimental data. What I'm talking about is black hole thermal emission, or Hawking Radiation. The time scale is denoted by "t".
t_evaporate = (5120)(2)(3.14)(G^2)(m^3)/(1.055E-34)(c^4)
t_evaporate = [(5120)(3.14)((6.673E-11)^2)((1.673E-27)^3)]/[(8.1E33)(1.055E-34)]
t_evaporate = (18822.72)(4.453E-21)(3.75E-80) ~0
Well, all of this together proves it. Even if somehow the black hole was created at rest, it couldn't gobble anything up to get bigger. Even if it somehow could, it would instantaneously evaporate before it could get anything. So, absolutely nothing will or can happen. Even if it could, the energies of the LHC are so far below even this example, it's absurd to believe otherwise. One more thing, the gravity of neutron stars is far more intense than Earth's, if this kind of thing happened. We would have seen pulsars just up and disappear, or similar things of that nature as they were eaten from the inside out. The fact is, we don't see these things happening. So rest easy everyone!
If you made it this far, and listened to all this, I am truly grateful. I also didn't have the energy to proof read this, so I apologize for any grammar errors. Good night all.
