Part 2: Black holes

Normal black holes are astromical bodies formed in a manner of different ways. Stellar collapse is perhaps the most well known of these: when stars with a mass greater than about 2.71 solar masses (2.7 or more times as massive as the sun) die, they collapse. Even neutronium - the stuff of which neutron stars are made, the densest possible state of 'ordinary' matter, a substance at least, and up to 100 times as1 dense as an atomic nucleus - cannot withstand gravity on this scale. Matter is crushed beyond all sane limits, vanishing away and hiding itself beneath an event horizon as its density approaches infinity, the point at which mathematics rends asunder and space-time follows suit, leaving only a window unto the deepest of oblivions. The process has been known to cause sensible discussion to degenerate into pseudo-Lovecraftian prose, so try to stay out of the way.

There is, of course, no chance whatsoever of the LHC creating such an 'astronomical' black hole. To reach that kind of energy level, you first need to go through the energy levels including the "destroy the equipment being used" energy level and the "obliterate the entire planet" energy level. The "real worry" is that the LHC could produce one or more Micro Black Holes, or MBHs. A black hole can have essentially any mass, and is desrcibed as 'micro' (or 'mini' or 'quantum mechanical') if it is so small that quantum mechanical effects are significant to it2 - that is to say, at or below an atomic scale3 - let's say 0.1nm (the diameter of an 'average' atom)4. This is 10-10m.

Now, forming a black hole is a matter of compressing something beyond a limit called the Schwarzschild Radius. At this point, no known force of repulsion or exclusion principle can prevent the object continuing to collapse until it forms a gravitational singularity5, 6. This distance is also the distance from the resulting singularity out to the event horizon (the point beyond which nothing can return, and all possible paths (routes a particle can take) lead towards the singularity).

So, a MBH has a diameter ≤ about 10-10m, meaning its Schwarzschild Radius is ≤ 5 × 10-11m. The Schwarzschild Radius, or RSh is proportional to the mass, so we can find out how massive a MBH might be. The relationship is:
RSh = 2MG ÷ c2
Where M is the mass, G is the gravitational constant, 6.67300 × 10-11m3kg-1s-2,7 and c is the speed of light. Therefore:
M = (RSh × c2) ÷ 2G
M = (5 × 10-11m × (3 × 108)2m2s-2) ÷ (2 × 6.67300 × 10-11m3kg-1s-2)
M = 3.3718 × 1016kg

That is quite a lot. It's hard to put it into perspective, because http://www.sensibleunits.com/ has no suggestions, although continuation of the trend would indicate "6100 teaspoons of neutronium." Still not very meaningful. A little research indicates that the oceans weigh* in at 1.4 × 1021kg8. This would mean our MBH has a mass of, very roughly, one-fiftythousandth of the earth's oceans.

* - Yes, very bad pun. Not initially intended, but... not changed either.

Now, this large MBH is clearly (if you don't think so, keep reading, it'll become apparent) far too massive to be produced at any particle accelerator. The question is - could ANY black hole, regardless of size, be produced at the LHC?

This question can be rephrased as: can the smallest possible black holes be created at the LHC? Or further still, down to simply: what is the smallest size a black hole can have?

A black hole is, essentially, the result of compressing a quantity of matter past a certain limit. The limit is based on the mass of the object. If there is a certain "minimum mass" that an object can have, then the smallest black hole is one with this mass, compressed past the relevant point. Alternatively, there could be a smallest length - in which case the smallest black hole is one with this length as its Schwarzschild radius. The first suggestion may initially seem more sensible, however, both are pretty much equally valid ways of approaching the matter.

We need to find a minimum natural value. Natural values are those values that are absolute, and can be derived from nature. The best example is the speed of light, c, which is the maximum speed of the universe, at 3 × 108ms-1. Another is the gravitational constant, 6.67300 × 10-11m3kg-1s-2.

The value we want is either mass or length. Taking the two constants above, we can get a natural value in m2kg-1s-1, by dividing G by c, or in m-2kg-2s by dividing c by G, or in m4kg-1s-3 by multiplying them together. To get a simpler value, we can take the first one of the above and cube c, so that G ÷ c3 = 2.47148 × 10-36 kg-1s.

To get a natural unit for length, this needs to be altered. This requires that another natural unit be introduced: Planck's constant. While pretty much everyone is at least a little familiar with the speed of light, and most people have at least heard of the gravitational constant, Planck's constant - though no less significant than either of these - receives little recognition outside of physics. This may be because it is an associate of the shady realm of quantum mechanics, something that most people shy away from (possibly because extended exposure to it causes people to go a little bit loopy). So, a quick introduction: constant, meet reader. Reader, this is Planck's constant, symbol h, value equal to 6.6262 × 10-34. Planck's constant is the relationship between the energy of a photon (light particle) and its frequency; generally written E=hf or E=hv, depending on location and preference (I will use f, as v has many other uses and is less easy to associate with frequency).8 Planck's constant is equal to the energy of any given photon divided by its frequency. Energy is measured in Joules - in SI terms, kgm2s-2, and frequency is oscillations per second (s-1). So the unit for Planck's constant is kgm2s-1.

So our fundamental unit of area can be found: h × G ÷ c3, in m2. The fundamental unit of length - the Planck length, or lp - is only a square root away, giving
lp = √ (h × G ÷ c3)
    = 4.050 × 10-35m.

Now, here I should go to the trouble of doing the same thing for mass. For now, though, let it simply be said that the Planck mass is equal to 5.456 × 10-8kg (if you're wondering, then yes, there are more units named after Planck - for example, the Planck time is the time it takes light to travel the Planck length).

So, taking the Planck length as the Schwarzschild radius, what mass can we expect the resulting black hole to have?
M = (RSh × c2) ÷ 2G
M = (4.050 × 10-35 × (3 × 108)2) ÷ 2 × 6.67300 × 10-11
M = 2.7292 × 10-08kg.
As you had probably already guessed it would be, this is similar to the Planck mass as stated above.* And, as with the larger-end micro black holes discussed above, this is far too massive to be produced at the LHC.
*Note: it's much more similar if I multiply by 2 (or don't divide by two). If anyone can explain this, it would be much appreciated. I've checked using http://fall.cerrocoso.edu/studenthelp/astronomy/Schwarzschild/schwarzschild.htm, so I don't think this is a rounding error or misuse of excel. Is this just one of those 'near-enough' things?

To demonstrate this, consider:
E = mc2
E = 2.7292 × 10-08 × (3 × 108)2
E = 2,456,277,000 J
And you may recall from part one that the LHC can summon up one-millionth of a Joule. Or you could run this the other way, and find that the Schwarzschild radius of the LHC collsion energy is 1.647 × 10-44m. Particle collisions have what you might call an "accuracy" - the separation of the beams when they collide (head-on collisions being essentially impossible). To collide beams with enough accuracy to form even a possible (ie planck mass minimum) small black hole would require an accelerator as wide as our galaxy is thick9, 10.

It does remain a possibility that the LHC will produce micro black holes, however. Indeed, the Opposition likes to point out at every opportunity that

"If the scale of quantum gravity is near a TeV, the LHC will be producing one black hole (BH) about every second."11
This requires that there exist large dimensions in addition to the four with which we are familiar, and while this may seem to be less than likely (or sane), it is actually predicted by some versions of string theory. Indeed, the prediction of 'one black hole per second' can be seen as sensible. So, are such black holes a threat?

Time to quickly introduce Hawking radiation. Although never observed, most modern models of the universe predict the existence of what sounds impossible: black holes emitting radiation, and hence losing mass. The simplest description is thus:

  1. A vacuum cannot be completely empty, as then it would violate the Uncertainty Principle by having both a known state (0) and a known rate of change (0). Therefore, what looks like a vacuum in fact contains (the mathematical equivalent of) pairs of 'virtual' particles that pop into existence, each time as a particle and its antiparticle, then collide with each other and are annihilated, the energy they had to 'borrow' from the vacuum in order to exist being thusly repaid.
    Note: this is one explanation for the Casimir effect12. Less particle pairs fit between the plates than exist outside them, creating the inward force.
  2. At the event horizon of a black hole, it is plausible that such a pair could appear, one fall into the black hole and the other escape (or a mathematical equivalent thereof).
  3. The two particles cannot meet and annihilate. Their mass cannot simply come into existence unpaid for, so the black hole effectively has to cough up for them. While it has swallowed one, and hence gains back what it paid for that one, the other has escaped.
  4. The black hole loses mass equal to the mass of the escaped particle.
The black hole shrinks as it loses mass, eventually disappearing in a puff of radiation. For larger black holes, the effect is small. For smaller black holes, there is a greater chance of particle pairs being split up by the black hole rather than both swallowed/both escaping. The end result is that the observed temperature of a black hole (assumed to be in completely 'empty' space)
= (h × c3) ÷ (16 × π2 × G × M × k). 13
k is Boltzmann's constant, 1.3807 × 10-23 Joules per Kelvin 14.

So, assuming the LHC can create the Planck mass black hole discussed above, how long would such a black hole last? Well, the energy emitted from a (perfect) blackbody radiator such as the black hole is calculated using the Stefan-Boltzmann law, which says the energy emitted per unit area per second (P ÷ A) equals the fourth power of the absolute temperature multiplied by a constant σ, equal to 5.6703 × 10-8 J s-1 m-2 K-4. 15

So our black hole has a temperature T:
T = (h × c3) ÷ (16 × π2 × G × M × k)
  = 4.7600 × 1030 °K

And from this temperature, the energy emitted each second can be calculated.
P = A × σ × T4
  = A × 2.9111 × 10115

A = 4 × π × r2
  = 2.05794 × 10-68 m2

P = 2.05794 × 10-68 × 2.9111 × 10115
  = 5.99 × 1047 Watts

The calculations done above say such a black hole's mass is equal to 2,456,277,000 Joules. At the calculated rate of energy loss, a Planck mass black hole would evaporate in 4.1 × 10-39 seconds.

It is worth noting that the black hole is highly unlikely to be able to eat anything in this time, because:

  1. As the Opposition has claimed, the black hole would form nearly stationary relative to the earth, and inside the collider is a vacuum.
  2. The black hole has insufficient mass to attract any nearby material. For the black hole to gain mass, not only would Hakwing radiation have to be weak enough for the black hole to reach the chamber wall, and then be lucky enough to hit an atomic nucleus - solid matter is mostly empty space, of course.*

The Opposition do not see this as an obstacle. Indeed, one of their favourite hobbies is to point out that Hawking radiation has never been observed and is purely theoretical. This merely shows that they care more for fearmongering than for good science. They are perfectly willing to accept large new dimensions, based on nothing but some theories, but won't accept Hawking radiation despite most theories predicting its existence? Sure, there may be a physical model of the universe in which there are large new dimensions and no Hawking radiation, but since none of the Oppositon have constructed such a model, claiming that one exists and that everyone should act as if it does (and is correct) is akin to claiming that it is dangerous to get out of bed, as it risks destabilising the Atlantean HyperMatrixTM.

Really, micro black holes are probably the biggest risk in using the LHC. And as you can see, it isn't really a risk at all - no more than destabilising the Atlantean HyperMatrixTM is, anyway. There are, of course, even tinier risks. What are they, and should we be worried?

⇐ Back to part one                                                                                                   Forward to part three ⇒

Bibliography

1 - http://www.sciencedaily.com/releases/2008/01/080114162455.htm
2 - *sigh* I tried to avoid this, but straight information on MBHs is difficult to find on the intarwebs. If anyone knows of a better source for this,
please tell me. http://en.wikipedia.org/wiki/Micro_black_hole
3 - http://en.wikipedia.org/wiki/Quantum_mechanics
4 - http://hypertextbook.com/facts/MichaelPhillip.shtml
5 - http://www.skybooksusa.com/time-travel/timeinfo/schwarzc.htm
6 - http://scienceworld.wolfram.com/physics/SchwarzschildRadius.html
7 - http://scienceworld.wolfram.com/physics/GravitationalConstant.html
8 - http://scienceworld.wolfram.com/physics/PlancksConstant.html
9 - http://en.wikipedia.org/wiki/Micro_black_holes#Creation_of_micro_black_holes
10 - http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/980317b.html
11 - http://arxiv.org/abs/hep-ph/0106295
12 - http://physicsworld.com/cws/article/print/9747
13 - The Universe in a Nutshell
14 - http://scienceworld.wolfram.com/physics/BoltzmannsConstant.html
15 - http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan.html

* - Thanks go to RPGeek for suggesting I include this.

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