Black Holes at the LHC - What can happen?

Black holes sound like dangerous beasts, evoking voracious vacuum cleaners that suck in everything in their neighbourhood. If they are created at the LHC, that may mean the end of the world - or so one might think, with an intuition gained mainly from the exposure to special effects of science fiction movies.

The reality is more prosaic.

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 m², or 10 nanobarn (that's small but more than typical neutrino cross sections).

What happens if such a "naive" black hole passes through the Earth?

For simplicity, we can assume that the Earth is made up of iron, with a density of 8 g/cm³, or 8000 kg/m³. Since mass is essentially the mass of nucleons (the protons and neutrons in the nuclei of the atoms), and taking into account the proton mass, this density corresponds to a density of 5×10³⁰ nucleons per m³. But this means that on average, the black hole would travel 200 km before encountering a nucleon (200 km travel distance × 10^-36 m² cross section × density of nucleons of 5×10³⁰ nucleons per m³ equals one nucleon).

Nobody knows exactly what will happen when a tiny black hole hits a nucleon. On the scale of the black hole, the nucleon is about 1000 times larger in diameter, and a very dilute cloud of a few quarks and gluons. It may be that the black hole hits one of these partons, as they are called, thus disrupting the nucleon and carrying away a fraction of its mass. There is no theory to describe this, and there are all kinds of problems involved, as to what happens to confinement, colour neutrality, and so on. But whatever happened, in the end, the black hole may have gained, in the most extreme case, the mass of a nucleon.

Now, this is just one permille of the mass of the black hole, so it won't change its momentum, and just travel on along a straight path. After 100 encounters or so with nucleons, on average, it will have left the Earth, even if goes right through its centre. In addition one should keep in mind that scenarios in which a black hole can reach a thermodynamically stable endstate (though these scenarios are strongly disfavored for theoretical reasons), once the black hole would gain some mass it would no longer be in this stable endstate, and evaporate again until it reaches again its initial mass.

If the black hole starts at rest on the Earth's surface, it will fall through the centre of the Earth and engage in the oscillatory motion that is set as a problem in undergraduate textbooks, dealing with the free fall through a tunnel across the Earth.

But does the black hole start at rest, in the first place?

In fact, if the initial velocity of the black hole is larger than the escape velocity of the Earth's gravitational field, it will just escape into space. The escape velocity for the black hole is 11.2 km/s, the same as for any ordinary canon ball or satellite. This corresponds to a value of β = v/c ≈ 0.00004. That's tiny for typical high-energy collision kinematics, and most black holes produced will easily exceed this number by orders of magnitude. Earth' gravity can not trap these black holes created in the collision.

Even though the centre of mass frame of the colliding protons at the LCH is at rest with respect to the detectors and the surface of the Earth, this is generally not the case for the pair of colliding partons that creates the black hole. As a consequence, the black holes will have quite large momenta along the beam axis. Technically speaking, this momentum is expressed by a variable called rapidity y, where y = arctanh β. The black hole can become trapped in the Earth's gravitational field only unless its rapidity exceeds 0.00004 (for such a small argument, arctanh β is pretty much the same as β).

Now, here is a plot of the rapidity distribution of black holes from collisions at the LHC, assuming large extra dimensions.

Figure 5 from Black Hole Remnants at the LHC by Benjamin Koch, Marcus Bleicher and Sabine Hossenfelder, JHEP 0510 (2005) 053 (hep-ph/0507138)

Although this is not the initial rapidity distribution of the nascent mini black holes, the essential feature of the plot is clear: the typical rapidity of the black holes is of the order 1. For comparison, the rapidity of the 7 TeV colliding protons is 9.6. This means that just about one in 100000 tiny black holes produced will have a velocity along the beam axis smaller than escape velocity. But even these black holes will have some initial velocity in the direction perpendicular to the beam axis. This velocity is usually expressed by the "transverse momentum", and typically, it is also much bigger than the escape velocity. There are no tiny black holes created "at rest" in the detector!

In short: If tiny black holes were produced because large extra dimensions did exist in the necessary number with the necessary radius, and if they did not evaporate within 10^-26 seconds as expected (Hawking evaporation is considered a very robust prediction, so this scenario is not confirmed by well founded theories), most of them would have such a high velocity that they escaped the gravitational field of the Earth for good. Even if they travelled straight through the centre of the Earth, the few nucleons they can hit wouldn't change their momentum in an appreciable way.

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Tags: physics, LHC, black hole