Almanzo said:
Humanino: Assuming that the initial mass of the micro black hole would be equal to ten thousand proton masses, I calculate a Schwarzschild Radius of 1.2 * 10-50 meter. The impact parameter would not be appreciably larger, because the force of gravity is 1040 times weaker than the electromagnetic force, so the hole would not be able to draw anything in by its gravity. Neither would it be able to polarize or damage anything by its tidal force.
R= GM/c2, where G = 6.67 * 10-11 m3/kg*s2, c = 3.00 * 108 m/s, and M = 1.67 * 10-23 kg.
The above post contains a mistake, which I realized last night. The event horizon would indeed have an apparent radius of 10
-50 meters, but the impact parameter might be quite a bit larger.
Consider an electron in the outer shell of a carbon atom. It is 0.77 * 10
-10 m from the center of the atom, and feels, on average, the attraction of just one elementary charge, because the other charges are compensated for by the other electrons. It would feel a force F = Q
2/4*pi*epsilon*R
2, which comes to 2 * 10
-8 Newton. Dividing this by the electron's mass gives an acceleration of 10
22 m/s
2.
To feel the same acceleration from the gravity of a 10.000 proton mass (whether as a black hole or in any other form), one would have to approach it to a distance equal to the square root of that mass times G, divided by 10
22 m/s
2. This comes to 3 * 10
-28 meter. Still a very tiny distance, but rather larger than the Schwarzschild Radius.
I wonder, however, whether an electron might actually come within this distance of the hole, and what would happen if it did. If the electron is actually a point sized particle, as suggested by the Wikipedia article, it might be simple. The point would be drawn in, and as soon as it crosses the event horizon inwards, it can never cross it again outwards. But if the electron has a "classical" radius of 7 * 10
-15 meter, it would be huge compared to the impact region. Only a tiny part could enter this region, and the force felt by the electron as a whole would be correspondingly tiny. In that case we may be back to an effective size of 10
-50 m for the black hole, although this wouldn't matter very much, because the size of the electron now dominates its probability of encountering the hole.
But there is something bothersome about this disparity in size. What if such a huge electron encounters such a tiny hole? How can more than a tiny part of it cross the horizon? I wouldn't like to envision the hole digging a tunnel through the electron, or taking just a nibble of it. Electrons are, after all, elementary particles; they may be annihilated by positrons, absorbed into nucleons or swallowed by (large) black holes, but they probably cannot be disassembled. I also would not like to envision the electron being stuck to the black hole, like a huge but nearly empty balloon being glued with superglue to a grain of sand. Because it would be stuck by virtue of being partially inside. No information can however leave the black hole, so the confined part of the electron would have no way to communicate its location to the rest of it.
One might, of course, envision the electron being deformed. To deform it, the black hole's influence would have to overcome the Poincare stresses keeping it in shape, and these would probalby be 100.000 squared, or ten billion times stronger than the force keeping it in orbit around the nucleus. This could only happen in a radius of 10
-33 meter around the hole, which would therefore not be able to deform the entire electron.
Anyway, if the entire electron were to be deformed, one might envision two ways of doing so. It might assume an oblate "pancake" shape, which would (seen from afar) seem to wrap itself around the event horizon. But it would have to envelop the horizon kazillions of times, self-intersecting and becoming as thin as gossamer. The tidal force (the inhomogeneity of the gravity field) would in any event suggest an opposite effect; the electron assuming a prolate "cigar" shape. But a cigar slender enough to pass as a whole through the event horizon would have to be slender indeed. The word "needle" would be more appropriate, and one would have to think of a needle which, if it were scaled up to stretch across the observable universe, would still have a waist of subatomary size.
Both deformations would greatly increase the mass of the classical electron, by intensifying the electric field near its surface. Considering this, I realized that if a 10
-50 meter size black hole were to acquire an electron charge, the electric field around it would have a mass of several kilograms! Which would cause a new event horizon to form, surrounding the best part of the field, at a radius of 10
-22 m around the original hole. And even if the charge were later to be neutralized, the hole would retain this mass. A frightening idea; if this were actually to happen, there would be no "dormant" stage at all.
But I don't expect this to happen, because the hole could not suddenly acquire a charge. What might happen, is that it starts to grow on approaching a charge, and eventually attains the minimum size which a black hole with an elementary charge must have. This minimum size I calculate to be roughly 10
-36 m, corresponding to a mass of a few micrograms.
This leaves however the question where the mass (and hence the energy) came from. And I can see only four possibilities. (1) Energy might not be conserved. (2) Charge might not be conserved. (3) Micro black holes beneath microgram mass can exist, but cannot acquire charge. (4) Micro black holes beneath microgram mass cannot exist.
Personally, I don't believe (1) or (2) to be true. (But it is only a belief, nothing more. After all, people are talking about recreating the conditions of the Big Bang.) I would greatly prefer (4) to be true, but I can see no reason why. If (3) is true, which would be my guess, 10.000 proton mass black holes could be relatively harmless, because even neutrons are assembled from charged particles. They could only grow by encountering each other, and even if the LHC were to create trillions of them, the Sun would be in its grave before they did.
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