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It occurred to me in passing that the Lorentz contraction of a black hole from the perspective of an ultra-relativistic (Lorentz factor larger than about 10^16) particle could reduce the thickness of a black hole to less than the DeBroglie wavelength of the particle.

It would seem to me that under those conditions the particle would have a non-insignificant probability of tunnelling right through the black hole rather than being adsorbed by it.

Is this so? If not, why?

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2 Answers 2

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The important thing is the cross-sectional area of the horizon, and this is independent of Lorentz transformation, since the $y$ and $z$ coordinates are not changed.

Additionally, you can calculate that light will be captured by the horizon with non-zero cross section, and the geodesics ultra-relativistic particles will asymptote to the geodesics of null particles.

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I'm not getting why the cross-sectional area of the horizon matters. We are talking about the probability of the particle tunnelling through a thin potential barrier. The size of the barrier in the directions perpendicular to the direction of motion would seem to be irrelevant. To put it another way - if I were to replace the black hole with a normal matter wall, how would that differ for the question of tunnelling a ultra-relativistic particle through it? –  Benjamin Franz Jan 29 '13 at 17:26
@BenjaminFranz: thinking about it as a potential barrier is a bad way to think about it. In general relativity, the only important thing is the geodesics. You will get messed up if you take the classical picture of energy too seriously. And irrespective of that, the limit to null particles circumvents that issue. –  Jerry Schirmer Jan 29 '13 at 17:29
And above I should say, rather than the "classical picture of energy", "the idea of the energy of a single particle" –  Jerry Schirmer Jan 29 '13 at 17:39
Also, the literal wavelength of an incoming EM field can be pretty easily made larger than the size of the black hole. The absorption cross section will be nonzero. –  Jerry Schirmer Jan 29 '13 at 20:16
I'm not sure we aren't talking past each other. My question is about the quantum mechanical possibility of the particle quantum tunnelling 'past' the barrier posed by black hole without ever actually 'entering' it (ala the Klein Paradox). I'm happy either way - I just want a clear explanation of why it can or cannot. –  Benjamin Franz Jan 30 '13 at 1:32

Light (apart form high energy particles) can't even tunnel through a thin piece of metal let along a black hole!

If you packed enough energy into your particle, perhaps it would release gravitational waves on the way in, so some of the incoming energy would escape.

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