Mini black hole not Lorentz invariant?

[jmd_dk]

Let's say that we have a particle flying through space, at a collision course with a planet. As seen from an observer on this planet, the particle has an enormous energy, and its wavelength is just slightly bigger than the Planck length. As the particle falls down the gravitational well of the planet, it gets blue shifted enough to gain the last bit of energy, and the particles wavelength shrinks below the Planck length, which transforms the particle to a black hole. The person on the planet observes this black hole.

The whole scenario is also seen by a different observer, flying with great speed, also toward the planet, in the trajectory of the particle. In her system, the wavelength of the particle isn't that close to the Planck length, and the gravity of the planet surely isn’t enough to transform it into a black hole.
Now what does she see? Will the particle behave “nice” in her system, and not turn into a black hole? In that case, two contradictory histories are being observed. On the other hand, if all observers agree on the fact that the particle has turned into a black hole, that would be very weird indeed (because I can always make up some system for which the wavelength of the particle is less than the Planck length).

Any ideas about how to resolve this?

 

[Haelfix]

This should be in the SR/GR forum.

So one problem with this statement, is that it does not follow that a particle must turn into a black hole if 'its wavelength shrinks below the Planck Length'.

More generally it is extremely difficult to give a precise invariant statement in GR about when a black hole forms at all. Blackholes are global objects that involve event horizons, yet you are talking about a highly localized interaction. Of course the answer is clear in this case, no black hole will form simply by appealing to the principle of relativity..

Anyway, the most common statement for the conditions about when a bh forms goes by the name of the hoop conjecture, whereby a bh forms if enough stress energy is confined by a bounding region that is smaller than its Schwarzschild radius. This innocuous statement however is frought with mathematical subtleties.

A slightly more precise (but still not settled) formulation of this conjecture involves analyzing how gravitational shock waves form closed trapped surfaces.

 

[PAllen]

Another point is that a BH is frame and coordinate independent feature of the manifold. Thus, if something is not a BH in coordinates built from a frame at rest relative to the object, it is not a BH in any other coordinates.

While, as with much in GR, it is hard to be precise about this, terms in the stress energy tensor representing momentum and KE of the body as a whole end up not contributing to 'amount of curvature'.

 

[jmd_dk]

So one problem with this statement, is that it does not follow that a particle must turn into a black hole if 'its wavelength shrinks below the Planck Length

Are you sure? Can I just put an arbitrarily large amount of energy into an elementary particle (shorten it's de Broglie wavelength), without it becoming a black hole? If I somehow created a particle with energy E, and confined it within a radius r < 2E, why wouldn't it turn into a black hole?

Aparrently, such a particle is called a Planck particle:
http://en.wikipedia.org/wiki/Planck_particle

 

[humanino]

I think you should drop the BH question, because their is a simpler situation in which you may study the concepts of energy in GR vs QFT : does a falling electron radiate ? It is a classic question.