That would violate relativity. If there's no material around to define a laboratory frame, then you're free to transform into the rest frame of the particle. If it's a stable particle, it'll just sit still in that frame. Particle showers happen because two things with a high relative velocity interact.

Said another way, velocity isn't a property of an object. It's a property of a relationship between two objects (one of which might be an abstract reference frame, but since that only exists in our heads, the particle won't interact with it!).

It is supposed that in vacuum we have the so called 'virtual particles' that live so small time so we cannot identify them. So, why not? Vacuum is not the empty set...

Demetris: The vacuum state is Lorentz invariant. So if it happens in a vacuum with a moving particle, then it will also happen in a vacuum with a stationary particle. Which means you're just talking about normal decay of an unstable particle, which is not the same thing as the process that's called a particle shower. While you can think of radioactive decay as the interaction of a particle with the vacuum-state fields it's sitting in, referring to this as a "particle shower" is just not consistent with standard terminology.

Just because the vacuum isn't exactly nothing doesn't mean that the amplitude for colliding with it is nonzero. And it's easy to show that any nonzero answer is inconsistent with relativity.

Dear Bryan, can you formally define the concept of 'vacuum'?

@ Bryan - ok, this makes sense. As long as one can transform the observation to a rest-frame. In that respect I have some other questions:

- suppose there are two particles that move with near-c velocity relative to one another. I understand that a particle shower could happen ONLY if the particles were made to interact with each other ... otherwise one could put his lab-frame on either of the particles and argue that it's the other one that's moving with near-c velocity and it's the other one that should burst. Do I understand it right?

- regarding initiating a particle burst by a vacuum fluctuation - this would be possible ONLY if there was absolute space (and respectively 'pinned' vacuum fluctuations) to measure the velocity of the particle against - right?

- could a particle shower be initiated in an environment with critically high gravity?

As in - (if it is possible to attain that high a curvature of space-time outside of an black-hole's event horizion) could space-time curvature cause particle showers in such extreme conditions?

It is interesting, but I think "Yes" is also a correct answer.

In a strong electric field 1010-1012 V/m of vacuum gap over the polar cap surface of neutron star, the pair-production processes (gamma -->e+e-) in the strong magnetic field ~1012 G of neutron star produce an avalanche of e+e- particles like a particle shower.

However, by definition particle showers (cascade showers) is a result of multiple interactions of secondary particles in matter, for instance Extensive Air Showers (see wiki).