In a given frame S, consider an inelastic collision between a particle A and a fixed target B. In frame S, the relative velocity of A (and thus the kinetic energy) to an observer in frame S is not enough to produce a black hole. However, in another frame S', the relative velocity of A is high enough that the center of mass energy could produce a black hole when it collides with the fixed target B.
Does particle A produce a black hole in frame S? If so, then why can't that logic be extended to any collision, given that, for every frame that witnessess a collision that has a particular center of mass energy, there exists another frame where the center of mass energy is enough to produce a black hole?