Question about relative velocity and black hole formation in a fixed target experiment

Question

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?

Answer 1

If particle 1 (2) has 4 momentum $p^{\mu}_1$ ($p^{\mu}_2$), then in the center-of-momentum (colloquially, the center of mass), the total 4 momentum is:

$$ P^{\mu} = p^{\mu}_1 + p^{\mu}_2 = (E_1, \vec p_1) + (E_2, -\vec p_1) = (E_1+E_2, 0)$$

and its square

$$P^{\mu}P_{\mu} = (E_1 + E_2)^2 $$

is a Lorentz invariant.