Do two massive particles flying close past one another near the speed of light cause a black hole? [duplicate]

Question

Not too long ago, at CERN, protons with relative a huge energy were smashed into one another to find out about a "large" dimension in string theory. It was conjectured that if gravity could leak into this extra dimension, a mini black hole could be created with attainable energies. The two protons, upon impact, could turn into one. No black holes were detected.

Now, assume the protons have enough energy to form a black hole. What if the protons just miss each other and fly past one another close by? Can they momentarily form a black hole in which they get stuck subsequently? A short while a high amount of energy is concentrated in a small volume.

The same holds though for a single proton though, and in that case no hole will come to be. You can always imagine a rest frame for the proton, but not for the both at once.
A moving proton can cause frame dragging. Two oppositely moving protons flying past one another closely, will give opposite frame dragging. But will their combined energy cause a black hole?

As a side note, the combined kinetic energy depends on the frame of reference. In the COM, both protons have a smaller kinetic energy as seen from one of the protons. The kinetic energy seen from the (laboratory) COM is half the kinetic energy as seen from one of the protons.

The question of which this is a supposed duplicate doesn’t refer to quantum particles, which the proton is. In too of that, Philip Gibbs, in this article, says it's not the case that a black hole forms. (Thanks to @safesphere!)

Answer 1

The answer is that yes two very energetic particles colliding with each other could form a black hole, though unless large compact dimensions exist the energies needed are many orders of magnitude larger than are currently possible with modern colliders. In fact the particles could even be massless photons, in which case the black hole formed is known as a kugelblitz.

Whether kinetic energy does or doesn't contribute to the spacetime curvature around an object is a common source of confusion for beginners to relativity. The confusion arises because the curvature is related to an object called the stress energy tensor not just the mass.

There is a useful discussion of the stress energy tensor for point particles in the question: Intuitive understanding of the elements in the stress-energy tensor. In this context the important fact to remember is that the momentum of the particle(s) also contributes to the stress energy tensor.

To remove the contribution from the momentum we need to switch to the centre of momentum frame of the system, and for a single point particle this is the frame in which the KE is zero and therefore does not contribute to the curvature. By contrast for two colliding particles their KE is not zero in the COM frame and therefore does contribute to the curvature. The two situations are qualitatively different.

Now, you ask:

What if the protons just miss each other and fly past one another close by?

But if the two protons come close enough that their separation is less than the Schwarzschild radius due to the total energy (including KE) then they cannot fly apart again. Their trajectories will spiral inwards towards each other and they will inevitable collide. They cannot momentarily form a black hole that then disappears again as the particles fly apart.