Is the Planck mass the minimum required to form a black hole?

Question

I have heard this stated before, even that this is reason elementary particles do not collapse into a black holes no matter how much they are compressed and why we don't need to worry about them being created inadvertently by a particle-accelerator destroying this planet. Has this actually been proved? If so how?

Answer 1

There's a pop-science myth going around that Planck units are somehow known to be fundamental to physics. Like many myths, it's based on a kernel of truth: there are reasons to suspect that at the scale where things tend to be measured in Planck units, some new physics must appear (in particular that quantum gravitational effects will become significant). But no experiment has ever actually shown this, and there is no actual evidence that there's anything special about the Planck length, Planck time, or Planck mass. So far as we know, they're just units like meters, seconds, or kilograms -- the only advantage they have (so far as we know now) is that they can be defined without reference to arbitrary human scale constants.

Answer 2

What matters for creating a black hole is the density not the mass. This discussed in What is exactly the density of a black hole and how can it be calculated? though the equation for the density required is not especially informative:

$$ \rho = \frac{3c^6}{32 \pi G^3 M^2} \tag{1} $$

So to make some mass a black hole, and this applies whether or not it's a fundamental particle, we need to compress it until its density reaches this value.

For a fundamental particle we "compress" by localising it. Quantum objects are always delocalised over some region of space and their density is determined by how delocalised they are. In principle we can localise a particle to increase its density, but there is a wrinkle with this.

When we localise a particle we always increase its energy. The reason for this is a little involved, but a handwaving explanation is that the uncertainty principle requires:

$$ \Delta x \Delta p \ge \frac\hbar2 $$

Localising the particle means decreasing $\Delta x$ so $\Delta p$ has to increase, and that increases the energy. But Einstein's famous equation tells us that $E = mc^2$ so as we localise the particle and increase its energy we are increasing its mass and therefore increasing its density. That means at some point the density will increase above the critical value given in equation (1), and at that point the particle will form a black hole.

Note that it doesn't matter what particle we start with. Even for a neutrino the act of localising it would increase the mass and density.

Anyhow, we can calculate what size the collapse happens at, and it turns out to be about a Planck length, and the mass at which it happens is about the Planck mass. I say "about" because we would need a theory of quantum gravity to be sure what happens and no such theory exists at the moment.

Finally, you ask about experimental tests but the energies involved in this process are vastly greater than anything we can achieve in colliders. Unless some unexpected rears its head, e.g. large extra dimensions, we are not going to be experimentally testing this for the foreseeable future.