Black holes and the Planck length

According to Special Relativity, an object in motion relative to you appears shortened. The faster it is, the shorter it appears. This is effect known as Lorentz-contraction. According to General Relativity, an object that has a sufficiently high mass-density in a small volume collapses to a black hole. Does this mean that if a particle moves fast enough relative to you it turns into a black hole? No, it doesn't. But it's a confusion I've come across frequently. Take Craig Hogan's recent "paper", where he writes:

"[B]elow the Planck length... it is no longer consistent to ignore the quantum character of the matter that causes space-time to curve. Even a single quantum particle of shorter wave-length has more energy than a black hole of the same size, an impossibility in classical relativity..."

The wave-length of a particle depends on the motion you have relative to it. For every particle there is a reference frame in which the wave-length of the particle appears shorter than the Planck length. If it was true what Hogan writes, this would imply that large relative velocities are a problem with classical general relativity. Of course they are not, for the following reasons.

A black hole is characterized by the existence of an event horizon. The event horizon describes the causal connectivity of space-time. It's a global property. Describing an object from the perspective of somebody moving relative to this object is a coordinate transformation. A coordinate transformation changes the way the physics appears, but not the physics itself. It just makes things look different. You cannot create an event horizon by a change of coordinates. Ergo, you cannot create a black hole just by looking at a particle that is moving rapidly relative to you.

There are three points I believe contribute to this confusion:

First, one can take the Schwarzschild metric for a black hole and describe it from the perspective of an observer moving relative to it. This is known as the Aichelburg-Sexl metric. The Aichelburg-Sexl metric is commonly used to handle black hole formation in particle collisions. The argument about the Planck length being a minimal length makes use of black hole formation too. But note that in these cases there isn't one, but at least two particles. These particles have a center-of-mass energy. They create a curvature which depends on the distance between them. They either do or don't form an horizon. These are statements independent on the choice of coordinates. This case should not be confused with just looking at one particle.

Second is forgetting that black holes have no hair. Leaving aside angular momentum, they're spherically symmetric which implies there are preferred frames. Normally one uses a frame in which the black hole is in rest, which then leads to the normal nomenclature with the Schwarzschild radius and so on.  But you better don't apply an argument about concentrating energy inside a volume that you'd have in the static case to the metric in a different coordinate system.

Third is a general confusion about the Planck length being called a "length". That the Planck length has the dimension of a length does not mean that it behaves the same way as a length of some rod. Neither is it generally expected that something funny happens at distance scales close by the Planck length - as we already saw above, this statement doesn't even have an observer-independent meaning.

The Planck length appears in General Relativity as a coupling constant. It couples the curvature to the stress-energy tensor. Most naturally, one expects quantum gravitational effects to become strong, not at distances close by the Planck length, but at curvatures close to one over Planck length squared. (Or higher powers of the curvature close to the appropriately higher powers of the inverse Planck length respectively.) The curvature is an invariant. This statement is therefore observer-independent.

What happens in the two particle collisions is that the curvature becomes large, which is why we expect quantum gravitational effects in this case. It is also the case that in the commonly used coordinate systems these notions agree with each other. Eg, in the normal Schwarzschild coordinates the curvature becomes Planckian if the radius is of Planck length. This also coincides with the mass of the black hole being about the Planck mass. (No coincidence: there is no other scale that could play a role here.) Thus, Planck mass black holes can be expected to be quantum gravitational objects. The semi-classical approximation (that treats gravity classical) breaks down at these masses. This is when Hawkings calculation for the evaporation of black holes runs into trouble.

For completeness, I want to mention that Deformed Special Relativity is a modification of Special Relativity which is based on the assumption that the Planck length (or its inverse respectively) does transform like the the spatial component of a four-vector, contrary to what I said above. In this case one modifies Special Relativity in such a way that the inverse of the Planck length remains invariant. I've never found this assumption to be plausible for reasons I elaborated on here. But be that as it may, it's an hypothesis that leads to consequences and that can then be tested. Note however that this is a modification of Special Relativity and not the normal version.