Accelerating mass, black holes, frames of reference

I feel like I may have asked this before but search didn’t turn it up. Sorry if this is a repeat.

When you get enough mass together in a small enough space, it collapses into a black hole. (right?)

As an object’s speed increases in a frame of reference, its mass also increases in that frame of reference. (right?)

So you can create a black hole by accelerating any piece of matter sufficiently. (right?)

But the black hole will exist only in certain frames of reference, since its existence depended on its speed, which depends on frame of reference (right?).

Wrong. The notion of “relativistic mass” has been discarded, largely because it leads to misconceptions like yours.

Okay.

So what’s the grain of truth behind the misconception? What were people misunderstanding? What should we say popularizers were “really talking about” when they said things like what I said?

(Also, do I recall correctly that as a thing approaches the speed of light, its length tends toward zero?

Could the argument I gave be re-run then given that as the length becomes shorter, the density rises, eventually leading to gravitational collapse? Probably not, but why not?)

In Newtonian physics F = ma, i.e. force equals (inertial) mass times acceleration. Inertial mass of a body is a direct measure of it’s resistance to acceleration.

Relatvistic mass is (or was) an attempt to recreate the concept of inertial mass. However it really isn’t as simple as it seems because in special relativity not only is a bodies resistance to acceleration dependent on it’s speed, it’s also dependent on the direction the force is applied. The concpet of relativstic mass was popular in the eraly days of special relatvity, but it’s been almost completely disregarded now, only appearing in popularisations of the subject.

No is the answer, the density that matters is the density in the instaneous co-moving inertial frame.

Basically what happened that led to the concept of inertial mass is that folks noticed that there was funny stuff going on in the mechanics equations, and tried to attach the funny stuff to the mass part of the equations, when it’s really fundamentally attached to the space and time parts. For instance, take the formula for momentum: In the Newtonian limit, it’s P = mv. In relativity, it’s P = gammamv, where gamma is a factor dependent on the velocity that shows up a lot in relativity. You might say "Aha, let’s define a quantity M as M = mgamma, and now the equation is P = Mv, which looks like what we’re familiar with!", and then say that M is “really” the mass, and so the same physics applies as at low speeds. You’d be better off, though, saying "Let’s define a quantity u, such that u = gammav, and now the equation is P = m*u". This quantity u is something called the “proper velocity”, and like “relativistic mass”, it serves to make the equations look prettier. Unlike relativistic mass, though, it also makes the equations easier to use.

It’s still a good question, because:

A quickly moving particle is Lorentz contracted. Could it not Lorentz contract below the Schwarzschild radius?

The answer is: no, because the derivation of the Schwarzschild radius is only applicable in the particle’s rest frame. A proper relativistic derivation would show that the particle does not become a black hole. We know this because of the principle of relativity: the laws of physics are the same in all reference frames, and the particle is not a black hole in its own reference frame.

As we’re talking about a mass that is actually accelerating, it’s worth noting that for a body that is accelerating uniformly throughout in it’s instantaneous comoving inertial frame (ICIF) that the acceleration will actually cause tension in that body’s ICIFs. I.e. the physical effect of acceleration will be to tend to rip it apart rather than compress it.

Infact if we specifiy that the body accelerates in such a way that it’s length remains constant in it’s ICIFs (this type of acceleration we would call Rindler acceleration) there is a definite upper limit on the length that a body can be and accelerate in this manner (that length will be a function of it’s Rindler acceleration). Beyond that length the forces on the back end of the body diverge, much like the force required to keep an object stationery in Schwarzchild coordinates diverges at the event horizon of a black hole.