Cleaning up after Matt Parker

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This video Matt Parker posted a couple of days ago really pissed me off:

It’s cheating to say that a result arrived at through classical mechanics—in this case, that is the kinetic energy of a particle—is wrong because it’s only an approximation to relativistic physics. Of course that’s the case, and I swore at my phone when the “reveal” came.

By the end of the video, though, I had calmed down because I knew I could put together a quick post of my own by rewriting the infinite series equation he derived in a better form. Also, I could provide a general expression for the individual terms.

Let’s start with this result: that the kinetic energy of a particle (including relativistic effects) is

where m is the mass of the particle, c is the speed of light, and

is the Lorentz factor, with v as the velocity of the particle. The video gets to this result at about the 8-minute mark.

Matt then does a series expansion of the term to get

The leading 1 of this series gets canceled by the 1 that’s subtracted from γ, giving

for the kinetic energy. The leading term is the one we get from classical mechanics and the others are essentially zero unless you’re in a particle accelerator (which is discussed later in the video).

I don’t like this form for the equation. It’s cleaner if you factor out all the terms that give the expression the units of energy and then have a nondimensional expression afterward. Like this:

Isn’t this nicer? The expression in the parentheses is a function of the ratio of the velocity of the particle to the speed of light. If we call that

then the kinetic energy is

and it’s much easier to see why the terms after the 1 are vanishingly small for most situations—all the situations for which classical mechanics applies.

One last thing. Matt sort of explained how to calculate the terms of the expansion of γ in the companion video, but there was a lot of handwaving and he bailed out after the second term. It doesn’t take too much effort to show that the series expansion of γ can be written like this:

That means the kinetic energy, , is

You can confirm that these terms match the equation we saw earlier by plugging values of n from 2 through 8 into this expression. And now we can extend the series as far as we like, even though the additional terms add essentially nothing.