Ben (narrator/author of the LessWrong post)

However, as the runner wades they will kick a parcel of water along in front of their knees.

The total momentum associated with the fact that the runner is in motion is not entirely contained inside their skin.

Suppose that we place an obstacle in the runner's path.

They collide with it and come to a complete stop.

During the collision, we record the impulse on our obstacle and infer the runner's momentum.

Clearly, in the process of stopping the runner, we have also stopped the water that they were kicking along with them, so that the momentum such an experiment would detect is not simply, here's a formula, but some higher value, including the water parcel.

Indeed, it is not possible to run through water without pushing the water around.

So whenever we compare the situation of the runner to the counterfactual where they are standing still, we need to include the momentum of that pushed water.

The collision experiment is comparing exactly those states.

What if we posited imaginary ghost water that phased through the runner?

Then we could avoid thinking about the momentum of the water and keep things simple.

Yes, we could.

But the ghost water also wouldn't slow the runner down, so that is just the same as the runner on land example.

In the context of the photon in glass, we have the momentum in the electric and magnetic fields, like the momentum inside the runner's skin.

A photon in glass plucks the electron orbitals of the atoms as it passes them by, depositing some amount of energy and momentum into the matter that then springs back out into the electric field a moment later.

That momentary retention is the cause of the light slowing down in the first place, or at least, the momentary retention in the time domain Fourier transforms into the frequency domain as the slow down.

So, is it just as simple as the Abraham momentum?

Here's a formula.

Describes the electromagnetic field exclusively while the Minkowski momentum.

Here's a formula.