Ben (narrator/author of the LessWrong post)

By asking what we want the number for we sidestep that particular rhetorical spiral and keep ourselves arguing about physics, not definitions.

Momentum as a term is arguably overloaded with too many definitions.

Group velocity versus phase velocity.

In a material the phase velocity, how quickly the phase fronts of the electromagnetic field propagate, and group velocity, how quickly the energy of the field propagates, are not necessarily equal, and these two velocities imply two refractive indices, n subscript p and n subscript g, which are the slowdown factors of the two relative to vacuum.

That same paper from the last section tells you to use den subscript g in Abraham's form, dividing, for kinetic momentum and en subscript p, in Minkowski's, multiplying, for the momentum that goes with the uncertainty principle.

A clean-cut proof of Abraham's result.

There is a nice, simple, way of motivating that Abraham should be right.

Get a glass block, throw a photon at it, from the left.

For simplicity, assume that the photon transmits through the block without reflection at either edge.

This can be achieved using either anti-reflection coatings or post-selecting on a photon that transmits.

We will also assume the glass block is rigid so that it can be moved but doesn't support any vibrational modes, no wobbling like jelly.

We also assume the glass block is on a frictionless table so is able to slide either left or right.

As the photon enters the glass block either its momentum falls or increases

If the photon momentum falls then the glass blocks start sliding left to right by momentum conservation.

If instead, the momentum increases the glass blocks start sliding right to left.

In either case, the photon will soon exit the block from the other side and return the block to a standstill.

In total the glass will have either shifted slightly left or right depending on which momentum equation is correct.

Usually in physics, for an isolated system, the center of mass moves at a constant rate.

The photon and glass block together constitute an isolated system, and energy-mass equivalents can be used to give the photon a contribution to the center of mass.

The general principle that the center of mass moved a constant rate can only be satisfied by the glass block moving left to right, and only by it moving by exactly the distance predicted by the Abraham momentum.