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I'm starting with the Einstein box. It's the one that's floating in space and has photons moving from one side to the other. The idea being that every time a photon is sent from one side to the other, the recoil sets the box in motion until that photon strikes the other side breaking to zero.
The center of gravity never changes because as the emitting side looses mass, the absorbing side gains and that combined with the shifting position of the box towards opposite to the motion of the photons keeps the center of gravity stationary - as it must.
So, I now place a pane of glass floating in the middle of this box. As the light passes through the glass, it will travel a bit slower - and so its travel time to the opposite side will be delayed. This, in turn, will allow the box to continue moving a bit longer - so its displacement in the opposing direction will be greater than it was before. So, in order to keep the center of gravity stationary, the glass must become displaced in the direction of travel of the photon.
The center of gravity never changes because as the emitting side looses mass, the absorbing side gains and that combined with the shifting position of the box towards opposite to the motion of the photons keeps the center of gravity stationary - as it must.
So, I now place a pane of glass floating in the middle of this box. As the light passes through the glass, it will travel a bit slower - and so its travel time to the opposite side will be delayed. This, in turn, will allow the box to continue moving a bit longer - so its displacement in the opposing direction will be greater than it was before. So, in order to keep the center of gravity stationary, the glass must become displaced in the direction of travel of the photon.