This thought occurred to me after I began reading about the EM drive, and I know there are a lot of theories out there on how that works/doesn't work, I'm wondering why this solution wouldn't make sense. My background is in CS/Math, and I've only taken a few semesters of (Newtonian) physics so if this isn't reasonable at all I apologize.
Some things I believe to be true:
Photons always move at the speed of light. They have energy and momentum.
When photons are reflected, they impart a small amount of their momentum/energy into the reflector and are red shifted.
The speed of light is orders of magnitude faster than the speed of a pressure wave (sound) through every solid material.
(unsure) Reflected photons always impart their momentum as some kind of proportion of their total energy (a small percentage as opposed to a static amount).
So, I will try to explain as best as I can. I could make some clarifying diagrams if that helps. Suppose you have two mirrors facing one another, and also attached to the same vehicle. You shoot a photon (assume an energy of 1) at mirror one. The photon imparts a very small amount of momentum on it and it begins moving away. The photon bounces off with, I don't know, .9 energy now, leaving .1 energy with the first mirror and red-shifting itself. Normally if it weren't light going so fast, the first mirror's momentum would be transferred to the second mirror via a pressure wave through the vehicle, causing the relative energy between mirror 2 and the photon to be the same as mirror 1 and the photon prior to their collision. However, because it's going so much faster than the pressure wave the two mirrors are in a separate frame of reference so when the photon hits mirror two, it imparts the same proportion of its total energy, reflecting with I guess .81 using the same numbers as above. This means that when the two pressure waves meet somewhere in the middle, they are unequal (.1 vs .09 with above). Reflecting a photon back and forth continuously, you get asymptotically close to the photon's momentum reflection proportion (if that makes sense), and your vehicle actually moves, without reaction mass and without breaking conservation of momentum/energy.
Put a different way, if you're a static observer between initially stationary mirrors floating in space and you shoot a photon at mirror 1, which bounces off and hits mirror 2, and then is pulled out of existence, will mirror 1 be traveling away from you faster than mirror 2? I would think yes, since the photon lost momentum onto mirror 1 prior to hitting mirror 2.
Is this wrong? Correct? Impractical? Thanks in advance.
Edit: To clarify how the first bit is related to the second bit. Take the second example and attach the mirrors. When the photon hits the first mirror it will create a ripple that slowly propagates through the material connecting them. So until this ripple travels through the entire vehicle, you could consider the first mirror to be moving away from you and the second mirror to be static, which is the difference in the frame of reference of the two mirrors. After its first collision, the photon will hit the second mirror creating a second (presumably smaller) ripple before the first ripple has hardly propagated at all. If these ripples were equal, the vehicle wouldn't move, but because they are unequal, it does.