I'm not a physicist and I can't debunk this theory point by point, but actually, you don't need to know much physics to see how this must be wrong. Special Relativity requires conservation of momentum, in every observer's frame of reference. If someone claims to explain supposed non-conservation of momentum by invoking "Special Relativity", they are just attempting to trick you.
(As far as crackpot theories go, I'm actually surprised they didn't throw in a "quantum" or two for good measure.)
(Edit: I'm not saying that the experiment itself is invalid. It might be, I don't know. What I'm saying is the supposed theory[1] is necessarily wrong, because you can't get non-conservation of momentum out of Special Relativity without suggesting new laws of physics, which is not at all discussed in the "paper" I linked below.)
Like I said, the best test would be to put one in space. If you fire the thing up an the satellite's orbit changes, that's kind of a hard measurement to screw up. (Let it orbit with the drive off for a while as a control.)
No idea why you're being downvoted for this. You're completely correct as to why this device's claimed physics are bunk (and why, as I note above, the way it always gets tested is highly suspect).
(As far as crackpot theories go, I'm actually surprised they didn't throw in a "quantum" or two for good measure.)
(Edit: I'm not saying that the experiment itself is invalid. It might be, I don't know. What I'm saying is the supposed theory[1] is necessarily wrong, because you can't get non-conservation of momentum out of Special Relativity without suggesting new laws of physics, which is not at all discussed in the "paper" I linked below.)
[1] http://www.emdrive.com/theorypaper9-4.pdf