If the speed of light in a vacuum is constant in all frames, then acceleration to or beyond light speed is impossible. At least, that's my understanding of special relativity.As far as light in the ship if you think that's a problem you really don't understand the theory of relativity.
But in Star Trek, acceleration to and beyond the speed of light is possible, hence some premise of special relativity is false. One premise that might be false is that the speed of light in a vacuum is constant in all frames. But in that case, illumination in the ship would be weird.
My special relativity is that of an informed amateur, and my general relativity is that of someone who has never learned tenser calculus (ie none). So I don't really know what the warping is supposed to involve, or how it is supposed to be different from motion. When I watch Star Trek, it certainly seems to depict the Enterprise in motion.Star Trek uses an explanation that maybe, just possibly, might work. They don't accelerate in the same sense that we accelerate in our car when we press the accelerator, they warp the space-time continuum by expanding the area behind it and contracting the area in front.
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Star Trek uses technology that may be impossible but it does follow the laws of physics as we understand them.