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It has been a while since I've thought about physics, however, I remember something about how if you are on a train with no windows that is going perfectly straight and is perfectly smooth, there is no experiment that you could perform to prove that you are in motion or at rest. So I have a couple of questions related to this.

  1. Is this still held to be true? I was just wondering if anything in physics has come to light since I learned about this (about 10 years ago) that would allow an observer on the train to tell if they are moving.

  2. If this is true, why couldn't you accelerate a particle in different directions to a certain mass and measure how much energy it took to get it up to that mass. If you were moving in any direction, wouldn't you measure less energy in your direction of travel?

  3. Would having an outside observer affect the aforementioned experiment in any way?

CumminUp07
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Yes, it is held to be true, both in Galilean and Einsteinian relativity.

If you did the experiment you described you would discover that there was no difference based on direction in the energy it took to accelerate something to a given velocity.

If there was an observer compared to whom you are moving, then, as speeds approach that of light, they would observe different things than you would observe. Perfectly symmetrically, if you observed them doing the same experiments you would differ about things in the same way they do, because neither of you has any basis to assume that you are in some privileged 'rest frame' since such a thing does not exist(*). In particular you and the observer will differ about lengths & times in such a way as to make the laws of physics be the same to both of you.

The way that you differ in what you observe as speeds approach $c$ is described by special relativity, and it's exactly this kind of reconciliation between all observers measuring the speed of light to be the same in all directions and observers moving relative to each other that gave rise to this theory.

(*) No privileged rest frame exists according to the best theories we have, which theories have been very extensively tested. This does not mean it is absolutely impossible that someone will find that one, in fact, does exist, but this seems absurdly unlikely.