I know it would violate the principle of relativity. But have there been serious experimental attempts to contradict that principle?
If I am in a moving train without windows and totally sound proof, is there absolutely no way for me to determine that I am moving?
Can this be proved as a mathematical theorem given some other laws/axioms of physics?
I think the answer is that in theory, in an ideal situation, you can't tell. In practice though, ideal situations don't exist and you are always moving relative to something. As long as you decide what it is that the movement you wish to measure is relative to, you can measure it.
You gave the example of travelling in a sound-proof train without windows, and let's assume you want to detect movement relative to the earth's surface. You can't use visual light or sound as that has been blocked-out, but there would be vibration from the movement of the train. Let's assume though it is a maglev train and you can't detect the vibration. However, the train still lets in radio waves, so you could detect those (from a known source) and measure the Doppler effect. That would tell you your speed relative to the radio source. But let's assume you line the train with lead so that no radiation can get in. What else can you measure?
In reality, there is one force that you can't shut out and that is gravity. You will be moving inside a gravitational field. The earth's gravity varies across the surface of the earth, so you could detect that. You could also theoretically detect if you were moving towards or away from the sun by the variation in the tidal effect due to the sun's gravity and the earth's orbit.
You could move your test to a rocket in space and allow that rocket to free-fall (in orbit or not), in which case Einstein's equivalence principle says that you should not be able to tell that you are accelerating. However, in reality, there will be a gravitational gradient across the train and so, even though you don't feel any acceleration, you could work out that you were in a gravitational field and that therefore you must be going somewhere.
But in terms of theories and axioms, those support the idea that motion itself is not directly detectable. If you were not allowed to detect or interact with any outside influence, but only things produced inside your train, there is no experiment you can do that changes depending on your movement.
The principal of relativity essentially states the laws of physics are the same in all inertial frames.
As I understand it Galileo conducted (or it has been claimed he conducted) an experiment in which he dropped a rock from the top of the mast of a moving ship to show it would land at the base of the mast and not behind it.(How he was able to make sure the ship was moving at constant velocity is another matter).
As far as the mathematics go you have the Galilean transformations and Lorentz modifications already discussed below.
I suppose to absolutely prove the principle you would have to conduct experiments that demonstrate all currently known laws of physics are the same on the train when it’s moving at constant speed on a straight track as when it is at rest with respect to the track. Or at least conduct enough experiments to be convinced.
Hope this helps.
I think it's a pretty well-tested idea considering we experience it pretty much in our daily lives.
And actually, even though this is technically not correct since we're on earth which is a big ball with a gravitational field and we're neglecting acceleration, even if you're on a train moving at a constant velocity with windows and sound, you still wouldn't be able to prove that you were moving. All velocities are relative, that's the principle of relativity. This is getting a bit philosophical but how do you know it isn't the earth moving below the train that gives the appearance of motion (again ignoring that the earth is a big globe and other factors for sake of simplicity)? It's a question of your frame of reference. That's relativity.
As for the mathematics, what you're looking for is something called the Galilean Transformations, or something called the Lorentz Transformations which take into account the invariance of the speed of light. Those equations give you the ability to transform from one reference frame into another.
If you're looking for say, your position along the x axis in one reference frame compared to another in galilean relativity, while one reference frame is moving away from the other, the equation is $$x'= x-vt$$ Where t is time and v is the velocity.
In special relativity, where the speed of light is constant, this equation becomes $$x'= γ(x-vt)$$
Where $γ=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and c equals the speed of light. At low velocities, γ reduces to 1 and the whole equation reduces to the galilean transformation.
One chief difference between the lorentz transformations and the galilean transformations is that not only position changes between reference frames, but time does also, also known as time dilation.
As for why this principle of relativity is the case, i'm not really sure, it seems to be a fundamental symmetry of spacetime. There could be a quantum field theoretic explanation towards this but at this moment in time i am not aware of it.
