Why is the door that is farther away from the train observed to close
first?
By observe, in SR, we don't mean see, we mean essentially to record the time and place of events according to rods and (synchronized) clocks at rest. For example,
- Assume that, at the front and back of the train, there are identical
clocks that are synchronized in the frame of the train.
- Further assume that, at the ends of the tunnel, there are identical
clocks that are synchronized in the frame of the tunnel.
By the relativity of simultaneity, the clocks on the train are not synchronized in the frame of the tunnel where it is observed that the clock at the front is behind the clock at the back.
Symmetrically, the clocks in the tunnel are not synchronized in the frame of the train where it is observed that the clock at the entrance of the tunnel is behind the clock at the exit.
- Finally, assume that the length contracted train in the frame of the
tunnel just fits within the tunnel.
Thus, there is a moment, according to the tunnel clocks, that the contracted train is completely within the tunnel.
But remember, in the frame of the tunnel, the train clocks are not synchronized. In particular, since the clock at the front of the train is observed to be behind the clock at the back, it must be the case that, as recorded by the clocks on the train, the door at the exit of the tunnel closes earlier than the door at the entrance.
That is, in the frame of the train, the front of the train just reaches the exit, the door there closes for an instant without hitting the train and there is still a trailing portion of the train that is yet to enter the tunnel.
When the back of the train just clears the entrance of the tunnel, the door at the entrance closes for an instant without hitting the train and there is a leading portion of the train that has exited the tunnel.
As always, I recommend that you draw a spacetime diagram of this sequence of events to get a better 'picture' of how this works.
The crucial mistake you are making is to focus on the reception of the signals, not the events that generated them. Clearly Sally receives the blue and red signals at different times, and that would be true whether we were talking of light or sound. The difference is that the speed of light is constant in Sally's frame, so if the two signals arrive at different times, having travelled the same distance, they must have been emitted at different times. However, with sound, the speed of sound coming from the front of the train will be higher than the speed coming from the back of the train (assuming the sound is travelling through the air outside the train), so although the signals arrive at two different times, the events that caused the sound to be emitted can still be considered simultaneous.
Best Answer
Not in the meaning of the word "state" that you are thinking here. The issue is that there is no way for the bystanders and the train passengers to "know of the train movement". The relative velocity between the bystander and the passengers is a physical fact, but that relative velocity could be because the bystander is moving or because the passengers are moving or because both are moving. There is no possible way (even theoretically) to distinguish those cases. Therefore there is no way to set any absolute state of simultaneity because simultaneity depends on the reference frame.
There is an "absolute state of the universe", but simultaneity is simply not part of it. We typically don't use the word "absolute" to describe it, but instead use the word "invariant" or "covariant". "Absolute" has some bad connotations.
In the invariant description of the universe things are described in terms of coordinate-independent geometric objects called tensors. The tensors may be described with respect to some chosen basis, but they are themselves a geometric object that is independent of such descriptions.
Simultaneity is simply not a part of this tensor-based description of the universe. There is no "simultaneity tensor". The simultaneity concept itself is not part of the state of the universe in any invariant sense. Instead, what is invariant is causality. The universe "cares" that if A causes B then A must come before B, that is an invariant fact. But if A and B could not be causally related then the universe simply doesn't care which happens first. That idea that such non-causally-related events should have a temporal order is a human conceit, not a fact of nature.