Simultaneity in Newtonian mechanics How would Newtonian mechanics answer the train and moving light question?
The setup is:
A train is moving in the positive x_axis with speed c/2.  A person stands in the middle of the train.  There are two light bulbs at both ends of the train.  The light goes off at the same time (absolute time in Newtonian physics).  The person standing in the middle of the train would perceive both lights independently.
Outside the train there is a stationary observer.  Let's assume the train is already to the "right" of the observer (in x_axis) when the lights go off.  Would the stationary observer observe the rear light before the front light?
The reason why I am asking this is that the relativity of simultaneity is often attributed ONLY to special relativity.  Here, would Newtonian mechanics also predict that the stationary observer observes different simultaneity than the moving observer in the train?
 A: According to Newtonian Mechanics, the observer on the ground would see the light  travelling to the right at a speed $V=\frac{3c}{2}$, and the light travelling to the left will have a speed $V=\frac{c}{2}$. This is because we assume the Galilean transformations are true. Hence the two light beams appear to reach both sides simultaneously in either frame according to Newtonian Mechanics.
A: Under C19th Newtonian theory, light was usually treated using the ballistic emission theory of light, which would say that the speed of light generated by bulbs inside the train would by default be cTRAIN and not cPLATFORM.
So if the central observer sends a trigger signal to cut the lights, that signal travels at cTRAIN, both lights cut out, the final wavetrain signal travels at cTRAIN, and both lights appear to the central train observer to go off at the same moment.
If by this time, the train observer (with the train moving across our page from left to right) is already to the right of the platform observer, then the platform observer will see the rear light go off first for two reasons:


*

* Because it's closer, and

* Because the signals from the approaching rear of the train will (for platform observers) be travelling faster than those from the receding front end of the train. 

A: Rather than think in terms of what someone will observe, think in terms of what would be measured in their frame of reference assuming the frame to be equipped with suitably distributed clocks.
As an example, even in the Newtonian realm, suppose an someone is at the origin of his frame of reference and there are lamps at x = 10 million and x = 20 million miles that turn on at the same time. The light from the closer lamp will reach the observer before the light from the further lamp. He would observe them not to be simultaneous but they really are simultaneous in his frame of reference.
