# 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?

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.