In Einstein's famous train platform thought experiment, motion is indeed a crucial factor in determining the observation differential. The experiment, often referred to as the "twin paradox," involves two twins: one stays on a stationary platform (let's call her Alice), while the other (let's call him Bob) travels on a high-speed train.
The key concept in special relativity is that the laws of physics, including the behavior of light and the passage of time, are the same for all observers moving at a constant velocity relative to each other. This is known as the principle of relativity. However, when considering relative motion, there are two important aspects at play: time dilation and the relativity of simultaneity.
Time dilation occurs because the speed of light is constant in all inertial reference frames. When Bob travels at a high velocity relative to Alice, the light signals he observes from her (such as a clock's ticking) will appear to be slower than they actually are. This is due to the time dilation effect, where time appears to pass more slowly for the moving object (Bob) compared to the stationary object (Alice).
On the other hand, the relativity of simultaneity refers to the fact that the order of events can be perceived differently by observers in relative motion. For example, if Alice and Bob have synchronized clocks, when Bob passes Alice on the train, they will no longer agree on the simultaneity of events. Alice may perceive events to occur at different times compared to Bob due to the time it takes for light signals to reach her.
So, motion does play a crucial role in the thought experiment. It is the relative motion between the twins that leads to time dilation and the relativity of simultaneity, resulting in the observed differential in aging. The twin who experiences the acceleration and deceleration of motion (Bob) will end up younger than the twin who remains in a relatively stationary frame (Alice). It is the combination of motion and the constancy of the speed of light that leads to these effects in the thought experiment.