-A common thought experiment by Einstein often used to help explain his concept of sim​ultaneity involves a train car and a nearby station platform. If lightning strikes the fore and aft ends of a moving train car at the same time, then a definition of simultaneity emerges.

-Having grown up in the American Midwest, thunderstorms are a common occurrence. A story that weathercasters often tell is that when a lightning bolt is observed some distance away, then if the time is counted until the thunder reaches the observer, he or she can tell how far away the lightning strike occurred. So using this principle, I think a new type of simultaneity can be defined.

-Instead of focusing on the scorch marks left by the lightning strike on the train car, or left on the platform, I will concentrate on the audible thunder clap that occurs at the same time as the lightning strike.

-At the central location of the train car is a single observer, with a single clock, a single sound receiver, a single light sensor. They are all at rest relative to each other within the single reference frame attached to the moving train car.

-The light from the lightning reaches the central location (a distance L from either end) effectively instantaneously; and thus by definition simultaneously. The observer starts the clock when the light arrives. The light and sound start at the same time, but the sounds will arrive at the central location some time later, possibly at different times. Then, if it can be determined that the sounds were simultaneous, then this leads to the conclusion that the lightening strikes were simultaneous

-If she sees that the thunder sound arrives at different times she can assume that either the train car is at rest and the lightning did not strike simultaneously; or the train car is in motion, and the lightning strikes were simultaneous.

-Now the question becomes more philosophy than physics. She has in her mind that it is a windless day, so the air is not moving (nor the platform). So this leaves the only philosophically viable option available to her is that the train is in motion. She then tries to imagine what mathematical means can she use to determine the value of this velocity. Her friend is waiting on the platform (there is a reference frame at rest attached to it) but she will not consult with him.

-So she adds together the arrival times from the fore and aft ends, which are L / (c-v) and L / (c+v) in non-relativistic terms for sound (not [L / c] + [L / c], since the speed of sound is much less than the speed of light). This becomes [2Lc] / (c²-v²) which leads to her being able to find the velocity of the train, v.

-Once having this value she can make the comparison and correction, and make a call to the engineer to verify the magnitude and direction of this velocity. So she has found the velocity of the train, and also whether the lightning strikes were simultaneous, completely from within the reference frame attached to the moving train car.

http://en.wikipedia....of_simultaneity