Simultaneity

 A train is traveling at a velocity near the speed of light. A man raiding in the middle of the car (Observer A), sends two pulses of light directed toward the two opposite sides of the wagon. These pulses are emitted exactly at the same time just when the train is crossing a woman standing on the station (Observer B). The two observers can synchronize their clocks right at this time. The question is, how do they observer the events corresponding to the pulses arriving to the photocells located at the end of the train car? Notice that the length of the car is known to be L*.

Newtonian Mechanics Analysis:

Point of view of Observer A:

This observer does not even know, necessarily, that the train is moving implying that he will assign to both rays the same velocity c. Thus, the distance to be traveled by the rays is the same and equal to . Therefore, the time taking by each ray is

and   implying .

Notice that both rays reach their targets at the same time or simultaneously.

Point of view of Observer B:

Classically, Ray 1 travels the distance  plus the distance that the front of the car moves ahead during the time taken by the ray in reaching its target, . The total distance is . The speed at which this ray moves is . Thus, the time taken can be calculated from

Ray 2 travels the distance  minus the distance moved by the back of the car during the time taken by the ray in reaching its target, . The total distance is . The speed at which this ray moves is . Thus, the time taken is

Therefore,  which agrees with the result obtained by the observer in the train.

Semi-relativistic Mechanics Analysis:

Point of view for Observer A:

Just as in the classical case, this observer does not even know, necessarily, that the train is moving implying that he will assign to both rays the same velocity c. Thus, the distance to be traveled by the rays is the same and equal to . Therefore, the time taking by each ray is

and  implying .

Notice that both rays reach their targets at the same time or simultaneously.

Point of view of Observer B:

In coincidence with the previous analysis, Ray 1 travels the total distance . However, the speed of light is unique for all observers, c. Thus, the time taken can be calculated from the relation

Ray 2 travels the total distance  again at the speed of light, c. Therefore, the equation for the time taken can be written as

Therefore,  which implies that these events are not simultaneous for the observer standing on the station. Notice that for the observer in the train the events are simultaneous.

* Notice that this calculation does not account for any change in length associated to the motion. The purpose of the present calculation is to open the discussion for the derivation of such calculations.

Relativistic Mechanics Analysis:

The most significant change in the analysis comes from the fact that the observer on the station does not measure a length L for the train car. In fact, Observer B measures an Improper Length. Observer A measures a Proper length. In contrast to Observer A, Observer B cannot use a meter stick to determine the length of the car. Thus,

Observer A as well as Observer B cannot measure the time for these events with a single clock. Therefore, neither of the two observers, A and B, can measure a proper time and both times are improper. Observers traveling with the individual rays will be able to measure proper times implying that not a single observer can measure proper time for both rays.

Point of view for Observer A:

Just as in the classical and semi-relativistic case, this observer does not even know, necessarily, that the train is moving implying that he will assign to both rays the same velocity c. Thus, the distance to be traveled by the rays is the same and equal to . Therefore, the time taking by each ray is

and  implying .

Notice that both rays reach their targets at the same time or simultaneously.

Point of view of Observer B:

In coincidence with the semi-relativistic analysis, Ray 1 travels the total distance . However, the speed of light is unique for all observers, c. Thus, the time taken can be calculated from the relation

Ray 2 travels the total distance  again at the speed of light, c. Therefore, the equation for the time taken can be written as

Therefore,  which implies that these events are not simultaneous for the observer standing on the station. Notice that for the observer in the train the events are simultaneous.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com