The Moving Train

 

Einstein considered the issue of simultaneity by using the example of a train and its movement relative to the embankment on which the train was traveling. Figure 1 shows the relevant arrangement of the moving train example.

 

Figure 1

 

The train and bank have equally spaced positions assigned to them. On the train are positions a, b, and c, with b being equally distant from a and c. On the bank are positions A, B, and C, with B being equally distant from A and C. The train and bank are considered to be in different reference frames when the train is moving relative to the bank. With the train stationary, the positions line up as shown in the figure. If lightning strikes both A and C at the same time (simultaneously) while the train is stationary as in the figure, then observers at b and B (O_b and O_B) would both see the strikes as being simultaneous.  O_b would see the strikes occurring slightly later than O_B since the light has farther to travel to reach O_b.

 

Now suppose that the train is traveling at some velocity v with respect to the bank, and at the instant b is directly opposite B lightning strikes A and C in a manner such that O_B sees the strikes as being simultaneous. If part of the light immediately left the reference frame of the bank and entered the reference frame of the train, then O_b would also see the strikes as simultaneous. Otherwise, according to Einstein[1],

 

“Now in reality (considered with reference to the railway embankment) he [O_b] is hastening towards the beam of light coming from [C], whilst he is riding on ahead of the beam of light coming from [A]. Hence the observer will see the beam of light emitted from [C] earlier than he will see that emitted from [A].”

 

This would appear to be true. What is interesting about it is that it considers that light is traveling at different speeds in different directions with respect to an observer who is not in the same system as the light. Otherwise, since the light from either side was emitted at the same distance from O_b, the light from A and C would reach O_b at the same time and the strikes would appear simultaneous. Thus relative to O_b, the light from A is traveling slower than the light from C.

 

The animated picture in Figure 2 portrays the moving train with lightning hitting the two ground points when all triangles are aligned. It is assumed that the light remains in the reference frame of the bank until it reaches the central observer on the train.

 

Figure 2