Special Relativity Conceptual Question

[Dongus]

Homework Statement

Einstein developed much of his understanding of relativity through the use of gedanken, or thought, experiments. In a gedanken experiment, Einstein would imagine an experiment that could not be performed because of technological limitations, and so he would perform the experiment in his head. By analyzing the results of these experiments, he was led to a deeper understanding of his theory.

In each the following gedanken experiments, Albert is in the exact center of a glass-sided freight car speeding to the right at a very high speed v relative to you.

Albert has a flashlight in each hand and directs them at the front and rear ends of the freight car. Albert switches the flashlights on at the same time.

In Albert's frame of reference, which end, front or rear, is struck by light first, or are they struck at the same time? Which end is struck first in your frame of reference?

Homework Equations

None (conceptual question)

The Attempt at a Solution

So I can easily figure that since the train is at rest in Einstein's frame, and the speed of light is a constant in all frames, both ends will be struck at the same time for him.

The textbook reading elaborated on the relativity of simultaneity, and I figured from the textbook's thought experiment (involving a car with firecrackers at each end) that both observers must agree that the light hits each end of the train at the same time, and that this is made possible by having Einstein turn on the flashlight on the right before he turns on the flashlight on the left in the reference frame of the stationary observer.

It turns out that this is incorrect, and that instead of when Einstein turns on the lights being relative, it is relative when the light reaches the ends of the car.

My question is, how do I know which "kind" of relativity of simultaneity is the correct solution? I approached the problem the way I did because the example of a car with firecrackers at each end, given in the book, involved the assumption that both observers must agree on when the light from either firecracker reaches the center of the car after they explode simultaneously relative to a stationary observer, and yet here, I am instead supposed to assume that both observers agree on when the light is sent, rather than when it is received.

I have considered that it might not make sense for the observers to disagree on when Einstein turns on the lights, since Einstein would seem to be executing a different set of actions to each observer, but I figured that the situation is no different from if we were to replace the lights with exploding firecrackers that Einstein was simply observing.

It would be greatly appreciated if anyone could clear this up to me!

 

[phyzguy]

This is a good question, and there is a clear answer. The relativity of simultaneity has to do with two observers disagreeing on the simultaneity of two events which are at different points in space time. But in the case of Einstein turning on the two flashlights, these two events both happen at the same space-time point. Disregarding the distance between Einstein's left hand and his right hand, they both happen at the same point in space. Since they are at the same point in space, if they are simultaneous in one frame, they have to be simultaneous in every frame. The events of the light striking the two ends of the car, however, are happening at two different points in space, so it is possible for two different observers in different states of motion to disagree on whether or not they are simultaneous. Another way to look at it is the take the coordinates of the two events in the first case (Einstein turning on the two flashlights) in Einstein's rest frame, and say they both happen at (t,x,0,0). Applying the Lorentz transformation to these points, you see that the time coordinates are the same in any frame. The coordinates of the lights striking the two ends of the train, however, are something like (t+L/c, x+L,0,0) and (t+L/c,x-L,0,0). When you apply the Lorentz transformation to these coordinates, the two times can be different.

 

[Dongus]

This is a good question, and there is a clear answer. The relativity of simultaneity has to do with two observers disagreeing on the simultaneity of two events which are at different points in space time. But in the case of Einstein turning on the two flashlights, these two events both happen at the same space-time point. Disregarding the distance between Einstein's left hand and his right hand, they both happen at the same point in space. Since they are at the same point in space, if they are simultaneous in one frame, they have to be simultaneous in every frame. The events of the light striking the two ends of the car, however, are happening at two different points in space, so it is possible for two different observers in different states of motion to disagree on whether or not they are simultaneous. Another way to look at it is the take the coordinates of the two events in the first case (Einstein turning on the two flashlights) in Einstein's rest frame, and say they both happen at (t,x,0,0). Applying the Lorentz transformation to these points, you see that the time coordinates are the same in any frame. The coordinates of the lights striking the two ends of the train, however, are something like (t+L/c, x+L,0,0) and (t+L/c,x-L,0,0). When you apply the Lorentz transformation to these coordinates, the two times can be different.

Thanks! So if I understand correctly, the example of a car with firecrackers at each end allows disagreement on simultaneity of explosion but not of receiving at the detector because the detector is at a fixed point in spacetime, while the firecrackers are at different points - and the roles are effectively reversed in the Einstein example with there being a single point where light is emitted and multiple where it is received, meaning all observers must agree that both lights turn on at the same time. Does this mean that there could be slight disagreement between when the lights are turned on in a realistic example where the distance between the lights is factored in?

 

[phyzguy]

Thanks! Does this mean that there could be slight disagreement between when the lights are turned on in a realistic example where the distance between the lights is factored in?

Yes, as long as they are not at the same space-time point, different frames can disagree.

 

[PeroK]

Does this mean that there could be slight disagreement between when the lights are turned on in a realistic example where the distance between the lights is factored in?

Yes, exactly. If the source of the light on the train was two torches pointing in opposite directions, and the beams left the torches simultaneously in the train frame, then they would be slightly out of sync in the platform frame. The amount by which things are out of sync in one frame compared to the other is proportional to the distance between the events (in the direction of motion).

In the thought experiment, this situation is idealised to a single point source and hence to a simultaneous emission in both frames.