A question that probably has something to do with simultaneity

[zanadu]

There are two observers, A (stationary) and B (in the middle of a train moving towards A). At each end of the train is a lamp. At a point in time in the frame of reference of B, both lamps emit a pulse of light directed at B (middle of train). When the pulses hit B, they scatter in all directions. To B, the two pulses should hit him at the same time since the speed of the two pulses and the distance between B and each lamp should be the same. To A, the pulse from the lamp at the head of the train should hit B before the pulse from the lamp at the tail of the train.

I believe that something is wrong with this scenario, but am unsure of where I went wrong . This would seem like the common example used in high school textbooks to explain how simultaneity varies in different frames of reference, but I am not considering when the two beams reach A, but instead I am considering when A perceives the two beams to reach B.

Apologies beforehand if this question is juvenile; I am not too familiar with the special theory of relativity.

And this really isn't a homework question if anyone is wondering.

 

[Fredrik]

To A, the pulse from the lamp at the head of the train should hit B before the pulse from the lamp at the tail of the train.

Because you think the light that moves in the same direction as the train moves faster (in frame A) than the light that moves in the opposite direction? The main difference between special relativity and pre-relativistic physics ("Galilean relativity") is that the speed of light is always c in all inertial frames. You need to take that as the starting point, and derive other interesting things from it (including the correct rule for how velocities add up in special relativity).

Apologies beforehand if this question is juvenile

It's not, but there are lots of threads where very similar questions have been asked and answered before. So you could search for them. "Train" is probably a good word to search for.

 

[zanadu]

I believe in frame A the light from head of train hits B before light from tail of train not because the light moves faster, but because as the light travels, the train also travels, such that the light from the head of the train travels a shorter distance than the light from the tail of the train in order to hit B.

 

[HallsofIvy]

The two beams start at a given instant. If at that instant, the head and tail of the train are at the same distance from the middle of the train, the pulses of light will have the same distance to go and will get there at the same time (to an observer on the train) no matter how fast the train is going.

 

[DrGreg]

zanadu, you have it the wrong way round. You seem to be saying that A & B agree the flashes were emitted simultaneously but disagree when when the light reaches B. That is wrong. They agree when light reaches B but disagree whether or not the flashes were simultaneously emitted. According to B the emissions were simultaneous because the flashes traveled the same distance at the same speed. According to A the emissions were not simultaneous because one flash traveled a longer distance than the other, but still at the same speed.

We are talking about what A and B each retrospectively calculate must have happened in the past, not what they actually see occurring at B.

 

[zanadu]

The two beams start at a given instant. If at that instant, the head and tail of the train are at the same distance from the middle of the train, the pulses of light will have the same distance to go and will get there at the same time (to an observer on the train) no matter how fast the train is going.

what about the observer on the ground, the stationary one? Shouldn't he observe that one beam hits the middle first?

 

[Aaron_Shaw]

what about the observer on the ground, the stationary one? Shouldn't he observe that one beam hits the middle first?

DrGreg has spotted your problem. it looks wrong because you're asuming they were emmitted simultaneously in both frames and hit the middle at different moments. It's the other way round. If the guy in the middle gets hit by both together, then the stationary observer must have seen the rear lamp emit before the front in order that both observers may agree on the event of the guy on the train getting hit by both at the same moment.

 

[starthaus]

It's the other way round. If the guy in the middle gets hit by both together, then the stationary observer must have seen the rear lamp emit before the front in order that both observers may agree on the event of the guy on the train getting hit by both at the same moment.

Very true, if two spatially co-located events are simultaneous in one frame, they are simultaneous in any other frame. Mathematically:

$$dx'=\gamma(dx-Vdt)$$
$$dt'=\gamma(dt-\frac{Vdx}{c^2})$$

If the temporal and spatial separation are null in frame F ($$dt=dx=0$$) this implies null temporal and spatial separation ($$dt'=dx'=0$$) in any other frame F'.

 

[zanadu]

They agree when light reaches B but disagree whether or not the flashes were simultaneously emitted. According to B the emissions were simultaneous because the flashes traveled the same distance at the same speed. According to A the emissions were not simultaneous because one flash traveled a longer distance than the other, but still at the same speed.

We are talking about what A and B each retrospectively calculate must have happened in the past, not what they actually see occurring at B.

First, thanks for the all the replies.

Pardon me if I am mistaken, but summing up the replies, I have the following conclusions:

1. B will always be hit by both pulses at the same time, and will always observe both lamps firing up at the same time.

2. A will always observe the two pulses hit B at the same time, since co-spatial simultaneous events are so in all frames of reference.

3. A will observe the two lamps light up at different times.

So reconstructing the scenario:
A observes the scattered photons coming from B due to the incidence of both pulses of light, and concludes that the two beams of light hit B simultaneously. In turn, he will have to draw the conclusion that the two lamps fired up at different times, since in the frame of reference of A, the distance that light from lamp at head of train travels a shorter distance to reach B than light from lamp at tail of train.

So far there is no contradiction, but I am concerned about conclusion 3 above, as it does not seem to be always the case. Assume that the two lamps fire up at the precise moment observer B is aligned with observer A, i.e. both lamps are equidistant from A. In this case, the two lamps should be observed to fire up simultaneously in the frame of reference of A. Meanwhile, the conclusions 1 and 2 still seem to hold. This would in turn mean that A observes that the lamps light up simultaneously, as well as that both pulses of light hit B at the same time, which does not seem right.

Apologies in advance if there are any errors in my reasoning.

 

[Aaron_Shaw]

First, thanks for the all the replies.

Pardon me if I am mistaken, but summing up the replies, I have the following conclusions:

1. B will always be hit by both pulses at the same time, and will always observe both lamps firing up at the same time.

2. A will always observe the two pulses hit B at the same time, since co-spatial simultaneous events are so in all frames of reference.

3. A will observe the two lamps light up at different times.

So reconstructing the scenario:
A observes the scattered photons coming from B due to the incidence of both pulses of light, and concludes that the two beams of light hit B simultaneously. In turn, he will have to draw the conclusion that the two lamps fired up at different times, since in the frame of reference of A, the distance that light from lamp at head of train travels a shorter distance to reach B than light from lamp at tail of train.

So far there is no contradiction, but I am concerned about conclusion 3 above, as it does not seem to be always the case. Assume that the two lamps fire up at the precise moment observer B is aligned with observer A, i.e. both lamps are equidistant from A. In this case, the two lamps should be observed to fire up simultaneously in the frame of reference of A. Meanwhile, the conclusions 1 and 2 still seem to hold. This would in turn mean that A observes that the lamps light up simultaneously, as well as that both pulses of light hit B at the same time, which does not seem right.

Apologies in advance if there are any errors in my reasoning.

Your problem here is that you've made the assumption that A sees the lamps light up at the same time, AND the light hit B at the same time, AND that B sees the same. But this is not a possible scenario. If A does infact see the lamps light up at the same time, then he will NOT see the light hit B from front and rear at the same moment. Therefore neither will B. B will see the lamps light at diffeent times, explaining why he also does not see the light hit from front and rear together.

 

[zanadu]

If A does infact see the lamps light up at the same time, then he will NOT see the light hit B from front and rear at the same moment. Therefore neither will B. B will see the lamps light at diffeent times, explaining why he also does not see the light hit from front and rear together.

The problem is that B will always see the pulses from the lamps hit him at the same time in his frame of reference, as both the speed of the two light beams, and the distance of each lamp from him, are the same in his frame of reference.

 

[Aaron_Shaw]

The problem is that B will always see the pulses from the lamps hit him at the same time in his frame of reference, as both the speed of the two light beams, and the distance of each lamp from him, are the same in his frame of reference.

Why? Only because you've decided that they lit up at the same time for B. If you say that then you CANNOT also say that they lit up at the same time for A. Do you get what I mean? If you say both of those scenarios are true then you're making up your own laws of physics. And Scotty wouldn't approve.