# Can somebody read this and tell me the flaw in the argument?

1. Nov 18, 2007

### idea2000

Hi,

I was reading this guy's website about the Einstein train and I actually had thought of the same argument a long time ago, but I wasn't sure what to make of it. Can somebody read it and let me know what is going on?

http://homepage.mac.com/ardeshir/Einstein'sTrain.html [Broken]

Last edited by a moderator: May 3, 2017
2. Nov 18, 2007

### JesseM

Last edited by a moderator: May 3, 2017
3. Nov 18, 2007

### idea2000

Hi,

Thanks for your excellent reponse to my posting. =)

I think the point where the author of the website is confused about (and I am confused about) is, how do we know which frame of reference the light source is travelling with? If the lightning hits only the tracks and not the ends of the train, we would say that the light source is stationary with respect to the observer on the ground. But, if the lightning hits only the ends of the train and not the ground, we would say that the light source is stationary with respect to the traveller on the train.

In the Einstein train paradox, it says that the lightning hits both the ends of the train and the ground. In this case, which frame of reference is the light source moving with?

If light source is somehow stationary with respect to both the tracks and the train, then we would expect the traveller on the train to also see the flashes reach her at the same time.

4. Nov 18, 2007

### bernhard.rothenstein

relativity of simultaneity

"if two events happen at the same time-coordinate in one frame, then they must happen at different time-coordinates in another frame".
I would add: and do not take place at the same point in space then they must....

5. Nov 18, 2007

### JesseM

I don't see that. A "source" suggests an object with its own rest frame that is emitting light continuously, like a light bulb or a laser...in this case, the lightning strikes are just treated as to instantaneous events which send out light from the exact time and place they occur, but do not continue to send out light after that. An event can't really have a rest frame.

But beyond that, it actually wouldn't matter to the problem even if the light was emitted by a source with its own rest frame, for the simple reason that the speed a light wave travels in any frame is independent of the speed of the source in that frame. In other words, if there is one laser at rest on the tracks and another riding along with the train, and if both lasers are switched on at the moment they line up next to each other, then the fronts of both laser beams will travel at the same speed of c no matter which frame you're using, and any observer on the tracks or on the train will see the light from each laser turning on at the same instant.
No, all that matters is whether the two light beams depart simultaneously or non-simultaneously in her frame. If you had two lasers sitting on opposite ends of the train (so they were stationary with respect to the train), but they were turned on simultaneously in the frame of the observer on the tracks, then that would mean they were turned on non-simultaneously in the frame of the observer on the train, so the light from each would reach the center of the train at different moments.

6. Nov 18, 2007

### idea2000

Re:

Hi,

I quote from this post: "any observer on the tracks or on the train will see the light from each laser turning on at the same instant." I think this exactly the argument that the author of the website intended to use.

He is trying to say that if there are two lasers that are are switched on at the moment they line up next to each other, then both the observer on the train and the observer on the ground should see the lasers turning on at the same instant.

If we interpret the lightning striking the tracks and the end of the train as two sets of light sources, would we expect both observers to think that the light sources were switched on at the same time?

7. Nov 18, 2007

### JesseM

But in this quote I was talking about two lasers which are right next to each other when they are turned on (note that I said 'if there is one laser at rest on the tracks and another riding along with the train, and if both lasers are switched on at the moment they line up next to each other'), not lasers at different ends of the train. As bernhard.rothenstein pointed out, disagreements about simultaneity only happen for events that are separated in space, not events which happen at the same position in space.
Why do you think he's talking about lasers that are switched on when they line up next to each other? Both his words and his diagrams show he is talking about the light from two events which occur on opposite ends of the train.
If there is one lightning strike on the left end of the train and another lightning strike on the part of the tracks that is right next to the left end of the train at the same moment, then yes, both observers will receive the light from these two strikes at the same moment (which is fine since they each agree the strikes happened simultaneously, and they each think that the light from each strike moved towards them at c in their own rest frame). But he is talking about strikes that happen at the position of both the left and the right side of the train.

8. Nov 19, 2007

### idea2000

Hi Jesse,

Thanks again for all your help. =) I don't mean to frustrate anyone, I'm just slow in undestanding some of these things, so please be patient...

Okay, I think I might see what the problem with the author's argument is...

Let me just confirm...

We have produced two different interpretations as to what is going on in this course of our discussion.

Let me first summarize the two...

Interpretation 1:

If there is ONE lightning strike at the left end of the train and ONE MORE lightning strike on the tracks right next to the left end of the train, then both observers would say that these two lightning strikes happened at the same time.

If there is ONE lightning strike at the right end of the train and ONE MORE lightning strike on the tracks right next to the right end of the train, then both observers would say that these two lightning strikes happened at the same time.

If all four lightning strikes happen simultaneously (the two on the left and the two on the right) to the observer on the ground, then they should also happen simultaneously to the person on the train. This is equivalent to having two lasers set up on the left and two more lasers set up on the right.

Interpretation 2:

If however, there is ONE lighting strike that strikes both the left end of the train and the tracks next to the left end of the train and ONE MORE lightning strike that strikes both the right end of the train and the tracks next to the right end of the train (and the light source is considered stationary with respect to the observer on the ground), then the observers would disagree which lightning strike happened first.

I think the author's mistake is that he is treating the problem as if a SINGLE lightning strike produces TWO light sources -- one that moves with the train and one that is stationary with respect to the ground. He doesn't explicitly say any of this, but I think he might be confused because his treatment of the problem shows that he is indeed thinking this way. Try rereading his logic with this in mind, and let me know if this is what you see as well. If he follows thru with this interpretation, then he will conclude that both observers think the lightning strikes on both sides of the train happen simultaneously which is similar to Interpretation 1.

If this is indeed what he is thinking, then it is his interpretation of the problem that is wrong -- one lightning strike produces one light source and not two.

9. Nov 19, 2007

### JesseM

Wait, where did you get the idea that all four would be simultaneous with one another in the train-observer's frame? The two strikes on the left end will be simultaneous with each other, and the two strikes on the right end will be simultaneous with each other, but the two strikes on the left end are definitely not simultaneous with the two strikes on the right end in the train-observer's frame if the left strikes are simultaneous with the right strikes in the track-observer's frame.

My basic point is that adding multiple lightning strikes "next to each other"--i.e. at the same position in space and same point in space--doesn't change the problem at all, doesn't change when any observer will see light coming from that point in space, doesn't change what any observer will say about the simultaneity of strikes that happen at different points in time and space, etc. In the original thought-experiment there were only two lightning strikes at different points in time and space, we can call them lightning strike L on the left end and lightning strike R on the right end. In this original thought-experiment, if L and R were simultaneous in the track-observer's frame, they are non-simultaneous in the train-observer's frame. So in your altered version of the problem where we have two simultaneous strikes L1 and L2 on the left end and two simultaneous strikes R1 and R2 on the right end, absolutely nothing is changed...whatever was true of L and R in the original thought-experiment (for example, that they happen non-simultaneously in the train-observer's frame, or the times that light from each one reaches her) will still be true of L1 and R1, and true of L1 and R2, and true of L2 and R2, etc. All that matters is when and where events happen, it doesn't matter how many of them happen at those coordinates.

As an analogy, if I have a piece of graph paper with x and y axes drawn on, and I draw a dot A at position x=0,y=0 and another dot B at position x=3,y=4, then we can note various facts about these dots, like the angle of the line between them relative to the x-axis, or the distance between the two dots. In this case, using the pythagorean theorem we can see that the distance is squareroot[3^2 + 4^2] = squareroot[25] = 5. Now if we draw two dots A1 and A2 both at the same position x=0,y=0, and two more dots B1 and B2 at the same position x=3,y=4, all the geometrical facts are the same for A1 and B1, or for A2 and B2, or for A1 and B2...for example, the distance between A1 and B2 is still 5, it only depends on the position of the two dots.

So, adding more dots at the same position is basically irrelevant to any geometric problem we're interested in, and the same goes for adding multiple events at the same position in space and time. After all, when calculating how long it takes for the light from an event to reach some other position, you just find the time by taking the distance divided by the speed of light, and the distance depends only on the position of the event, not whether there were other events at the same position or any specific facts about the nature of the event (like whether it was the turning-on of a laser on the train or a laser on the ground).
Like I said, even if we explicitly defined the problem this way it would make no difference to the problem...if the event of the two sources turning on at the left end happened simultaneously with the event of the two sources turning on at the right end in the track observer's frame, then the events at the left end would happen at a different time from the events at the right end in the train-observer's frame, and all the facts about when the light from each event reached each observer would be unchanged as well.

10. Nov 19, 2007

### Mentz114

http://homepage.mac.com/ardeshir/Einstein'sTrain.html [Broken]
He makes an elementary error in the second paragraph of his argument -

Wrong. Because the light takes a finite time to reach her, by the time it does, she is no longer half way between the burn marks, and so the distances are not the same.

Last edited by a moderator: May 3, 2017
11. Nov 19, 2007

### JesseM

This is not an error, because he was talking about the distance the flashes move in her rest frame. Where the burn marks are is irrelevant (if the lightning hit the ends of the train instead of the ground next to the ends of the train do you think this would change anything?), all that matters is how far each flash was from her at the moment the flash happened.

12. Nov 24, 2007

### idea2000

Hi,

Thanks for the replies, I think I know what's going on now!