Simultaneity agreeing in different frames

[DAC]

If a train is moving towards two simultaneous flashes and is equidistant from them, they will be seen to be simultaneous. Meanwhile a stationary footbridge observer watching the same train approach, also sees the flashes as equidistant, therefore simultaneous. Can someone explain how observers in different motion can agree on simultaneity/

 

[PeterDonis]

Can someone explain how observers in different motion can agree on simultaneity/

They don't. The scenario you have specified is not possible as you state it. It only appears possible because you have made an error in your specification. You have to be very careful in relativity when specifying scenarios; my suggested rule is that, to be sure you have specified the scenario consistently and completely, you must be able to give the coordinates of every event of interest in some inertial frame (any frame will do, as long as it's the same one for all events).

For your scenario, the events of interest would be: light flash #1 is emitted from its source; light flash #2 is emitted from its source; the train observer receives light from light flash #1; the train observer receives light from light flash #2; the footbridge observer receives light from light flash #1; the footbridge observer receives light from light flash #2. If you try, you will see that it is impossible to assign coordinates to all these events such that all the statements in your OP are true.

 

[DAC]

They don't. The scenario you have specified is not possible as you state it. It only appears possible because you have made an error in your specification. You have to be very careful in relativity when specifying scenarios; my suggested rule is that, to be sure you have specified the scenario consistently and completely, you must be able to give the coordinates of every event of interest in some inertial frame (any frame will do, as long as it's the same one for all events).

For your scenario, the events of interest would be: light flash #1 is emitted from its source; light flash #2 is emitted from its source; the train observer receives light from light flash #1; the train observer receives light from light flash #2; the footbridge observer receives light from light flash #1; the footbridge observer receives light from light flash #2. If you try, you will see that it is impossible to assign coordinates to all these events such that all the statements in your OP are true.

Thanks but if the flashes are simultaneous as in AE's thought experiment, and both observers are equidistant from the flashes, why aren't they seen as simultaneous?

 

[PeterDonis]

if the flashes are simultaneous as in AE's thought experiment, and both observers are equidistant from the flashes, why aren't they seen as simultaneous?

By AE's thought experiment, I assume you mean the one with the lightning flashes, train, and embankment? Here are the coordinates of the events of interest for that case, in the frame in which the embankment is at rest. Coordinates are given as (x, t), and we assume that the train has length 2 in the embankment frame, the embankment observer is at x = 0 in this frame, and the train is moving in the positive x direction with velocity v in this frame. The spacetime origin (0, 0) is assumed to be the event at which the train observer and the embankment observer are momentarily co-located (i.e., passing each other).

Light flash #1 emitted from source: ( 1, 0 )

Light flash #2 emitted from source: ( -1, 0 )

Light flash #1 received by embankment observer: ( 0, 1 )

Light flash #2 received by embankment observer: ( 0, 1 )

Light flash #1 received by train observer: ( v / (1 + v), 1 / (1 + v) )

Light flash #2 received by train observer: ( v / (1 - v), 1 / (1 - v) )

Notice that, first, both light flashes are received by the embankment observer at the same event, (0, 1). Since the sources of both flashes are equidistant from that observer (they are both a distance 1 from him), he judges them to be simultaneous since he receives them at the same event.

By contrast, the train observer receives light flash #1 at one event, and light flash #2 at another, later event. (Note that it is "later" not just in the coordinate time of the embankment frame, but according to the train observer's own clock, that is, according to the invariant ordering of events along the train observer's worldline.) He judges the sources of both light flashes to be the same distance from him (a distance ##\gamma = 1 / \sqrt{1 - v^2}## according to him), so that means that he judges flash #1 to have happened before flash #2--i.e., they are not simultaneous for him.

From the coordinates above, it is easy to see why the flashes are not simultaneous for the train observer: as Einstein said, the train is moving towards the first flash and away from the second, so naturally it is going to receive the first flash before it receives the second. Since the flashes are equidistant from the train observer, that means the first flash must have happened before the second, according to the train observer.

 

[DAC]

Sorry Peter, I am not following.
If both flashes are simultaneous, and both observers are equidistant from the flashes, both observers must see the flashes at the same time.

 

[PeterDonis]

I am not following.

That's because you're not reasoning correctly.

If both flashes are simultaneous

That's not something we start out knowing. It's something we figure out. So you are reasoning incorrectly right from the start. See below.

both observers are equidistant from the flashes

This part is fine.

both observers must see the flashes at the same time

You have it backwards. The fact that the embankment observer sees the flashes at the same event, and the train observer sees them at different events (the first flash, then the second), is a starting assumption of the problem. Look at the events I specified. That specification of events is the specification of the problem. We look at that specification of events, and then deduce that the embankment observer judges the flashes to be simultaneous while the train observer does not. We don't start from the assumption that the flashes are simultaneous, and then deduce what the observers "must" see.

 

[PeterDonis]

The fact that the embankment observer sees the flashes at the same event, and the train observer sees them at different events (the first flash, then the second), is a starting assumption of the problem.

Let me expand on this a bit. It would also be possible to specify a different scenario, one in which the train observer received both flashes at the same event, while the embankment observer received them at different events. (We can stipulate that both observers would still be equidistant from the flashes in this scenario.) The event coordinates for this scenario would be different from the ones I gave, because it's a different scenario. In that different scenario, the train observer would judge the flashes to be simultaneous, but the embankment observer would not.

What is not possible is to specify any set of events such that both the train observer and the embankment observer receive both flashes at the same event, and are also both equidistant from both flashes. (Try it!) That is why it is impossible for the same pair of flashes to be simultaneous for both observers.

 

[Janus]

Sorry Peter, I am not following.
If both flashes are simultaneous, and both observers are equidistant from the flashes, both observers must see the flashes at the same time.

In Einstein's example the flashes are only simultaneous in the embankment frame.
The embankment observer is equidistant from the flashes and sees them at the same time, thus the flashes originated at the same time.
Also, the light from one flash hits the train observer before it reaches the embankment observer, while the light from the other flash passes the embankment observer on its way to the train observer, and this is true according to someone standing on the embankment or riding the train. Both frames also agree that the lights meet at the embankment observer.
Therefore, the train observer does not see the light from the flashes at the same time, and because he is equidistant from the origins of the flashes, this can only happen if the flashes did not originate at the same time in his frame.

Thus in the embankment frame the lightning strikes occur simultaneously, but they do not do so in the Train frame.

 

[DAC]

AE states the flashes are simultaneous. It is not something we therefore have to establish.

In the moving frame as the light moves to one flash and away from the other, the flashes are not simultaneous. But if the light moves equally, which it does to both flashes it is simultaneous in both frames.

 

[SlowThinker]

If a train is moving towards two simultaneous flashes and is equidistant from them, they will be seen to be simultaneous. Meanwhile a stationary footbridge observer watching the same train approach, also sees the flashes as equidistant, therefore simultaneous. Can someone explain how observers in different motion can agree on simultaneity?

Interestingly, from your description, I imagined something like this:

Apparently, Peter is imagining something very different. Can you show the layout you have in mind?

 

[DAC]

The train is moving towards lightning flashes at the front right and front left of the train. The train is

Interestingly, from your description, I imagined something like this:
View attachment 91326
Apparently, Peter is imagining something very different. Can you show the layout you have in mind?

The same, apart from the flashes being between the train and footbridge, which makes no difference.

 

[Nugatory]

The same, apart from the flashes being between the train and footbridge, which makes no difference.

We just finished a whole long thread about exactly this scenario... And the answer hasn't changed...

Relativity of simultaneity doesn't mean that observers in motion relative to each other must necessarily disagree about the simultaneity of ALL events. It means (and this is the key difference from classical physics) that they do not agree about the simultaneity of some events.

What part of the explanation in that thread did you not understand?

 

[PeterDonis]

AE states the flashes are simultaneous.

No, he doesn't. AE states that the embankment observer observes the two flashes at the same event. That means, in effect, that the events have the coordinates I gave. Which means the flashes are simultaneous with respect to the embankment, but not the train. Which AE also says.

If you are going to willfully misinterpret what AE says, productive discussion is impossible.

 

[PeterDonis]

This is the second thread on the same topic, and as Nugatory says, the answer has not changed. Thread closed.