I Spacetime diagram

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1. Jun 9, 2017

Pushoam

The images have been taken from page no.74,75 and 76, special relativity , A.P.French,1968
I understood that A1' and C1' are not simultaneous in S -frame.
But I don't understand the principle on whose basis it is claimed that A1' and C1' are simultaneous in S' -frame.

Here, the simultaneity is defined by the line A1 C1 in S -frame. I didn't understand this ,too.
The simultaneity should be defined by two points A1 and C1 in S -frame. Shouldn't it?

Can anyone please explain these two points?

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2. Jun 9, 2017

Pushoam

I have understood it now.
A,B,C are at rest in S' frame and B is in the middle of AC.
So, the light emitted by B will reach A and C at the same time.
Hence, A1' and C1' are simultaneous in S' -frame.

But I haven't understood the second question.

3. Jun 9, 2017

pervect

Staff Emeritus
The logic in the second diagram is the same as the first. You're using the same method, which can concisely be described as the midpoint method, to determine if two events are simultaneous or not, but in a different frame of reference.

The midpoint method says that if B is the midpoint between A and C, a signal emitted at B will arive "at the same time" at events A and C. The tricky part is to realize that the notion of "at the same time" is frame dependent, which is what the exercise is trying to show. If you assume that "at the same time" has some meaning independent of the frame of reference used, and you try to assume that the speed of light is equal to "c" for all observers, you'll wind up with a paradox. One of the two assumptions has to go. Many people get stuck here, and cant let go of the notion that simultaneity depends on the frame of reference, so they basically wind up rejecting the important notion of relativity, that the speed of light is "c" for all observers.

But if you can accept the idea that simultaneity is frame dependent, the seeming paradox disappears.

4. Jun 9, 2017

Pushoam

This I understood, pervect.
What I don't understand is :
Why is the simultaneity defined by the line A1 C1 in S -frame instead of the two points A1and C1?

5. Jun 9, 2017

Vitro

Because simultaneity is not limited to the two events, there can be an infinite number of events which are simultaneous. That's what that line represents, the set of all possible events in spacetime simultaneous with $A_1$ and $C_1$ in frame $S$. Simultaneity on the space-time diagram is represented by lines parallel to the $x$ axis.

6. Jun 10, 2017

Ibix

Because all events on that line are simultaneous, not just those two. In fact, all pairs of events on any line parallel to that one are simultaneous.

7. Jun 10, 2017

timmdeeg

Possibly its confusing that the $x'-$axis parallel to $A_1' C_1'$ in frame $S'$ is missing.

8. Jun 10, 2017

robphy

The next diagram in sequence [in the attachments] constructs the x'-axis.
I think the point of the complete passage is to construct the x'-axis for S', starting from "simultaneity according to S' ".

9. Jun 10, 2017

timmdeeg

Ah, which I'd overlooked, thanks.