How is Simultaneity Defined in Spacetime Diagrams by A.P. French?

[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?

 

[Pushoam]

But I don't understand the principle on whose basis it is claimed that A₁' and C₁' are simultaneous in S' -frame.

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.

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?

 

[pervect]

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 can't 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.

 

[Pushoam]

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

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?
Please, explain me this.

 

[Vitro]

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?
Please, explain me this.

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.

 

[Ibix]

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?
Please, explain me this.

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.

 

[timmdeeg]

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

 

[robphy]

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

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' ".

 

[timmdeeg]

The next diagram in sequence [in the attachments] constructs the x'-axis.

Ah, which I'd overlooked, thanks.