Can someone help me interpret this spacetime diagram?

[Vitani1]

Homework Statement

Borrowed from "A first course in General Relativity" by Schutz. This is question 12 of chapter 1.

Relevant Equations

S^2=S'^2

My question in specific is understanding what this line EB exactly represents. This was borrowed from the book "A first course in general relativity" by Schutz. There is a question on page 30 (number 12) which asks the following:
"Use the fact that the tangent to the hyperbola DB in Fig. 1.14 is the line of simultaneity for O¯ to show that the time interval AE is shorter than the time recorded on O¯’s clock as it moved from A to B. "

I see that there exists an event in the unbarred observer's frame with coordinates (C,F) corresponding to an event in the frame of O' with coordinates (B,0). In order to do the above proof I assume I need to know some information about the x-coordinate in observer O's frame at this time E but this diagram doesn't (I think) give this information. Certainly do not give me the answer because this is still homework. I'm mostly curious about this line EB.

 

[Vitani1]

Actually I was able to prove that the time to pass from points A to C in O's frame is shorter than the time in required in the frame of O ' prime to pass from A to B. This by extension would imply that the time to go from A to E is less than this path A to B as well. In any case can anyone help me with understanding what this line EB is referring to?

 

[vela]

It's a line of simultaneity for the O' frame.

 

[Vitani1]

What you are saying, then, is that in O' frame this line is the line by which when crossed happens to be the point in spacetime when this observer sees O with the same coordinates?

 

[Vitani1]

Oh - it says that in the book. Thanks.

 

[vela]

What you are saying, then, is that in O' frame this line is the line by which when crossed happens to be the point in spacetime when this observer sees O with the same coordinates?

No. It's a line where $t'=\rm constant$, so an observer at rest in O' would say that events E and B happened at the same time.

 

[Vitani1]

Would observer O say the events happened at the same time at event C?

 

[vela]

Yes, an observer at rest in frame O would say that C and B happened at the same time.

 

[Vitani1]

Thank you.