sadegh4137 said:
hi
I read this statement in site
www.askamathematician.com and I couldn't understand it.
is it possible to explain this?
"if measurements A and B are taken enough apart, they will be ?space-like separated? according to SR, meaning that neither event precedes the other. Some observers will correctly believe that A happened first, others will know that B came first, and SR says that nobody is wrong. Time doesn?t work the way we usually think it does, so watches won?t agree for observers moving relative to each other. "
Your article does an excellent job of explaining so I'm not sure I can offer anything more but I'll try. I'm going to draw some spacetime diagrams for you. The article explains what a spacetime diagram is so I won't go into details about that.
In order to make things a little simpler, I'm going to show three observers, all traveling at different speeds and we will consider the first event A to occur when they happen to coincide. The second event is labeled B in the diagrams. Each observer measures the time of the event by sending a radar signal at an appropriate time which hits the second event B and bounces off it and returns to the observer. Each observer considers the time at which the radar signal hits event B to be half way between when he sent the radar signal and when he received its echo.
I'm using the speed of light (and the radar signals) to be 1 foot per nanosecond and because of the way that I have draw the diagrams, the radar signals will travel along 45-degree diagonals (as the article pointed out).
In the first diagram, focus your attention on the blue observer. The dots mark off 1-nanosecond increments of time. At the dot labeled with the blue 0, he sends the black radar signal which propagates upward and to the right, reflecting off of event B and arriving back at the blue observer at his time of 12 nanoseconds. (Don't be confused by the fact that the red observer has previously sent his radar signal which then propagates "in parallel" with blue's and finally with green's.) Therefore, he concludes that event B occurs at his time of 6 nanoseconds which is the same time as event A.
Now do the same thing for the red observer who is traveling to the right at 0.6c. He emits his black radar signal at his dot marked with the red 0 and receives its echo at his time of 15 and so he concludes that event B occurred at his time 7.5 nanoseconds which is before his time for event A.
Finally do the same thing for the green observer who is traveling to the left at 0.8c. He emits his black radar signal at his dot marked with the green 0 and receives its echo at his time of 20 and so he concludes that event B occurred at his time 10 nanosecond which is after his time for event A.
Note that all of these measurements were made in the same Inertial Reference Frame (IRF), the rest frame of the blue observer. Key to this working is the fact that the dots for the other two observers are Time Dilated according to their speeds and spaced out accordingly.
But we can use the Lorentz Transformation process to redraw the diagram in the IRF in which the other two observers are at rest. First the rest frame for the red observer:
Please note that all the information that was contained in the first diagram is also contained here, the only difference being in the coordinates. The diagram for the rest frame of the red observer clearly shows that event B occurs before event A.
Finally, the rest frame for the green observer:
Note that in the diagram for the rest frame of the green observer, event B occurs after event A.
Also, please note that the reason why each observer comes to a different conclusion about the ordering of the two events is because they each are assuming that their own radar signal takes the same amount of time to reach event B as it takes for the echo to get back to them.
