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Hmm - there's a problem here already, as the inertial observer experiences the longest proper time between events,After i figured out how to show length contraction in this topic. I tried to use a similar way to show time dilation in Minkowski diagram.Time dilation means that time interval between two events is the shortest in the frame in which those two events happen in same place. We call this frame its rest frame.

You are missing at least two lines on your diagram, and I don't see them in in any subsequent posts, so I'm guessing that you're still missing something. I'd suggest adding three more lines, and two more points:

One important pre-requisite, which I'll call PRE Make sure that the Lorentz interval, also called the proper time, between the origin and point 1 is the same as the Lorentz interval between the origin and point 1'. I think you addressed this by saying that points 1 and 1' should both be on the same hyperbola of (ct)^2 - x^2 = constant.

The first line that you are missing is the set of points that are simultaneous with event 1 in the (t,x) frame. Hint: this set of points is parallel to the x axis.

The second line is the set of points that are simultaneous with event 1 in the (t', x') frame. Hint: this set of points is parallel to the x' axis. The important thing to note is that this set of points is DIFFERENT that the set of points of our first line. Otherwise, we won't be using this line.

The third line that you are missing is the set of points that are simultaneous with event 1' in the (t', x') frame. Hint: this is parallel to the x' axis, similar to line 2, but not similar to line 1.

So those are the three lines I suggest you add - now come the points, and the meaning of time dilation on your graph.

In the (t,x) frame, line 1 represents the set of points simultaneous with the event you labelled as "1". Find point 2, where this line intersects Ranja's worldline. Time dilation says that point if point 1' occurs after point 2, because using the simultaneity conventions of the (t,x) frame, the clock in the (t', x') frame is ticking slowly. (Be sure that the precondition PRE is met!)

In the (t', x') frame, line 3 represents the set of points simultaneous with the event you labelled as 1'. Find point 3, where this line intersects Ziga's worldline. Time dilation says that point 1 comes after point 3. Again, be sure the precondition PRE is met.