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Let me use my favorite actors, Bob and Alice. Bob is the stationary observer; Alice is the moving one. Both are in inertial frames (or however one prefers to phrase it). We look at Alice's direction of movement and create a coordinate system for her such that her motion is completely in the positive X direction. We then set up Bob's axes to be parallel to each of Alice's.

There is some point at which Bob's and Alice's x coordinates are the same—as determined by Bob. For Bob, all the x coordinates on his worldline are 0. We set his clock time (t) to 0 at this point. For Alice, the situation is the same: all the x' coordinates on her worldline are 0 and we set her clock time (t') to 0.

We can now use the spacetime diagram to solve problems related to Bob and Alice, so this is very useful. It's like a geometric version of the Lorentz transform.

One pitfall is that when Bob and Alice have the same x coordinate, it is very tempting to think that they are at the same spot (at least, I keep falling into that trap). Some problems are actually stated that way and then the spacetime diagram is drawn, but the spacetime diagram looks no different if Bob and Alice are separated by wide distances along the y and z axes. While I can say that both Bob and Alice have a clock time of 0 at this magic location, I cannot say that they can observe each other's clocks to read 0 without adding that their y and z separation is 0—this information is not included in the diagram.

Another pitfall that I just ran into is in determining if two events have spacelike, timelike, or lightlike separation. I was thinking that I could just look at the angles made by a line connecting the two events on a spacetime diagram. Thinking about it, I decided I couldn't because we cannot ignore the y and z coordinates in making the determination (at least, I don't think so).

Neither of these are the diagram's fault but they are easy traps to fall into. Are there other things which spacetime diagrams mislead us about?