I was thinking about the twin paradox in Special Relativity and I thought I understood it fine, but when I view the "paradox" in a certain way, I get confused. So we have two twins, John and Jim. Let A denote the point in spacetime where Jim leaves earth in a rocket ship, travelling at relativistic speeds (the exact speed is not important for my problem). Suppose that John was at the launch site and that he waited at the launch site the entire time for Jim to return (suppose Jim's rocket ship is capable of safely landing at the exact same launch site). Let B denote the point in spacetime where Jim returns to the launch site and is reunited with his brother John. Then the world line of John and Jim both pass through A and B, although their world lines are different. In fact, the segment of John's world line with endpoints A and B is a straight line, while segment of Jim's world line with endpoints A and B will be a curved line.
Because we are in $\mathbb{R}^4$, the metric is flat and thus the length of John's straight line with end points A and B is shorter than the segment of Jim's worldline with end points A and B. The proper time of John's straight line is given by its length and the proper time of Jim's wordline segment with end points A and B is also given by its length. So it appears that the time experienced by John is shorter than the time experienced by Jim, but this is obviously not true!
Where is the fault in my reasoning?
