Is this spacetime diagram for the twins paradox correct? The link at question is here: http://www.einsteins-theory-of-relativity-4engineers.com/twin-paradox-graphical-solution.html

My question in specific is the loop in Jim's worldline. I know that typically in the Twins paradox with a instantaneous acceleration or inertial frame jump, the difference in ages is explained by the gap in time the the travelling twin doesn't experience due to this changing of frames. But is this loop how it would look with a constant acceleration, or is the author of this picture incorrect? Is a closed loop like this even possible without going faster than the speed of light?
Thanks for any help.
 A: Even though the author doesn't specify what math they did, it's pretty straightforward to tell that they don't know what they're doing. Relativity is extremely permissive about what coordinate system we use, but when we have an acceptable coordinate system and then do a change of coordinates to get a different one, there are certain requirements. The functions expressing the new coordinates in terms of the old ones must be smooth, and they must also be one-to-one. The fact that Jim's world-line crosses itself tells us that whatever coordinate system the author used, it wasn't one-to-one. So whatever they did was just plain wrong.
It's fine to try to do treatments of the twin paradox in this style. Special relativity can handle accelerating frames of reference (contrary to what some people say). However, it can be a little tricky to get it right; counterintuitive things can happen; you have to be careful about your mathematical assumptions; the description can be nonunique; and there is no guarantee that you will end up with a single coordinate chart that covers all of spacetime. The most common description is referred to as the Rindler coordinates. If someone wanted to do a better presentation in this style, probably a nicer way to do it would be to let Pam have constant proper acceleration. Then the transformation would simply be the transformation from Minkowski coordinates to Rindler coordinates. There is also a treatment in this style in Hewitt, Conceptual Physics.
The danger in this style is that impressionable people will get the idea that there's only one way to do it, or that all kinds of presentation-dependent facts are "real." We can never say whether a certain event for Jim and a certain event for Pam are "really" simultaneous. At best they are simultaneous according to a certain convention defining simultaneity. This was in fact one of the basic insights leading to Einstein's 1905 formulation of relativity: that simultaneity is a matter of convention.
