In the previous two pages on the "twin paradox", we have looked at inertial frames, where the period of Pam's acceleration was so small that it could be ignored. In this version of the twin paradox, Pam accelerates all the way to the turnaround point and back. She maintains a positive acceleration for the first quarter (halfway to the turnaround) and a negative acceleration for the second and third quarters, which bring her back to the halfway point again. From there on Pam accelerates positively again in order to bring her to a stop near Earth.
We have chosen the turnaround point at 4 light years from Earth (near the star Alpha Proxima) and found that an acceleration of 0.8g will bring Pam back in approximately 12 Earth years. She will reach a maximum relative speed of 0.924c at the halfway point (on both the outbound and the inbound legs), resulting in Pam aging only 8 years during the 12 year voyage (Earth time).
The left graph is the usual hyperbolic motion of constant proper acceleration, viewed from Jim's inertial frame near Earth. The blue bullets on Pam's worldline shows how she ages relative to Jim, for a total of 8 years during Jim's 12 years of reference frame time. The green worldline belongs to a hypothetical star near Alpha Proxima that is at rest relative to Earth.
The right graph shows the same scenario from Pam's accelerating frame of reference. Her acceleration and the reversal thereof have interesting effects on the worldlines of Jim and the star, as viewed by Pam. We see sudden changes in the slope of the worldlines, as well as loops in spacetime! A sudden change in the slope of a worldline indicates a sudden jump in apparent velocity of the object. Loops indicate apparent spacetime movement backwards in time. How can this be?
Let's take the two issues one by one. The sudden changes in the slope of the worldines of the right hand graphs happen where the acceleration reverses (e.g. at the red 3-year marker). A positive acceleration means that the relative velocity increases and so does the Lorentz contraction. As Pam flies away from Earth at increasing speed, the distance between Pam and the Earth appears Lorentz contracted more and more. The result is that the apparent speed of recession of Earth is less than what it would have been without Lorentz contraction. Therefore the (magnitude of the) slope of the red worldline from 0 to 3 years is less than what it would have been.
When Pam's spaceship starts to decelerate, the relative speed between her ship and Earth starts to decrease. The Lorentz contraction then becomes less and it appears as if Earth is moving away from her at a greater speed. In fact the apparent speed eventually becomes superluminal, with a slope "flatter" than that of light (in this case, with a slope of less than -1). This is not a "real speed", but simply as things appear from an accelerating frame of reference. One must remember that even the speed of light is not the same in all directions when measured in an accelerating frame of reference.
The second interesting effect is the apparent time reversal of the right-hand curves. This comes from the differences in the definition of simultaneity between Pam's accelerating reference frame and the Jim/star inertial frame. As Pam picks up speed relative to Jim and as the distance between them increases, their clocks get more and more out of synchronization. The dotted lines connecting Jim's frame to the star frame for corresponding times illustrate that - they are lines of synchronization for Jim's frame, while Pam's lines of synchronization are all horizontal. So, no clocks go backwards in time; it is just when Pam projects Jim's clock along his line of simultaneity that the "loops in time" appear.