The socalled "twin paradox" remains difficult to wrap one's head around. The graphical solution to the "paradox" presented here goes a long way towards demystifying it. The secret of this solution lies in the vivid illustration of the accelerations involved, rather than just viewing inertial frames. It clearly points towards the asymmetries in the reference frames of the respective twins.
Let's just recap what the "paradox" is all about. Pam sets off on a long, fast round trip while Jim, her twin brother, remains on Earth. When Pam returns, they find Jim to have aged more than Pam due to the velocity time dilation that Pam experienced relative to Jim. To put figures to it, let's round the distance to Alpha Proxima to 4 light years (ly) and say Pam quickly accelerate to a speed of twothirds of the speed of light. The round trip will last 12 years of Earth time. Pam experienced a time dilation factor of around 0.75, so she would have ages only 9 years during her voyage (three quarters of Jim's 12 years).
No problem, except that one may ask: Pam could have considered herself as stationary and that it was Jim that flew away from her. In such a case Pam should have ended up older than Jim. This is the essence of the "paradox". The solution lies in the asymmetry due to the fact that Pam did the physical acceleration at the start and end and also at the turnaround point, while Jim remained at rest (or inertial) for the whole time. This is a correct interpretation, yet it can also be confusing, because it has been proven that acceleration per se does not influence the rate of clocks. The bulk of the voyage is in any case inertial (cruising at constant speed), so where does the difference in the ages of the twins come from?
In the variant of the
twin paradox presented here, Pam accelerates or decelerates all the way to the turnaround point and
back. She maintains a positive acceleration for the first quarter
(halfway to the turnaround) and then decelerates until the turaround point. The return trip is essentially just the reverse of the outbound leg.
The round trip to Alpha Proxima and back (8 ly distance) is chosen to again last about 12 Earth years, which requires an
acceleration/deceleration of 0.8g (80% of normal Earth gravity) as measured by the spaceship accelerometers. This is called "proper acceleration" and is not the same as the coordinate acceleration as observed by Jim. Relativistic effects will cause the coordinate acceleration to be less than the proper acceleration that Pam feels.
Pam 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 math for calculating this is slightly complex, but do not worry about it. The graphical representation below is much more userfriendly!^{[1]}
The left graph shows how Pam gradually accelerates and decelerates at 0.8g, as viewed from Jim's inertial frame near Earth. She reaches about 60% of light speed after one year of Jim's time and 92% of light speed after three of Jim's years, which is the halfway point.
The blue
bullets on Pam's worldline show how she ages relative to Jim. At the halfway point, her calendar reads only 2 years, meaning a
total of 8 years during will elapse for the total voyage. In this type of diagram, the more the worldline deviates from the vertical, the more the spacing between the "yearring" markers is stretched. If the slope becomes 45 degrees (the speed of light), the spacing is stretched to infinity, i.e., no aging takes place. At the 92% of light speed that Pam reaches, the "time stretch factor", also called gamma", is about 2.55.
The
green worldline belongs to a hypothetical star near Alpha Proxima that
is at rest relative to Earth, meaning there is no stretching of the time scale. The star time runs in step with Earth time, or more technically, they have a common inertial frame of reference.
The righthand graph shows the same scenario
from Pam's accelerating frame of reference. Her acceleration (and the
reversal thereof) has 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 3year 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 slope if 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 form 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 righthand 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.
Interestingly, the green star worldline does not have a loop, but still displays the "backwards time" characteristic (see the main graph above). In effect, the bottomright green curve (from 0 to 6 years) is a leftright mirror image of the red curve from 6 to 12 years. This is simply because Pam's turnaround happens at the location of the star, but it happens 4 ly away from Jim. If Pam would have
turned around at Jim's location and headed out to the star again, the green
worldline would have displayed an identical loop to the red one.
How does this solve the "twin paradox"? Recall that the "paradox" is about viewing the situation from the two different inertial frames and apparently getting different (paradoxical) results. The above analysis shows that the results are the same when viewed from either Jim's or Pam's frame of reference  Jim ages 12 years while Pam ages 8 years. The fact that Pam accelerates and Jim stays inertial makes the situation asymmetrical and there is no valid argument for a paradox. The inclusion of the acceleration phases in the graphs makes this a lot more visible.
Do you agree? Just click "Reply" below and let us know...
Jorrie
[1] The math can be found in the downloads from this web page. The main graph is also available on this page, where it is viewable on a larger scale than what the CR4 editor allows.

Re: Twin Paradox  a New Graphical Solution