Here are some Spacetime Diagrams to illustrate the Inertial Reference Frames (IRFs) discussed in this thread. I use a velocity for the two traveling sisters of 0.6c so that the Relativistic Doppler shifts will be factors of 2 to make it easy to keep track of them in the diagrams. Each triplet sends out a signal each month to the other two triplets. The sisters travel for 12 months and then turn around and spend another 12 months coming home. When they meet, they have both aged 2 years but their stay-at-home brother has aged 30 months.
I am taking the triplet scenario from post #5 in which I show Stella as a thick black line with dots every month of her Proper Time. She sends out signals shown as thin black lines. She is traveling to the right for a distance of 9 light-months and then turns around.
Ursula is shown as a thick red line with dots every month where she sends out a signal shown as a thin red line. She travels to the left for 9 light-months and then comes back.
Terence stays home and is shown as a thick blue line with dots every month. His signals are shown as thick yellow lines going in both directions. I do it this way because there are times when his signal overlaps with a signal from one of his traveling sisters and I wanted to be able to show both signals traveling together.
The purpose of these diagrams is to show how Time Dilation, Length Contraction, Relativity of Simultaneity and Relativistic Doppler all work in the different IRF's. The first three are different in each IRF but Relativistic Doppler is the same in each IRF. Once I have set up the scenario in the original IRF, I simply use the Lorentz Transformation to create the other IRF's.
Time Dilation is the ratio of the Coordinate Time (marked by the grid lines) to each triplet's Proper Time (marked by the dots). Time Dilation is dependent on the speed of an observer relative to the IRF, not to another observer. At a speed 0.6c the Time Dilation factor is the gamma factor which is 1.25. In the last two diagrams, there are times when some of the triplets travel at 0.882c (the relativistic addition of 0.6c + 0.6c) and for them the Time Dilation factor is 2.125.
The Relatistic Doppler factor is also dependent on relative speed but this time it is not between an observer and the IRF but rather it is between two observers, keeping in mind the time delay between when one observer sends the signal and the other observer receives the signal. The Relativistic Doppler factors at 0.6c are 2 for approaching and 0.5 for receding. At 0.882c they are 4 for approaching and 0.25 for receding.
Note that relative direction matters for Relativistic Doppler but not for Time Dilation.
Ok, we start with the IRF in which Terence remains at rest and in which the two sisters start and end up at rest:
Note that during the outbound leg of both sisters' trips, they receive signals from Terence at one-half the rate that they send them and they receive signals from the other sister at one-quarter the rate that they send them. As soon as they each turn around, they start receiving the signals from Terence at double the rate that they send them but they receive the signals from each other at the same rate that they send them because they are both traveling at the same speed when both sending and receiving. Then near the end of their trips, they start seeing the signals from the other sister at four times the rate they are sending them.
Terence, on the other hand, receives signals from his two sisters at one-half the rate he is sending them for most of the time they are away but near the end, he starts receiving them at double the rate he sends them.
Between the sisters and their brother, the Doppler shifts are either 0.5 or 2 but this ratio switches near the end of the scenario for the brother but at the turn-around point half way through the scenario for the sisters. So the brother sees his sisters time running slow for more than half the scenario which accounts for why they end up with less time accumulated than he does.
Please read the previous posts and links for additional information on how to interpret these diagrams.
Now for the IRF in which Stella is at rest during the outbound portion of her trip and in which Ursula is at rest during the inbound portion of her trip:
Note that this IRF makes no difference to the Doppler shifts of the received signals as received by each triplet but it does make a difference to the Time Dilation that each triplet is subject to. For example, it is Terence who is traveling at 0.6c and subject to a Time Dilation of 1.25 while his sisters spend some of their time at rest (no Time Dilation) and some of their time traveling at 0.882c and subject to a Time Dilation of 2.125.
You can see the effect of Length Contraction if you look at the distance between one of the sisters at the moment of turn around compared to their brother. That distance contracts from 9 light-months to 9/gamma = 1/1.25 = 7.2. Look at the Coordinate Time of 12 months and observe that Stella (black) is at a Coordinate Location of 0 while Terence (blue) is at -7.2 months. Similarly, at Coordinate Time of about 25.5 months, Ursula (red) is at -22.5 light-months while Terrence (blue) is at -15.3 for a delta of 7.2 light-months.
The Relativity of Simultaneity is demonstrated all over these diagrams when you compare pairs of events in the two IRFs. For example, in the original IRF, both sisters turned around at Coordinate Time of 15 months but in this IRF, Stella (black) turns around at Coordinate Time of 12 months while Ursula (red) turns around at a Coordinate Time of 25.5
months.
Finally, the IRF in which Stella is at rest during the inbound portion of her trip and in which Ursula is at rest during the outbound portion of her trip:
Hope this helps. Any questions?
) who stayed at home.
