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Your space-time diagram is only drawn for the rest frame of the Earth. In order to get a true picture of what is going on, you need to draw the space-time diagrams for the other two frames as well.Sorry, not understand.

This is, I think, an spacetime diagram of the scenario I proposed in my original post.

3-4 and 4-5 are straight in my point of view, I´m wrong?

View attachment 211955

Here's your diagram redrawn with markers showing how time passes for each of the three participants according to the Earth frame. I gave the two spaceships a velocity of 0.6c relative to the Earth. (I chose this value because it makes the later diagrams easier to read.)

The blue line is the Earth observer, the green line is our traveling twin and the red line the inbound spacecraft. 0 marks where the Earth and traveling twin cross paths (I did not extend the lines prior to this point, even though you can imagine them as being so.)

If each number represents 1 year.

For every year the Earth photo ages, the traveling twin's photo ages 0.8 years. In four years earth time, the two space ships meet, and the traveler's photo has aged 3.2 years. The incoming ship takes a photo of this photo. The new photo (of a 3.2 year old photo) also ages 0.8 years for every Earth year, so that upon its reaching Earth it ages 3.2 years while the Earth photo ages 8 yrs. The numbers on the red line represent the 3.2 years that the traveler's photo aged, plus the additional aging of its appearance during the return leg. (I need to note here that the 5 on the red line is misplaced, it should be level with the 5 on the green line. I didn't catch it right away, and with the software this was drawn with, the only way to fix it was to to go back and start from scratch. It is only a small mistake and doesn't really effect the general gist of the example.)

Now we look at events as they occur for the traveling twin. For that we use the following space-time diagram. (keep in mind that these are the exact same events as in the last diagram, just drawn according to a different frame.)

Note that the Earth and traveling twin still pass each other when their respective time marks are zero. Now however, it is the Earth photo that ages 0.8 years for every year that the traveling twin's clock ages. When the traveling twin meets the incoming ship it the photo has aged 3.2 years and the Earth photo has aged only a little more than 2 1/2 years. The relative speed between the traveling twin and the incoming ship is ~0.882c. in the same direction as the Earth is traveling with respect to the traveling twin. Thus according to the traveling twin, the other ship has to chase after and catch up to the Earth. This is take a lot longer by the traveling twin's clock than it took between his passing Earth and meeting up with the other ship. During that time, even though it is aging slower than his photo, the Earth photo will age an additional 5.5 years for a total of 8 yrs. Since the photo on the inbound ship is moving at 0.882c relative to the traveling twin, his will measure this photo as aging at a rate of 0.47 years for every one of his years and will measure it as having aged 3.2 yrs upon reaching Earth. We end up with exactly the same answer as we got according to the Earth frame.

Finally, we look at the incoming ship frame.

The thing to note here is that by this ship's time reckoning, the Earth and the traveling twin pass each other almost 7 years before he meets up with the traveling twin's ship. In that time, the Earth photo ages over 5 years at 0.8 years per year and the traveling twin's photo ages 3.2 yrs at 0.47 years per year. It takes an additional 3.2 years for him to meet up with the Earth, while the Earth photo ages to 8 years.

No matter which of the three frames you are working with, you end up with the same results, even though they will disagree as whose photo was aging slower throughout the exercise.

It also illustrates what people having been saying about the difficulty with defining "at the same time" in Relativity.

In the Earth frame "at the same time" that the traveler's photo has aged 3.2 years, the Earth photo has aged 4 years.

However, according to the traveling twin, "at the same time" that the traveler's photo has aged 3.2 years, the Earth photo has aged 2.56 years.

And, according to the inbound spaceship, "at the same time" that the traveler's photo has aged 3.2 years, the Earth photo has aged 5.44 years