unaccelerated twin paradox in SR

O2 is moving at about 0.8660 lightspeed relative to O1, and O3 is moving at -0.8660 lightspeed relative to O1. That makes both of their time dilation factors exactly 2. Using (u+w)/(1+u*w) to get O2 and O3's relative velocity, we find it to be 0.9897, which implies a time dilation factor of 7.

Clocks O1 and O2 are synchronized to t=0 when they pass close by. Clocks O2 and O3 are synchronized to t=2 when they pass close by (at a time "simultaneous" (according to O1) with O1's t=4).

Note that the above diagram shows all three reference frames in one colorcoded diagram. The coordinate axes of each observer are shown with the time axis as a thicker line (red=O1, green=O2, blue=O3), and the space axis of each observer as thinner lines drawn through events of interest. For example, the three vertical red lines are drawn through the three crossover events, when O1 and O2 cross, then when O2 and O3 cross, then when O3 and O1 cross.

Note that whether we choose to view this as a diagram of O2 going out, passing a clock value to O3, and O3 coming back, or whether we choose to view this as a diagram of a single O[23] making a trip out and back, the intervals, events, and axes drawn would all be the same. Note that, because this O[23] is actually "in" two different coordinate systems, O[23] can't just use time dilation to calculate how much time elapses for O1. O1 can use time dilation alone to account for all of O[23]'s time, but not the other way around.

In particular, on the diagram above, the time between O1's t=1 (where the green space axis crosses the red time axis) and O1's t=7 (where the blue space axis crosses the red time axis) is skipped because of the turnaround. The formula for this skip is ux/(1-u^2)^.5, where u is O1 and O2's relative velocity, and x is the distance from O[23] to O1. Plugging in these values, we get 0.9897*1.732/(1-0.9897^2)^.5, or 12 seconds of O2 or O3's time skipped. Thus, while O1 can calculate O[23] total trip time by just taking 8*(1-0.8660^2)^.5=4, O2 or O3 must account for the skipped time, and calculate by (2+2+12)*(1-0.8660^2)^.5=8. Therefore, we see that each observer agrees about the time experienced by the other observer(s).

The formulas for time dilation and "time skipping" can be found in several places, one of which is the Feynman Lectures on Physics, volume I, section 15.

Below is a picture of the same situation, but "zoomed out" to show the event on O3's coordinate system simultaneous with the O1/O2 crossover. In addition, the time values for each observer at each crossover point are displayed.

Below is the same situation, but with O2's coordinate axes drawn as perpendicular.

And finally, below is the same situation, but with O3's coordinate axes drawn as perpendicular.