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Astronaut Spud Nick is space-traveling from planet X to planet Y at a speed of 0.60c relative to the planets, which are at rest relative to each...

First, what do you mean when you say "when he is precisely halfway between the planets as measured in the planet frame" and "the explosions are simultaneous in the frame of the planets"?


The only option I see is: before the spaceship neared the first planet, observers on both planets synchronized their clocks (the speed of these clocks is the same because the planets are in rest relative to each other), measured the speed of the spaceship, and estimated when it would approach the first planet. Then they computed how much time it would take for a spaceship to travel half of the distance and wrote down that time. And, finally, they explode bombs (actually, emit light) when their clocks show this computed time.


From the planets' point of view, the half of the flight will take `1/(0.6)` hours.


Now let's look on this from an astronaut's point of view. He travels between planets with the same speed `0.6 c.` The distance between the planets for him is less than it is for observers on the planet by the Lorentz factor `sqrt(1-0.6^2)=0.8,` so it is `2*0.8=1.6` light-hours.


Moreover, from its point of view, the planets' time is slower by the same factor. While observers on the planets wait `1/(0.6)` hours, for him it is `1/(0.8*0.6)` hours, and he travels `1/0.8=1.25` light-hours during this time. The remaining distance for him is `1.6-1.25=0.35` light-hours.


Finally, light travels at the speed `c` in all frames, so an astronaut will see the flash from the second planet after `0.35` hours and from the first after `1.25` hours. The difference is `0.9` hours, or 54 minutes.

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