Questions about a thought experiment involving time dilation

[Muzaz]

TL;DR Summary

My own thought experiment about time delation that puzzeled me.

I thought I understood relative time but got confused with this scenario. There are three observers, A, B, and C, each with precise atomic clocks to track each other’s time. Initially, A and B are on Earth, so their clocks match. Observer C is outside the solar system, moving slower based on cosmic background radiation (CBR), so time appears slower in C’s frame. From C’s view, the solar system is moving toward them.

Next, B launches from Earth in the opposite direction to the solar system’s motion (as seen by C), causing B’s clock to run slower relative to A due to increased speed. Meanwhile, C sees B’s speed decrease, meaning their clocks should start to sync.

If this continues, A passes C with the solar system, and B eventually reaches C’s location and speed. Now B and C are in the same frame and observe their clocks ticking together. Yet A sees B’s clock running slower and C’s running faster.

How can B and C be synchronized while A sees different times for both?

I understand relativity, but it seems I can't understand how this is possible: B/C are having the same frame of time with eachother but not when oberved by someone else.

Wouldn't this also mean that for example radioactive isotopes that were formed in different frame of time somewhere in spacex but happen to end up in earth may indeed have (slightly) different halving times.

 

[PeroK]

TL;DR Summary: Observer C is outside the solar system, moving slower based on cosmic background radiation (CBR), so time appears slower in C’s frame. From C’s view, the solar system is moving toward them.

I don't understand what you mean here. The time dilation between A/B and C will be symmetric.

TL;DR Summary: Next, B launches from Earth in the opposite direction to the solar system’s motion (as seen by C), causing B’s clock to run slower relative to A due to increased speed. Meanwhile, C sees B’s speed decrease, meaning their clocks should start to sync.

If this continues, A passes C with the solar system, and B eventually reaches C’s location and speed. Now B and C are in the same frame and observe their clocks ticking together. Yet A sees B’s clock running slower and C’s running faster.

A always measures C's clock to be running slow. Not faster.

TL;DR Summary: I understand relativity, but it seems I can't understand how this is possible: B/C are having the same frame of time with eachother but not when oberved by someone else.

Unfortunately, I think you've missed the first postulate of SR, which essentially says that all inertial motion is relative.

Velocity-based Time dilation is always symmetric. Whereas, you seem to be assuming that there is an absolute reference frame defined by the CMBR. This is a misunderstanding.