Undergrad Questions about a thought experiment involving time dilation

[Muzaz]

TL;DR

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.

 

[Nugatory]

There are three observers, A, B, and C, each with precise atomic clocks to track each other’s time.

No matter how precise it is, a clock will only measure its own time. When we say “This event happened at time T” we really mean “This event happened at the same time that our clock read T”. Thus when we say that “clock A is running slower than clock B”, we are saying something like “At the same time that clock A read $T_{A1}$ clock B read $T_{B1}$. Later, at the same time that clock A read $T_{A2}$ clock B read $T_{B2}$. $T_{A2}-T_{A1}$ is greater than $T_{B2}-T_{B1}$; therefore clock B is running slower than clock A”. But clearly this will depend on what we mean by “at the same time”, and because of relativity of simultaneity that depends on which frame we want to use to analyze the problem. (If you are not familiar with relativity of simultaneity, learn about it - it is essential to understanding special relativity and especially resolving apparent contradictions like the one in your original post).

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.

Yes, if we choose to describe the situation using the frame in which C is at rest with its definition of “at the same time” then A’s clock will be running slow and B’s will be even slower. However, if we choose to describe the situation using the frame in which A is at rest, then it will be B and C which are running slower; and if we choose the frame in which B after acceleration is at rest then A will be running slower and C even slower.

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.

Two identically prepared sample of radioactive material at rest relative one another will have the same decay rate and halving times. If we choose to describe the situation using a frame in which they are moving (necessarily at the same speed, if they are at rest relative to one another) the decay rate will be lower and the half-life greater than if we describe the situation using a frame in which they are at rest.
In fact, we can use them as clocks - just say that they tick once every halving and they could be my clocks A and B above.
However, depending on their history they may not be be equally decayed when we place them side by side to compare; depending on how they moved around before they met one could be younger than the other. This is just the classic Twin Paradox using radioactive decay clocks instead of biologically aging twins. However, this age difference is not the result of time dilation, and trying to understand it in terms of time dilation will just confuse you.