How Does Acceleration Affect Time Dilation in Twin Space Travel Scenarios?

[ghwellsjr]

NOTE: I compiled this response right after you made your post but by the time I got ready to upload it, many others had provided similar responses so I wasn't going to post it, but since you are still struggling with these issues, I thought it might be helpful to you.

You misread the crucial part, in the first scenario it is the accelerated twin that travels from A to B but in the second scenario the accelerated twin is the one that stays at A.

I fully understand your two scenarios and I'm trying to help you see that in SR you can choose any inertial reference frame to analyze scenarios. You seem to think that only an inertial frame in which the sun or some other massive body is stationary is valid. But I chose to analyze both your scenarios from a Frame of Reference in which both twins start out at rest. In your first scenario, the sun is also at rest in my chosen FoR but in your second scenario, the sun is moving in my chosen FoR which is a different FoR from the first one.

The question I am trying to address is whether the crucial part is who has undergone acceleration from the original inertial reference frame or who is traveling with the highest velocity with respect to the sun...

The sun is not crucial in either of your scenarios.

As you say "it is being at a higher speed for a longer time that causes less elapsed time". Yes but higher speed in relation to what?

In relation to your arbitrarily chosen reference frame.

True, in the classical twin test it does not matter. It both twins take off in the same direction at speed k*c with respect to the sun, this frame is the inertial frame.

I'm not sure if you mean a frame in which the sun is at rest or a frame in which the twins are at rest after they accelerate but you can use either one, it's your choice (or any other one for that matter). I would choose the latter.

Now one twin takes off at a speed of (b-k)*c in the opposite direction, all speeds measured in relation to the sun.

Since you are defining speeds in relation to the sun frame, I would have to calculate what these speeds were in my chosen frame if you were asking for quantitative results, but since you only care about which twin elapses less time, I don't need to worry about this.

Then he stops and chases after the first twin at a speed of (b+k)*c until he reaches him. Now it happens that it does not really matter what the numerical value ok "k" is. One twin spends all the time at velocity k*c and the other twin spends half the time at velocity (k+b)*c and the other half at velocity (k-b)*c. As long as (k+b) < 1 (choosing otherwise we would have superluminal speeds) the twin that has accelerated will always experience more elapsed time, no matter how we choose k and b.

The accelerated twin, the one that did the chasing, will experience less elapsed time, not more.

But, given a certain b, will the difference in elapsed time always be the same, no matter how we choose k? (maybe this is actually the case, then I have to rethink things...)

As is obvious from my choice of frame, the value of k won't matter.