Relativity - time dilation

Question

I'm learning about relativity and I'm having some issues with it and the twin paradox. I found many questions and answers on this subject but they did not answer my specific problem.

In my thought experiment I added a third twin - triplet. So here it goes:

Let's have triplets $A$,$B$ and $C$. sitting still at same point in space.

They want to test relativity so they do this experiment..

They all reset their clocks, and $B$ and $C$ start moving together away from $A$ at some ridiculous speed $v_{365}$ which coincidentaly has lorentz factor $365$. they move at uniform motion in only one direction (let's say acceleration time nears 0).

State 1: $$\begin{align}t(A)=t(B)=t(C)&=0 \\ v(A,B) = v(A,C) &= v_{365}; \\ v(B,C) &= 0\end{align}$$

$t(A)$ being A's clock, $v(A,B)$ being velocity at which $A$ and $B$ are moving away from each other, etc..

Now after some time, let's say when $t(A) = 1\text{ year}$, brother $B$ starts to decellerate to full stop. It also shows up that brother $C$ was sleeping through whole first part of the experiment. But now he wakes up and sees that his brother $B$ is accelerating away from him in the opposite direction than was planned but that doesn't bother him.

State 2: $$\begin{align}t(A)&=1\text{ year} \\ t(B)=t(C) &\stackrel{?}{=} 1\text{ day} \\ v(A,B) &= 0 \\ v(A,C)=v(B,C) &= v_{365}\end{align}$$

Now after another while lets say when $t(A)=2\text{ years}$, brother $C$ sees that brother $B$ isnt stopping any time soon so he decides to match his speed.. (brother $C$ stops now from point of $A$)

State 3: $$\begin{align}t(A)&=2\text{ years} \\ t(B)&= 1\text{ year }1\text{ day} \\ t(C)&= 2\text{ days} \\ v(A,B)=v(A,C)=v(B,C)&=0\end{align}$$

That's it.. now they stand still some light year away from each other, but they can communicate and compare their clocks..

here are my questions:

1] Is my assesment of time dilation effects correct in this scenario? if not which state contains first error and why?

2] If my assesment is correct, then from point of brother $C$ it was brother $B$ who was accelerating away, yet it was also brother $B$ who aged when they got in same speed - therefore it seems to me its always the one who is matching speed who doesnt age - is this a general rule?

3] If answer to questions 1 and 2 is yes, then can this effect be used to gather infomation that would take much longer to extract in normal speeds?

Regards

Answer 1

To develop a reliable SR intuition, you should become comfortable with spacetime diagrams and the distinction between coordinate time $t$ and proper time $\tau$.

Choose a reference frame (coordinate system) in which A remains at the spatial origin; the coordinate time and A's proper time (A's "wristwatch" time) can be chosen to have the same value.

Let's denote the coordinate time in this frame $t_A$ and the proper time of A, B and C, $\tau_A, \tau _B$ and $\tau_C$ respectively.

Now, in the set-up you describe, we have:

$t_A = 0$: $\ \ \tau_A = \tau_B = \tau_C = 0$.

$t_A = 1$ year: $\ \ \tau_A = 1$ year, $\ \tau_B = \tau_C = 1$ day.

$t_A = 2$ years: $\ \ \tau_A = 2$ years, $\ \tau_B = 1$ day $+ 1$ year, $\ \tau_C = 2$ days.

The coordinate time is read directly from the spacetime coordinates of the events. The proper time is calculated along the world lines in the spacetime diagram.

While the world line for A is along the time axis of frame A, the world lines for B and C have kinks where the accelerations occur.

Importantly, there is no inertial reference frame in which the world lines for B & C do not have kinks. This is crucial to understanding why the elapsed proper times for B and C are less than A's proper time.

A remains in one inertial frame while B & C make "jumps" (boosts) to a frame moving with respect to the A frame and then back. The lines of simultaneity for B & C rotate at each kink.

Answer 2

1] Is my assesment of time dilation effects correct in this scenario? if not which state contains first error and why?

It seems fine to me. I am assuming $t$ refers to self-time everywhere, and that simultaneous events are defined by observer A.

2] If my assesment is correct, then from point of brother C it was brother B who was accelerating away, yet it was also brother B who aged when they got in same speed - therefore it seems to me its always the one who is matching speed who doesnt age - is this a general rule?

Yes. The initial acceleration of B away from C is not important: it just sets up the initial conditions in which B and C are moving at different velocities. At this point they both agree that they are of the same age.

What is important is who can trust their special-relativity calculations regarding the other guy, after some time has elapsed. C sees B moving away and he can compute B's time using dilation. But once C decelerates he can throw his calculations in the trash because special-relativity does not apply to him: he needs general-relativity. B's calculations about C are fine though, all the way to the end.

Perhaps you already understand this, but I want to emphasize that in general-relativity (i.e. if your space was not flat) you would be more restricted, and would generally have to get everyone to meet in the same place in order to compare their clocks. In special-relativity you can get away with what you did, because two observers at the same velocity can work out their relative age by sending light signals back and forth.

3] If answer to questions 1 and 2 is yes, then can this effect be used to gather infomation that would take much longer to extract in normal speeds?

I don't see how.