Is the Twin Paradox a Logical Mistake in Time Dilation and Relativity?

[High Overlord]

Hi, my first post on the forums. I've known about twin paradox for a while, so when we learned special relativity at school in September, it wasn't anything weird. Math is elegant, unlike quantum physics. But a few weeks ago, I started wondering something. It might be just an ordinary logical mistake, so go easy on me.

Twin A is the one in the spaceship, and twin B stays with his wife and kids on Earth.

We will take Earth as the reference point.
In this system Twin A travels at relativistic speed v. Due to time dilatation he is aging slower, and after eg. 10 Earth years he will age eg. 6 years.

Now here is a thing I can't wrap my mind around:
To twin B, spaceship will be a reference point, and Earth will be moving away at -v. This means that time on Earth will be slower than his. So, when he returns from his voyage, people on Earth should be younger than him.

I've tried formulating this in proper time equation, in order to find an error, but square doesn't make a difference between -v and v. Will I get a different result if I use coordinates?

 

[Bill_K]

The key is what happens at the turnaround point. When the ship changes its velocity and heads back to Earth, its reference frame changes to a new one. Yes, people on Earth age more slowly in the new frame also, but there is a big change in going from one frame to the other, a large apparent change in Earth's time as seen from the ship.

 

[High Overlord]

So, if I understood it correctly, SR math is right. While they travel at CONSTANT high speeds, they will both see the other one age slower. But a real difference is when Twin A changes speed. This is then a consequence of general relativity?

 

[Dale]

It is still special relativity. But yes, that is the difference.

 

[bobc2]

So, if I understood it correctly, SR math is right. While they travel at CONSTANT high speeds, they will both see the other one aging slower. But a real difference is when Twin A changes speed. This is then a consequence of general relativity?

No. No general relativity is involved here. The essential difference is the path through space-time that the twins have traveled. The traveling twin took a short cut to the future point in space-time. We can explain the theory of space-time distances in more detail, but here is a sketch depicting the point I'm trying to make:

 

[High Overlord]

Thanks everyone
@bobc2 I am aware of space-time interval, my problem was with the relativity and equivalence of frames of reference. I presumed that both systems were INERTIAL, but as Bill K pointed out, they are not. Even if acceleration was instant, in that 1 moment the system would change speed, and not be inertial. And that thing decides which one is younger.
Now, the last thing. If I get this correctly, mod please lock this thread.
If we were to compare them at distance, without twin A coming back, we would get that they are the same age, because present of one twin is in the future of other, and vice versa. The twin paradox can be only observed when we bring the other one back.
Is this correct?

 

[bobc2]

Thanks everyone
@bobc2 I am aware of space-time interval, my problem was with the relativity and equivalence of frames of reference. I presumed that both systems were INERTIAL, but as Bill K pointed out, they are not. Even if acceleration was instant, in that 1 moment the system would change speed, and not be inertial. And that thing decides which one is younger.

The one that accelerates certainly gives you the tip off of who will be younger. That's because having accelerated, he is certainly taking a shorter path than the twin who does not accelerate. The acceleration just serves to get him on a short return path (a short path is the only thing available for getting back home). The time spent during the turn-around can be made insignificant. The aging is accumulating along the inertial segments. This final accumulation is what will be compared to the stay-at-home twin.

If we were to compare them at distance, without twin A coming back, we would get that they are the same age, because present of one twin is in the future of other, and vice versa. The twin paradox can be only observed when we bring the other one back.
Is this correct?

Not necessarily. What frame (or other criteria) are you going to use to compare ages? It's obvious how to compare ages when they reunite. This discussion could go off in a number of directions.

 

[High Overlord]

@bobc2
Sorry for wasting your time, I'm an idiot.
My calculations were pointing out that we couldn't know which one is younger. And I found the answer why.
I was using speed 0 for the first reference frame. No problem, that is correct. For the second one I should use speeds v or -v. I used them both. One for one way, other for the way back. And this makes it THE SAME frame as in the first case. I have just googled space time diagrams for the other 2 cases(v and -v), and twin A takes a shortcut in those cases, too. Thanks bobc2.

 

[1977ub]

This discussion could go off in a number of directions.

I wonder how these conversations would go if people encountered barn/pole paradox first. :) Understanding relativity of simultaneity leads one to walk a bit more carefully over the territory of clock/length contraction for relatively moving observer.