How to solve twin paradox in special relativity

They used many mathematical tweak to solve this paradox but it is still impossible to solve (at least philosophically).
I extended this paradox: If two twin brother Alice and Bob born in planet A. And at the same time both of the twin make a trip to other planets, planet B and C that have similar distance to  planet A but in opposite direction that is Alice go to planet B while Bob go to planet C and both of them have a same velocity relative to the planet A. Then after they returned to planet A who was young?

According to special theory of relativity both of them must observe time dilation of other twin. But when realizing this situation is similar for the twin. Then it is impossible to decide who was younger after returning to planet A. But if there is no twin younger after the trip, then why we used time dilation formula? 

Is there a gravitational difference between planets A, B, and C?

The key part of the twin paradox is the shift in reference frame.

For the original paradox, A stays on Earth, while B runs off to another planet and returns home.
If you can understand that, then the understanding of this modified version will come easily.

For the twin on the moving ship, their speed makes the other twin appear to age slower, as each snapshot of the twin's life takes longer and longer to reach them, in effect causing a Doppler shift.
The relative motion does the same for the observer on Earth.
Then on the return journey, the opposite happens, with them both appearing to age much faster.

The key difference is in how long each of these periods are.
Consider when B gets to the other planet.
They receive an image of A just a short time after they left.
But this image of B only reaches A shortly before B arrives home.
This means for the entire duration of the return journey (half the time for B) they are seeing A age much faster than themselves.
But for A, it is only a small portion of time that they see B age faster for.
This leaves B much younger when they reunite.

With your example, where we have another go to a different planet (which I will say is C), the same thing happens between A and C, with A being much older than C when C gets back home.

The only additional complexity is how B and C see each other.
Lets assume that it is symmetric, with both going at the same speed, in exactly opposite directions, travelling for the same distance, then turning around and coming back.
So now what happens on the outbound journey?
They both see each other ageing much more slowly, even more slowly than A.
This continues all the way until they turn around.
This turn around point is where it gets more complex.
For a portion of this (in fact most of it) they would see A ageing rapidly, while the one in the other ship ages at the same rate as them (but still remains younger).
This continues until they observe the one in the other ship turn around, at which point they start to appear to age rapidly, more rapidly than A is ageing.

This means that A will be the oldest (who stayed on the planet), while B and C are both younger than A, but the same age.

If you remove some of the symmetry then they can get different ages.

They used many mathematical tweak to solve this paradox but it is still impossible to solve (at least philosophically).
I extended this paradox: If two twin brother Alice and Bob born in planet A. And at the same time both of the twin make a trip to other planets, planet B and C that have similar distance to  planet A but in opposite direction that is Alice go to planet B while Bob go to planet C and both of them have a same velocity relative to the planet A. Then after they returned to planet A who was young?

According to special theory of relativity both of them must observe time dilation of other twin. But when realizing this situation is similar for the twin. Then it is impossible to decide who was younger after returning to planet A. But if there is no twin younger after the trip, then why we used time dilation formula?

We use dilatation formula to calculate how big is the difference between each brother and the people on their planet.

The key part of the twin paradox is the shift in reference frame.

For the original paradox, A stays on Earth, while B runs off to another planet and returns home.
If you can understand that, then the understanding of this modified version will come easily.

For the twin on the moving ship, their speed makes the other twin appear to age slower, as each snapshot of the twin's life takes longer and longer to reach them, in effect causing a Doppler shift.
The relative motion does the same for the observer on Earth.
Then on the return journey, the opposite happens, with them both appearing to age much faster.

The key difference is in how long each of these periods are.
Consider when B gets to the other planet.
They receive an image of A just a short time after they left.
But this image of B only reaches A shortly before B arrives home.
This means for the entire duration of the return journey (half the time for B) they are seeing A age much faster than themselves.
But for A, it is only a small portion of time that they see B age faster for.
This leaves B much younger when they reunite.

With your example, where we have another go to a different planet (which I will say is C), the same thing happens between A and C, with A being much older than C when C gets back home.

The only additional complexity is how B and C see each other.
Lets assume that it is symmetric, with both going at the same speed, in exactly opposite directions, travelling for the same distance, then turning around and coming back.
So now what happens on the outbound journey?
They both see each other ageing much more slowly, even more slowly than A.
This continues all the way until they turn around.
This turn around point is where it gets more complex.
For a portion of this (in fact most of it) they would see A ageing rapidly, while the one in the other ship ages at the same rate as them (but still remains younger).
This continues until they observe the one in the other ship turn around, at which point they start to appear to age rapidly, more rapidly than A is ageing.

This means that A will be the oldest (who stayed on the planet), while B and C are both younger than A, but the same age.

If you remove some of the symmetry then they can get different ages.

Why do you puzzle? All solved for you for a long time! Satellites of an echelon two, solved this problem long ago having applied as an argument of The etheric theory of Lorentz. That's all. This practical solutions of your question. Also there is no paradox. It is necessary to read scientific books - Sorry. Wagged the tail.

They used many mathematical tweak to solve this paradox but it is still impossible to solve (at least philosophically).
I extended this paradox: If two twin brother Alice and Bob born in planet A. And at the same time both of the twin make a trip to other planets, planet B and C that have similar distance to  planet A but in opposite direction that is Alice go to planet B while Bob go to planet C and both of them have a same velocity relative to the planet A. Then after they returned to planet A who was young?

According to special theory of relativity both of them must observe time dilation of other twin. But when realizing this situation is similar for the twin. Then it is impossible to decide who was younger after returning to planet A. But if there is no twin younger after the trip, then why we used time dilation formula?

I didn’t read any of the responses so this may have been covered.

The twin paradox already has a solution. Both twins see the other as moving but only one twin felt an acceleration. The one that felt the acceleration to get up to a moving velocity is the twin with the moving reference frame.

I didn’t read any of the responses so this may have been covered.

The twin paradox already has a solution. Both twins see the other as moving but only one twin felt an acceleration. The one that felt the acceleration to get up to a moving velocity is the twin with the moving reference frame.

Both experience acceleration, they have similar situation.

That’s not the twin paradox then.

Edit: after reading the OP more, this is indeed not the twin paradox.

That’s not the twin paradox then.

Edit: after reading the OP more, this is indeed not the twin paradox.

It is still twin paradox. Because we can use time dilation formula like in original version of twin paradox.

It is still twin paradox. Because we can use time dilation formula like in original version of twin paradox.

I would say it actually isn't a paradox (at least assuming it is symmetrical).
In this case, they both age the same amount, due to the same time dilation.
Sure, they observe each other age differently and at different rates over the course of their trip.

The key part of the paradox is that the twins started at the same age, but ended up with different ages.

I would say it actually isn't a paradox (at least assuming it is symmetrical).
In this case, they both age the same amount, due to the same time dilation.
Sure, they observe each other age differently and at different rates over the course of their trip.

The key part of the paradox is that the twins started at the same age, but ended up with different ages.

But they still observe other twin have a slow time rate. Are they age different or still at the same age?

But they still observe other twin have a slow time rate. Are they age different or still at the same age?

As I said before, they initially observe the twin age slower.
Then they observe them age the same.
Then they observe them age faster.

The fast and slow periods cancel each other out and they end up the same age.

Is there a gravitational difference between planets A, B, and C?

Assume they are similar planets, so there is no need to include gravitation in calculation.

Quote from: fjr66 on July 09, 2019, 04:14:14 AM

But they still observe other twin have a slow time rate. Are they age different or still at the same age?

As I said before, they initially observe the twin age slower.

Prove it. Your claiming hey initially observe the twin age slower does not magically make them so. Prove it mathematically.

The fast and slow periods cancel each other out and they end up the same age.

Most of the time they use time dilation formula, why they end up with the same age?

Because they had the same velocity and acceleration as each other. You need a difference to get a different time dilation.

Most of the time they use time dilation formula, why they end up with the same age?

For one, the symmetry of the situation demands it.
But the key thing is they don't just use the time dilation formula one way.
Initially they see each other ageing slower. At the end they see each other ageing faster.
These 2 periods cancel each other out.

For one, the symmetry of the situation demands it.
But the key thing is they don't just use the time dilation formula one way.
Initially they see each other ageing slower. At the end they see each other ageing faster.
These 2 periods cancel each other out.

Even if we go to negative direction, so velocity have negative sign, its square still positive in lorentz factor formula. So it is will end up with the same amount of time dilation.

And what determined the twin seing other twin aging slower or faster?

Even if we go to negative direction, so velocity have negative sign, its square still positive in lorentz factor formula. So it is will end up with the same amount of time dilation.

And what determined the twin seing other twin aging slower or faster?

The same kind of argument was used in the original version of the paradox.
Both twins are travelling at some velocity relative to the other twin and thus should both appear to age slower.
The problem arises from using different reference frames, i.e. the twins turning around.
If you consider the reference frame of observer A initially moving away from the planet, they will age normally while twin B ages slower. But the twins change velocity and twin A will age slower as it heads back to the original planet, while twin B will age normally on the way back.
This means both twins experience a time when they age slower and a time when they age normally.

But the key thing is they don't just use the time dilation formula one way.

Even for the original twin paradox, they don't use time dilation formula one way. But in that case, they end up with different age.

Even for the original twin paradox, they don't use time dilation formula one way. But in that case, they end up with different age.

Yes, because of the asymmetry.
The observer on Earth only sees the observer in the distant ship ageing rapidly for the return journey which appears to go very quickly.
For the observer on the ship, they see the observer on Earth ageing rapidly for the entire journey which takes half the apparent time for them.

Quote from: fjr66 on July 10, 2019, 02:10:43 AM

Even for the original twin paradox, they don't use time dilation formula one way. But in that case, they end up with different age.

Yes, because of the asymmetry.
The observer on Earth only sees the observer in the distant ship ageing rapidly for the return journey which appears to go very quickly.
For the observer on the ship, they see the observer on Earth ageing rapidly for the entire journey which takes half the apparent time for them.

So time dilation does not have physical meaning, it is just mathematical consequence of relativity. Each observer can say other time was slower than him.

So time dilation does not have physical meaning, it is just mathematical consequence of relativity. Each observer can say other time was slower than him.

It does have a physical meaning, it is just a lot more complex than most people think.