[Halc]

My question is, why the separation period of the twins is measured lower in B's frame compared to earth's, while both of them are equally inertial. The only difference I can find is that planet X is stationary in earth frame while moving in B frame.

The end of the separation period is defined by event R (reunion) which is on B's worldline (and A's as well), but not Earth's. In any frame, B is 48.99 years old at that event. In Earth frame, Earth is age 50 at an event simultaneous with R.  In B's frame, 48 years have gone by on Earth when the reunion takes place.

Is that what you're asking?  The separation period is lowest from A's POV actually.  Earth's frame doesn't really count since Earth is not present at event R.

None of this has to do with the frame of Planet X, since X's frame does not effect the reunion.  If both A and B assume frame X when they arrive together, then we can do it in X's frame, but I'm talking about the frame of B enroute when referencing B's frame above.

[Janus]

Same as the symmetry breaker before: A is not inertial for the duration of the exercise and B is.

My question is, why the separation period of the twins is measured lower in B's frame compared to earth's, while both of them are equally inertial. The only difference I can find is that planet X is stationary in earth frame while moving in B frame.

If you are asking why B measure's 48.99 yrs between passing Earth and meeting up with A at planet X, it is because in B's frame the spatial separation between Earth and planet X is ~9. 798 light years.  So, according to B, the Earth passes by at 0.2c, with planet x trailing 9.798 light years behind at the same velocity.  Thus it takes 9.798 ly/ 0.2c = 48.99 yrs between Earth passing and Planet X passing as measured by B.

[hamdani yusuf (OP)]

Let's make the situation more symmetrical by adding planet Y, which is stationary in B frame and located 10 light years behind, measured in B frame. How far is the distance between B and planet Y when measured by earth observer?
Let's say the earth observer also has a twin, called D. When B is passing by and A jump to earth, D jump into B's space ship. After 30 years in the ship, D return to earth at 0.5c measured in B frame. Will he arrive on earth at the same time as planet Y passing by earth?

[Halc]

Let's make the situation more symmetrical by adding planet Y, which is stationary in B frame and located 10 light years behind, measured in B frame. How far is the distance between B and planet Y when measured by earth observer?

Still 9.8 LY. They're moving in Earth frame, so the distance is contracted.

Let's say the earth observer also has a twin, called D.

No C?  What happened to poor Charlie? Run over by a bus at birth. Tragic...

When B is passing by and A jump to earth, D jump into B's space ship. After 30 years in the ship, D return to earth at 0.5c measured in B frame. Will he arrive on earth at the same time as planet Y passing by earth?

All in Bob's frame: Dewey is stationary for 30 years, then travels 10 LY to planet Y at 0.5c, so it takes 20 Bob years to get there. 50 years total.  Earth (with Alice) is moving steadily to Y at 0.2c, so it also takes 50 years to do it. So yes, they get there at the same time, and I didn't need to invoke relativity to figure that out. Relativity is only needed to figure out their ages when Dewey is reunited with Alice and they can grieve over Charlie's grave together.

[hamdani yusuf (OP)]

No C?  What happened to poor Charlie? Run over by a bus at birth. Tragic...

The letter c is already occupied frequently for speed of light. D was chosen because it's right before E, which stands for Earth observer in previous scenario.

All in Bob's frame: Dewey is stationary for 30 years, then travels 10 LY to planet Y at 0.5c, so it takes 20 Bob years to get there. 50 years total.  Earth (with Alice) is moving steadily to Y at 0.2c, so it also takes 50 years to do it. So yes, they get there at the same time, and I didn't need to invoke relativity to figure that out. Relativity is only needed to figure out their ages when Dewey is reunited with Alice and they can grieve over Charlie's grave together.

In the end of the journey, D will be reuniting with his twin, E when planet Y is passing Earth. Meanwhile, A will be reuniting with B when they are passing planet X.

[Janus]

Let's make the situation more symmetrical by adding planet Y, which is stationary in B frame and located 10 light years behind, measured in B frame. How far is the distance between B and planet Y when measured by earth observer?
Let's say the earth observer also has a twin, called D. When B is passing by and A jump to earth, D jump into B's space ship. After 30 years in the ship, D return to earth at 0.5c measured in B frame. Will he arrive on earth at the same time as planet Y passing by earth?

From the Earth frame:
Y is 9.798 ly away as B passes, and is approaching at 0.2c.  It will take 48.99 yrs to reach Earth.
D has jumped on to B's ship, an stays for 30 yrs (B's time)., This takes 30.62 years by Earth's clock.  This means that at that point B and D will be 0.2c x 30.62 = 6.124 ly from Earth.
D heads back towards Earth at 0.5c relative to B as measured by B,  this equates to (0.2c-0.5c)/(1-(0.2c)(0.5c)) = -0.333...c
(minus sign represents direction back towards Earth.)
6.124 ly/ 0.333c = 18.372 y
Added to the 30.62 yrs equals 48.99 yrs.  The same amount of time that passed between B passing and Planet Y arriving.

In B's frame:
Y is 10 light years away when he passes Earth and D jumps aboard. After 30 yrs of flight, B and D are 6 ly from Earth. And Y is 4 ly from the Earth.  With D moving at 0.5c towards the Earth and the Earth receding at 0.2c, the closing speed between D and the Earth, according to B is 0.3c,  at which rate, it will take 20 years for D to reach the Earth. It also will take 20 yrs for planet Y to cover 4 ly at 0.2c.  So according to B, it takes 50 yrs between his passing Earth and both D and Y to meet up at the Earth.   Meanwhile, according to B, the clock on Earth has been time dilated and accumulated 48.99 yrs. The exact same amount of time as determined by the Earth.

[Janus]

The letter c is already occupied frequently for speed of light. D was chosen because it's right before E, which stands for Earth

I'm pretty sure that people wouldn't have been confused by using "c" for the speed of light and "C" for the extra participant. And even if you messed up here and there and used "c" when you meant "C", the context in which it was used would have made it clear at to what you meant.

[hamdani yusuf (OP)]

With D moving at 0.5c towards the Earth and the Earth receding at 0.2c, the closing speed between D and the Earth, according to B is 0.3c,  at which rate, it will take 20 years for D to reach the Earth.

That doesn't seem like relativistic velocity addition.
To make the scenario perfectly symmetrical using those numbers, in B frame D should aim to planet Y which is 10 lightyears away. At speed of 0.5c, it will take 20 years to reach planet Y.

[Halc]

With D moving at 0.5c towards the Earth and the Earth receding at 0.2c, the closing speed between D and the Earth, according to B is 0.3c,  at which rate, it will take 20 years for D to reach the Earth.

That doesn't seem like relativistic velocity addition.

It's not.  Relativistic velocity addition is used in situations where you add Y's velocity relative to X, and Z's velocity relative to Y.  In other words the velocities are measured in different frames. Here, all velocities are relative to B's frame, so regular arithmetic is used to add/subtract them. Here, Janus was expressing the difference in velocities as measured in B's frame. Notice that he said 'according to B'.  To compute that same relative velocity relative to either Earth or D would require the relativistic addition.

To make the scenario perfectly symmetrical using those numbers, in B frame D should aim to planet Y which is 10 lightyears away. At speed of 0.5c, it will take 20 years to reach planet Y.

Right, and if you look at how I did the math at the bottom of post 43, you see I computed it just like that: how long Dewey takes to get to Y.

[Janus]

With D moving at 0.5c towards the Earth and the Earth receding at 0.2c, the closing speed between D and the Earth, according to B is 0.3c,  at which rate, it will take 20 years for D to reach the Earth.

That doesn't seem like relativistic velocity addition.
To make the scenario perfectly symmetrical using those numbers, in B frame D should aim to planet Y which is 10 lightyears away. At speed of 0.5c, it will take 20 years to reach planet Y.

As point out by Halc, it isn't needed here.
Example, You have Alice, Bob, and Charlie.
Alice "stays home", while as measured by her, Bob and Charlie both head off in opposite directions at 0.5c
According to Alice,  With Bob going in one direction and Charlie in the other, the distance between them is increasing at a rate of 0.5c + 0.5c = 1c . After 1 hr they will be 1 light hr apart from each other ( Bob will be 0.5 light hr from Alice in one direction, and Charlie 0.5 light hr from Alice in the the Other.) etc.
For Bob to get the relative speed between himself and Charlie, he adds the 0.5c he measures relative to Alice, to the 0.5c Alice measures as the relative velocity between herself and Charlie, using relativistic velocity addition.
(.05c+..5c)/1(1+0.5c(0.5c)/c^2) = 0.8c
The difference between Charlie's velocity and Alice's velocity as measured by Bob is 0.8c-0.5c = 0.3c
After 1 hr ( by Bob's clock) Bob measures Alice to be 0.5 light hr away, and Charlie as being 0.8 light hr way. A difference of 0.3 light hr.

[Bill S]

One thing about this so-called paradox that is worth a little thought touches on the very basics of how we think about time travel. 

Dan and Emily are twins, and Dan makes the obligatory journey, after which he will be 20  yrs younger than his sister.  Fast forward to the point of return in (say) the year 2150.  Dan has aged 20 yrs less than Emily; he is at the stage she would have been at 20 yrs earlier.  It is easy to think of this as his having travelled 20 yrs into his sister’s past, but is that the wrong way of looking at it?  Both are at the same point in time – 2150.  In what sense has Dan travelled into the past? 

Consider another scenario.  This time there is no journey into space, but in 2150 Dan uses a time machine to travel back 20 yrs.  He meets Emily.  She is as she was in 2130.  He has travelled 20 yrs into her past and she is younger, not older, as is the case in the former scenario. 

Dan comes back from space younger than Emily.  Pop Sci books and on-line discussions provide the "hitch-hiker" with abundant explanations for that.  However, if, instead of the high-speed journey, Dan had been placed in some sort of “stasis chamber” in which his development and ageing had been halted for 20 yrs, the result in 2150 would be the same as in the first scenario.  He would appear to be 20 yrs younger than Emily.  So, would we claim that he had time-travelled?       
 

[Janus]

One thing about this so-called paradox that is worth a little thought touches on the very basics of how we think about time travel. 

Dan and Emily are twins, and Dan makes the obligatory journey, after which he will be 20  yrs younger than his sister.  Fast forward to the point of return in (say) the year 2150.  Dan has aged 20 yrs less than Emily; he is at the stage she would have been at 20 yrs earlier.  It is easy to think of this as his having travelled 20 yrs into his sister’s past, but is that the wrong way of looking at it?  Both are at the same point in time – 2150.  In what sense has Dan travelled into the past? 

In no sense has he traveled into Emily's past.  He just experienced 20 yrs less time than she did between their separation and rejoining.  Relativity never claims that Dan travels into the past, only that the time interval he measured between the two events was shorter.  Time passed differently for the two.  If Dan left Emily when both their calendars read 2100, then when they met up again, Emily's calendar would read 2150, while Dan's would read 2130.
When They meet up again, it is 50 yrs into Emily's "future" and 30 yrs into Dan's future.
When you state that they are are the same point in time, 2150.  you are implying some type of absolute universal nature to time, and that 2150 is the "real" time.  It is only 2150 by the Earth clocks and this no more the "real time" than the  2130 according Dan's clocks.  It is just that when they meet up again, Dan agrees that by the Earth clock it is 2150 (twenty more years passed for Earth than did for him.)
There is the time as measured by Emily, and there is the time as measured by Dan. and there is no more meaning to time than that in this scenario. 

Consider another scenario.  This time there is no journey into space, but in 2150 Dan uses a time machine to travel back 20 yrs.  He meets Emily.  She is as she was in 2130.  He has travelled 20 yrs into her past and she is younger, not older, as is the case in the former scenario. 

Again, the SR twin scenario does not claim any travel "into the past".

Dan comes back from space younger than Emily.  Pop Sci books and on-line discussions provide the "hitch-hiker" with abundant explanations for that.  However, if, instead of the high-speed journey, Dan had been placed in some sort of “stasis chamber” in which his development and ageing had been halted for 20 yrs, the result in 2150 would be the same as in the first scenario.  He would appear to be 20 yrs younger than Emily.  So, would we claim that he had time-travelled?   

Once again.  SR does not imply "time travel" other than the time travel we all experience everyday from this moment to the next.  What SR does say is the there is no universal meaning to "the passage of time".  Every inertial reference frame measures time by its own independent standard, and that is the only meaning to "time" there is.
It's like the notions of left and right.  Everyone's "left" and "right" is unique to them.  If we are standing next to each and not facing the same direction, my "left" will not be the same as your "left'.   And there is no "universal" concept of "leftness".

Putting Dan in a stasis chamber, while he stays at rest with respect to Emily might, for Dan, "seem" the same personally, but only If Dan were not allowed to measure what was happening outside his chamber.  It might "mimic" the end result of Relativistic effects, but it wouldn't be the same. 

[Jaaanosik]

The Special Relativity is reciprocal.
The time dilation is reciprocal and therefore the twin paradox is a logical fallacy.
It is easy to prove, because Einstein introduced light propagation frame dragging with the postulate about the light propagation.
Jano

[Jaaanosik]

Here is a quick explanation of the problem, train car moving at 0.866c, there are train (red lines) and platform (blue lines) observers:


The red arrow up is crossing y-direction at speed c in the train frame
… and because the light beam is frame dragged then the y-component speed in the platform frame is c/2.

Copy and paste, replacing red and blue, changing frames.

The blue arrow up is crossing y-direction at speed c in the platform frame
… and because the light beam is frame dragged then the y-component speed in the train frame is c/2.

s - platform second, s' - train second... for better readability

There is no length contraction issue. The red arrow moves along x’=0cs’, it leads straight to time dilation definition.
The same goes for the blue arrow up, it moves along x=0cs and it leads straight to time dilation definition in the reciprocal way.



The events:
AB – red arrow up on the train -> 1cs'.
AE – blue arrow to the right -> 2cs – AB time dilation.
AC – blue arrow up on the platform -> 1cs.
AD – red arrow to the left -> 2cs' – AC time dilation.
AE – blue arrow up and down on the platform -> 2cs – the full round trip on the platform.
AF – 4cs' – time dilation of the full platform round trip AE.

The two observers see each others time to tick slower. They do not agree on time,
Jano

[A-wal]

There is never a paradox.  All observers will always agree which twin aged more if they are separated and then brought back together again, no matter what the scenario in which this occurs.

There is always a paradox with STR: the paradox is found in the issue of which clock was ticking faster during either one of the legs of the trip. STR has both clocks ticking faster than each other at the same time, and that's a mathematical impossibility.

The Special Relativity is reciprocal.
The time dilation is reciprocal and therefore the twin paradox is a logical fallacy.
It is easy to prove, because Einstein introduced light propagation frame dragging with the postulate about the light propagation.
Jano

Not true. Time dilation is reciprocal only with inertial motion. In every twin type scenario at least one of them has to accelerate which will desync their watches (unless they both accelerate at the same rate).

Doppler shift also cause two watches in inertial motion to each run slower from the other's perspective, nothing to do with relativity. This is direction dependant though, they'll each run faster from the other's perspective if they're moving towards each other.

Doppler shift is usually ignored because it will always cancel itself out once the twins meet back in the same frame as each other again but any question as to what an observer would actually experience in that situation needs to take it into account.

[Bill S]

Thanks Janus.  I think that helps to confirm my suspicion that when PS authors use this sort of thing as “evidence” for past-directed time travel, they are probably promoting their own ideas, rather than established scientific thought.

Interestingly, Wolfson, when talking of future directed TT, says:  “If you don’t like the Earth you find 100 years in the future, there’s no going back.  You’re either stuck where (or, rather, when) you are, or you can take your chances on a jump further into the future.  But you can’t go back.”   

[Jaaanosik]

Not true. Time dilation is reciprocal only with inertial motion. In every twin type scenario at least one of them has to accelerate which will desync their watches (unless they both accelerate at the same rate).
...

Please, support this statement: "Time dilation is reciprocal only with inertial motion." by quoting some textbooks.
Good luck!

The clock desynchronization is part of the Special Relativity,
Jano

Edit: Check this video, linked on the first page of this thread.

The acceleration has nothing to with the time dilation.

[A-wal]

Not true. Time dilation is reciprocal only with inertial motion. In every twin type scenario at least one of them has to accelerate which will desync their watches (unless they both accelerate at the same rate).
...

Please, support this statement: "Time dilation is reciprocal only with inertial motion." by quoting some textbooks.
Good luck!

The clock desynchronization is part of the Special Relativity,
Jano

Clock desynchronization happens with Doppler shift too, that causes no contradiction just as the time dilation caused by relative motion causes none.

In every case two observers that are in the same frame will always agree on the amount of time that has passed on each clock.

In instances where like the twin scenario the twin that accelerated will be the one who experiences less time passing so their watch will be behind the other's once they meet back up.

There's no need to quote any textbook, this is all well known in sr.

[Jaaanosik]

Well, let us analyze the video with some additional actions:



The observer A sends a light beam for a round-trip in the y-direction to his mirror at 1cs distance in the y-direction at EVENT I.
The round-trip is 2s for the A observer.
The B observer is watching this round-trip.
Question 1: How much time elapsed for the observer B?
Jano

[A-wal]

Well, let us analyze the video with some additional actions:

I'm not watching any videos.

If you have a scenario where one observer makes a journey and then returns back to the same frame as the other observer (which looks to be the case with observer A making the 'journey', in other words A is the one that changes frames) then less time has passed on observer A's watch once they return.

Then both observers agree on the amount of time that's passed on A's watch and both agree on the amount of time that's passed on B's watch. They agree that it's less time on A's watch. B's watch was time dilated from A's frame while they were in inertial motion relative to each other but B's watch sped up from A's perspective every time they (A) changed frames, enough that A's own watch is behind B's once they meet back up despite B's watch being time dilated from A's frame while they were in inertial motion relative to each other.