Question

Help resolving twin paradox.

Yes... one of the most confusing apparent paradox to solve :P. Now in many websites and books including the famous Resnick Halliday, they have conveniently solved it by saying one twin undergoes accelerations creating a gravitational field which speeds up his time and bla bla ..

But there many others which say that is wrong. That it is possible to just use inertial reference frames, neglect the accelerations (which I feel makes sense, cause technically I could have 0.000000000001g acceleration to get to the speed needed, and also to reverse the thrusters...

Answer 1

.. and head back to earth, and in this case, the effects of accelerations would truly be negligible compared to the effects of high speeds during the uniform motion. So I wen through a lot of explanation and I finally think I get a hang of it.

And So I want to understand in great detail.. therefore I try to explain it to walls. But the walls keep asking me questions that confuse me :P. SO I need some help with this.. I ll put my explanation forward in the next post and also tell where I need help

Answer 2

So we have two twins,
Speedy and Resty. (to make it quite obvious :P)

Speedy is going to go to a star which is 6 light years away from the earth. Speedy's ship can go at 0.6c.
So clearly from Resty's reference frame.
Distance to be covered = 6 ly, Speed = 0.6 ly, so the trip would take 10 years, and the round trip 20 years.

Now going to speedy's reference frame, we need to use the length contraction. The lorentz factor turns out to be 0.8 giving distance between the earth and the star to be just 6*0.8= 4.8 ly. (I chose these numbers from a website :P)

So according to him, he is at rest, and the star which is 4.8ly away from him is coming towards him at speed 0.6c, so it would take the star 8 years to reach to him and then 8 years to go back making the total journey last 16 years making him ultimately 4 years younger than Resty.

Now all we have to prove, is that in both reference frames, either of them see each other's watch to change according to relativistic time dilation and still we get the same result and no paradox. Each one has its own method, I use one of them.

So when they were together their watches were in sycn and set to zero. They promised as every passes, they would send a photo(n) of their watch (and these photo(n)s would travel at the speed of light) <--- see what i did there? ain't I smart or what? :D

Now comes my problem.
Time dilation, if we just use lorentz transformation, we see that the time dilation should be of .8. But This only works when both are close to each other right? here since light itself takes time to travel, there would additional time dilation and this would depend on the distance right? (that is my doubt number one)

okay.. so now I want to find out, when each one receives a photo of their twin, WHAT IS THE TIME IN THEIR OWN WATCH. This is where I am having difficulty.

Now when 1 year is up on earth, Rusty sends a photo, towards speedy. At this very moment, from Rusties point of view, speedy is 0.6light years away. But i want to know when Speedy finally catches this photo, how far he is from Earth (as seen by rusty) and also what is Speedies time? (i know he catches this photo even before he reaches the star.. you ll see how I know this :P)

Now skipping 2nd and 3rd , 4th is interesting. When 4th year is up, Speedy is at 2.4 ly away (4 times 0.6). He still has to cover 3.6 ly in addition to reach the star, which means he will still take another 6 years to reach it (ofcourse, cause 4 years already up, and Rusty calculated 10 years for the one way trip)
But the 4th photo(n) also takes exactly 6 years to reach from earth to the star right? cause its 6 ly away. Which means... WHich means, when speedy reaches the star, he immediately receives the 4th photo, which shows Rusty's watch being 4 years, but in his own watch 8 years would be the time (as we calculated earlier) so according to speedy, rusty's watch has slowed down by a factor of 2.
See this is what I mean, he is seeing a time dilation of 2. Is this is because of the large distances?
So that means when he is close, he would see time dilation by a factor of (1/0.8 = 1.25) and as he moves further and further it becomes more and ultimately when he is at the star (and STILL MOVING WITH 0.6c.. not to forget that) the dilation becomes 2?

So I want help to calculate when and where does Speedy catch the 1st 2nd and the third year photo(n)s.. plzzz :-/

Answer 3

@Kainui

Answer 4

@Vincent-Lyon.Fr

Answer 5

I think i found the asnwer..

What I said is wrong.. the factor will always remain 2. The factor depends on the speed (and maybe also the direction of the velocity vector, 2 is maximum when it is along the line joining, and when it is perpendicular it is min giving that 1.25 ?)
IF this is true... my headaches are gone.. !!!!!
Someone plzzz confirm???

Answer 6

@UnkleRhaukus @Compassionate

Answer 7

Oh wait.. its not the doppler shift? It has something to do with the fact that the distance is changing?
Oh I dunno someone reply plz :P

Answer 8

I thought the recognized solution to the twin paradox was that the dude on the spaceship accelerates at some point in time, taking off and heading back toward Earth. So, the space dude isn't in an inertial frame.

Answer 9

Special relativity can't explain this, to my understanding.

Answer 10

It can.. and I finally understand how :D.. we don't need accelerations to explain it

Answer 11

Oh, sorrry, I guess I didn't read the initial post well enough xD

Answer 12

..

Answer 13

@Mashy
Are you going to reveal your insight about the twin paradox??

Answer 14

@gleem
Ofcourse ofcourse

Leme continue from where we left of, so Speedy has reached the star, and 8 years are up on his watch, and he just received the 4th photo, which shows 4 years being up on Rusty's watch. So speedy is happy, because from speedys reference Rusy was moving and so he predicted that Rusty's watch must be slower and that is indeed what he found (twice as slow)

Now lets look at Rustys reference. Rusty starts receiving photons sent by speedy. Since speedy has reached the star in 8 years, he must have sent 8 photos till now, (8th photo he just sent). And rememebr Rusty's watch must be now at 10 years (according to our earlier calculation). This 8th photo takes 6 years to travel all the way from speedy to rusty (cause the star is 6 ly away as seen by Rusty right?).
So Rusty has to wait 6 more years. And finally when he gets the photo his watch will read 16 years and sees speedy's photo which shows 8 years on his watch. ANd he is happy, cause even he sees that Speedy's time is dilated (by the same factor 2, which should make us happy)

you might think this is nice and symmetrical, but it is not.

For Speedy, Rusty's watch has been slow for 8 years or half of his journey

For Rusy, Speedy's watch has been slow for 16 years.. which is more than 3 quarters of his journey. (remember Rusty has to wait a total 20 years , and 16 years already up)


Now from each others reference frame, the other twin starts making a return trip. Lets start with Rusty's reference frame
Rusty's watch is 16 years. He just received the photo of Speedy being 8 years and just reaching the star. So Rusty says.. my gosh ,, my bro took 16 years to reach the star. (of-course he may be clever enough to realize that information takes 6 years to travel.. but lets ignore that :P)
But he knows that whatever happens in next 4 years he is going to come back. And we know that in these 4 years, Speedy will be sending 8 more photos, or in other words Speedy would be advancing 8 years. SO without having to do any further calculation, we can see, that Rusty sees speedy's watch to become faster (by a factor of 2).
When 4th year is up, Speedy reaches home, and sends his last photo, which shows 16 years on his watch and rusty compares it to his own watch (which shows 20 years) and thinks that Speedy MUST BE NOW YOUNGER by 4 years. (Remember this last photo takes zero time, cause Speedy and Rusty meet up, But Rusty is facing the other way, to build the suspense :D)


Now the moment of truth. Speedy's reference frame.
So Speedy has reached the star.. or in better words the star has reached speedy in 8 years, and right now he has seen Rusty being just 4 years advanced. Again without having to do much math, we know as Rusty makes his return trip (remember that is what Speedy sees, speedy is at rest :P) Rusty will have sent 16 more photos until they meet. And these 16 photos Speedy would receive in just 8 years (because speedy has to wait 8 more years for the Earth and Rusty to come back to him). And so Speedy sees rusty's watch advance twice as fast (that should make us happy, both have seen the same dilation and speeding up of each others watches).
Now what will Speedy finally think?
First 8 years.. Rusty advanced only by 4 yars
But next 8 years, Rusty advanced by 16 year.
And so Speedy agrees, that when they meet up, his watch is 16 years , and Rusties watch is 4+16 =.... Equals... EQUALLSsssssssssssss 20 Years.. SO he AGREES that HE IS 4 YEARS YOUNGER!!!!!
THEREFORE.. they both open their eyes, and GWALLLAAAAA.. NO MORE A PARADOX..
We didn't need any accelerations to figure this out, and if you think about it, we didn't much math either..

So the main reason why people think its a paradox is

a) People think the situation is symmetrical. But its not, due to length contraction, the symmetry is immediately broken.

b) People think that the travellers watch is always slow as seen by the observer, which we have clearly seen is not true. It depends upon the direction of the speed. If the speed is perpendicular to the line joining the two (no radial component) then they are right, otherwise due to doppler shift what they see is very different from the lorentz contraction factor.

But we did see, both of them are happy, because the same laws work in their reference frames. When their twin is moving away, time dilation of a factor of 2, and when they are coming towards, time speeding up by a factor of 2, so when they discuss, again its impossible for them to conclude who was in motion (Relativity still holds, concept of no absolute motion still holds)

Answer 15

@Mashy

True the situations is not symmetrical. But I have an observation about your explanation which I believe is incorrect. When Speedy receives the fourth transmission You said He interpreted the signal as four years having elapsed on Earth when He received it and compared it to his elapsed time of 8 years. This implies instantaneous transmission of information. Speedy knows that it takes 6 years for a signal to reach the star where He is at. Why didn't You add this to Speedy's calculation of Rusty's elapsed time?. Rusty also knew that He did not need to send a fifth signal because He know Speedy had reached the star already. Also a time dilation of 2 is not consistent with a Lorentz transformation.

Let me suggest an alternative explanation. We started off with a goal of sending Speedy to a star 6 l.y. away from earth at a speed of 0.6c. Speedy and Rusty know that Rusty expects it to take 10 of his years for Speedy to get there and Speedy expected it to take 6 of his years to get there because at his speed he saw the distance to the star as only 4.8 l.y. from him due to length contraction of a factor of 0.8. In this situation the question that is begging is what does one see happen to the time in the others reference frame. Both are moving relative to the other at 0.6c. You can't both slow down at the same time
But wait. Rusty know the distance to the star and it is fixed. He knows Speedy's clock must be slowing due to time dilation and He sees Speedy getting closer to the star. Speedy knows his distance to the star dependent on his speed. The faster he goes the shorter the distance. He also know that the distance from Rusty to the start is fixed. Even though Rusty appears to be receding at a speed of 0.6c. Thus Speedy know that He is move relative to Rusty in Rusty's reference frame. The symmetry is broken. I think You must decide on one reference frame to start and stick with it through the discussion. And it seems to imply that a reference point is needed for any meaningful discussion since no reference point means not defined length scale for the two frames to compare.

Answer 16

" The faster he goes the shorter the distance. He also know that the distance from Rusty to the start is fixed. Even though Rusty appears to be receding at a speed of 0.6c.

Speedy is not going anywhere.. If you use speedy's reference frame.. he just sees the star coming towards him and the earth (along with rusty) receding away. For all he knows, the distance between earth and star depends on the speed at which they are both moving (with respect to Speedy of course). So this reference frame is as good as Rusty's! (meaning Speedy wouldn't feel something fishy to conclude that he is actually in motion, cause remember there is no such thing )

" Thus Speedy know that He is move relative to Rusty in Rusty's reference frame. The symmetry is broken"

I don't quite get what you mean, both of them know they are in motion with respect to the other twin, so how does that break the symmetry.


Also for your other question, ofcourse it takes 6 years for the info to travel, and although I used discrete time intervals to send and receive the info, you would see what I meant by time dilation of factor of 2, if they were receiving the information continuously
Suppose speedy and rusty had a huge telescope to keep looking at each other's watches, then they wouldn't see each other's watches to be slowed down by factor of 0.8 but instead they would see it slowing down by a factor of 2. They would actually see that. (of course I understand this is due to the fact that info takes time to travel, but that is what they see right? :D).

Answer 17

Why didn't Speedy add six years to the info He received from Rusty?

Answer 18

Also both Speedy and Rusty know that Rusty is not moving relative to the star while Speedy is. Appearance is irrelevant to the application of the Lorentz transformation.

Answer 19

So you are basically saying, if he added that info, he would see, that it his time which is really dilated?

Answer 20

I don't think speedy can conclude that Rusty sees the distance to be 6 ly between the earth and the star.
Because in speedy's reference frame, the distance just turns out to be 4.8 ly..
its not that speedy sees it to be 4.8 and figures out.. that this is lorentz contraction.. It would feel all normal to him right?

Answer 21

if speedy adds up that 6 years.. he will figure out that rusty's watch reads 10 years while his watch is reading 8..

but that destroys it.. speedy should somehow also conclude Rusty's watch is slower.. whats going on? :-/

Answer 22

aha ha that is the paradox right? that is if you let Speedy apply the Lorentz transformation from his frame. Speedy and Rusty are twins who started our on Earth knowing that the distance to the star is 6 l.y. and knowing Speedy's speed is 0.6c. Speed knows that he is moving relative to Rusty and not vice verse.

Answer 23

No.. the contraction is not the problem

If i threw a ruler towards you at near speed of light.. you will see the ruler to be contracted..
Seeing that contracted ruler, do you say that .. "oh no.. I am moving .. and since the ruler is contracted, the ruler must be at rest? "

no right? :D

Answer 24

So .. speed doesn't have to think he is moving.. he might as well think that the whole star and the earth system is moving, and it is THEIR MOTION that has led to the contraction of the space in between.. (just as the ruler contracted when you saw it.. because THE RULER IS IN MOTION as SEEN BY YOU)

Answer 25

but m quite baffled with the problem of "speedy being smart and factoring in the 6 years time lag.. when the star reaches him, giving him the clue .. that his time has been dilated.. "
That should have some fix :P

Answer 26

I think the problem is the proper application of the Lorentz Transformation. When can it and when can't it. Speedy is a physicist as is Rusty so they know what to do.

Answer 27

Regardless.. they both must be able to conclude the other twin is in motion.. and that the other twins time is dilated..

Think of it.. if you were the space twin.. you would be seeing my watch on earth.. and until the star is reaching you.. you would see my time to be slower.. twice slower..
so taht way it is dilated.. but the problem comes.. once you factor in that time gap..

Answer 28

I got the solution

The problem was.. you said speedy has to add 6 years as a correction for the lag..
But that is wrong

There is no real and apparent distance. There is only one real distance, and that is what you see in your reference frame.

So speedy doesn't think the actual diestance between the earth and star is 6 ly, he actually thinks it is 4.8 ly. THAT IS TRUE distance for him

So now lets, see, the star just hit him, and he saw the earth 4.8 ly away and his watch reads 8 years, he just received the 4th year pic. Now as you said he wants to add the correction. What should he add? Not 6.. according to him, since rusty sent that pic when he was 4 years old, he must have sent it when he was half way through that journey (cause 4 years is half of 8 years), and that means he was at 2.4 ly away.
So the photon only had to travel 2.4 ly, which means he adds 2.4 ly to 4 = 6.4 ly

and guess what.. 8 *0.8 = 6.4 ly.. so he concludes.. that Rusty was time dilated, becase HE IS IN MOTION ALONG WITH EARTH AND THE STAR
MUHAHAHAHAHAHAHAHAHAH

Answer 29

@Mashy

"So the photon only had to travel 2.4 ly, which means he adds 2.4 ly to 4 = 6.4 ly"

You added 2.4 l.y (a distance measure), to 4 y ( a time measure) can you do that?

Answer 30

@Mashy

In this case You can. Looks good. Now in terms of Lorentz transformations can you show the reason why we have an apparent paradox? Can you also reason why most texts cop out with the acceleration ploy instead of showing as you did the that proper time dilation is seen from Speedy's point of view.

Answer 31

That was a typo.. I mean't the photon had to travel 2.4 ly, which means he has to add the time delay of 2 YEARS (not light years) :P..
so 2.4+4 = 6.4 YEARS!

Answer 32

"Now in terms of Lorentz transformations can you show the reason why we have an apparent paradox?"
I feel like I am in an interview.. xD xD

I think apparent paradox is when people don't understand the concept of relativity well :P

Now you can clearly see that from Rusty's view.. 10 years in his watch = 8 years in Speedy;s watch (in good agreement with lorentz)

And also from Speedy's reference frame.. 8 years in his watch = 6.4 years in Rusty's watch
(in good agreement again with Lorentz)

But its not in good agreement with common sense.. cause how can both be true?

And I think the answer lies in Relativity of simultaneity ( I always wanted to say that in an explanation.. it sounds so cool when you say that xD)

I think when we say most textbooks we are tlking about introductory textbooks, and i think they don't wanna scare off viewers by having a rigorous approach like this and at the same time, a book which introduces spl relativity and doesn't talk about twin paradox won't sell I think, so they mostly take the easy way out using acceleration and tell 'its a job for general relativity' .. and when you hit general relativity you totally forget that twin paradox was not solved properly hahaha :D

Answer 33

So it seems you really haven't resolved the paradox since what Speedy thinks should be on Rusty's watch isn't correct. Namely Rusty watch reads 10 years elapsed but Speedy thinks 6.4 years have elapsed. In fact you showed that Speedy sees Rusty as younger just the opposite of the Twin Paradox or have I missed something again.

Answer 34

Speedy sees Rusty to be younger.. which is correct, because from Speedy's point of view, Rusty is moving away
The paradox is solved because although either of them see their twins to be younger, eventually when they meet up both agree that only one has aged more :P

Answer 35

@Mashy

I don't want to seem like a pain but I must be having some sort of mental block. I thought I had a handle on this but I have become more confused during this discussion. If time dilation is the slowing down of the observed time interval of an event occurring in the moving system, the actual time of the event in the moving system called the proper time. the proper time scale in Speedy's frame is the same as Rusty's. When Speedy's speed reaches 0.6c Rusty sees Speedy's time scale slowed by 0.8. So then why shouldn't Speedy see the same effect with regard to Rusty ie a time dilation of .8 of Rusty's proper time which has not changed since Speedy left for the star. Why shouldn't Speedy see Rust'y elapsed time for Speedy's journey as 8 yrs?

Answer 36

"have become more confused during this discussion."
that makes two of us xD

" Why shouldn't Speedy see Rust'y elapsed time for Speedy's journey as 8 yrs?"

Speedy sees Rusty's time to be dilated by a factor of 0.8. But the reason Speedy doesn't see it to be 8 years, because, speedy takes 8 years to reach the star.. So when you multiply 8 by 0.6 you get 6.4 and that is what speedy sees Rusty's time to be (after taking care of the time delay that is)

Answer 37

@Mashy
You did a nice job of showing that time dilation occurs as if Speedy where standing still and Rusty where moving at -0.6c. and I understand that. But ....... I'll not bother you about this again although I will continue to try an reconcile my issues. I will get back to you when I have some new angle on this problem which could be never. Bye

Answer 38

No no.. give me your buts.. I want your buts. .. M not bothered at all.. I want to a make a video on this... and I have never studied relativity before.. so yea.. I need to solve all buts xD

Answer 39

@Mashy

I just can't get my mind around this issue. Let me tell you I was introduced to SR many years ago and everything seemed OK. I know the basics. In reviewing material on simultaneity that I had previously "understood" (perhaps with the help of an instructor) I found myself misinterpreting the explanation . That is the explanation i.e. words did not convey in my mind the intent of the writer.. In other words the writer had a good picture of what he was explaining but the words and way hey put it did not convey that to me. After staring at it and stating to my self over and over that he is correct (many authors give the same explanation in this case the observation of what is simultaneous in one frame and not in another using a moving railroad car and simultaneous lightning strikes in the stationary frame) I think I got his picture. I recommend you study it carefully and understand it fully before anything else. SR is so counter intuitive that you find your mind trying to correct subconsciously for that, causing all sort of problems including misapplications of the Lorentz transformation.

If you haven't yet check out the Wikipedia web-site on SR which is actually not too bad and calls you attention to other paradoxes and controversies surrounding SR. Not all believe.

Tread carefully through this subject.

Answer 40

I did read about the simultaneity part and also working with Minowski space-time graphs and stuff..
And its very true what you said, because sub consciously we tend use time as an independent reference, rather than Speed of light.. (because of the way we are used to in Newtonian mechanics)
The more you dig deeper in relativity the more you tend to get amazed as to how Einstein must have figured it all out.. :D

Answer 41

@Mashy

Let me give you a few of my thoughts about Einstein. He had a lot of guidance in formulating his theory including Maxwell and Lorentz. He was determined to reconcile a lot of inconsistencies in Physics at the time even if it took radical hypothesis to do it. That the velocity of light is the same in all frames was almost unreasonable at least at that time. But it solve a lot of issues. Seeing the consequence of that hypothesis i.e the predictions of SR and believing that "the math will lead you to the truth" - The Theoretical Physicist Mantra ( which I might add seems to always be correct if the initial assumptions are correct and you do the math correctly) He had no doubts.

If the theory explains a lot of inconsistencies and in addition predicts things that seem ridiculous don't reject the theory until you have checked out the predictions for no matter how ridiculous the predictions for if they are found to be correct the theory is substantiated (at least until proven otherwise).

Interestingly Einstein was not disturbed by the Twin Paradox calling it "curious".

Answer 42

That is great info

the students I teach are highschool level physics, and when I do relative velocity concept, I always tease them with the concept of relativity (as a bonus lecture) which blows most of their minds..
And when they ask me why or how it is true.. I give simple explanation

The first postulate.. which basically says all inertial frames are indistinguishable making rest and relative .. needs speed of light to be constant in vacuum.
Otherwise you could always calculate the speed of light in your reference frame and if you found it out to be other than 'c', you would conclude you are in motion (which is not allowed, cause there is no absolute rest or motion) :D. Isn't that a very intuitive approach to why c is a constant?
I always wondered whether Einstein had this idea!