6.2 Twin Paradox

Key Takeaways

welcome back to a20 special relativity in this section we talk about the famous twin paradox it's probably the most famous paradox in special relativity i want to get to the bottom of this and understand really where there is a conflicting or contradictory statement in this story let me just first say that this year is personal to me i do have a twin brother and you can see three pictures of myself and my twin brother here you know very little on our first day of school in germany you get a little box of candy i mean when you go to schools to make it more attractive to actually learn and study and then a picture which is probably already about 10 years old so what you can take away from here is clearly moving clocks are slow it turns out that my twin brother lives in the very same village in germany where i grew up where we both grew up why while i traveled the world constantly and funds me on the road um between france and geneva and switzerland and the united states and again i think there is no dispute here it can be seen from this picture that your professor looks much younger i even have a more recent picture this is two years ago the german chris candle market where i asked my brother to take this picture for this class for age 20 and again i think the answer to the question is clear professor clutter has aged less all right on a more serious note um we want to quantitatively understand and analyze the situation and we use bob and alice again in this situation here bob stays local and alice has a spacecraft and she moves with a velocity of 0.66 times the speed of light a gamma factor of 1.25 her travel takes her to a distant star which is in this example three light years away from bob measured by bob um the journey takes her on bob's clock five years and the return takes another five years she doesn't spend much time she wants to go home as quickly as possible so if you analyze this from alice's perspective we see that for alice the journey takes four years and the distance traveled for her in the spacecraft is zero okay from bob's perspective the journey as seen by alice is only two point three point two years long and so we find that there's already a conflict if we add the times together both ways the inbound and the outbound waste um 6.4 years is not equal to 10 years so there's already a contradictory statement in the story but the key to the understanding of this problem is that alice in order to return has to change reference frames and there you do have to re-synchronize the clocks if you want or add a specific extra factor and we'll go back to this and we look at space-time diagrams so the time as seen by bob is 3.2 years for the outbound journey and then 3.6 years in order to re-synchronize the clocks on the return on the turning around and then 3.2 years on the return which makes 10 years and so that observation of bob of alice is in agreement with bob's own clock all right so we saved the day here so let's look at space-time diagrams the outbound journey is is shown here you see i've got it in addition to bob's reference frame i plotted alice's reference and it makes it easier to understand what's going on so we see in alice's reference frame the journey takes four years if you then go back to the position in which bob is two points three point two years have passed okay so this is if he adds the time at the time when alice arrives we go back to the position of bob 3.3 years have passed and we then turn around and ask the very same question at that time and still four years we go back in the other direction now to bob we are already um much further ahead two points three point two years blood plus plus three point six years okay and then the journey continues and we add another 3.2 years to the journey so when alice and bob reunite l is aged by 8 years 2 times four and bob aged by ten years so the question now is is there maybe a paradox here is it possible that we miss somehow that why and try to understand why is this problem not symmetric why can i not just use the other reference frame and just declare that alice stayed stationary in her in a spacecraft where bob moved away with earth and then came back why are those two things not consistent and the answer is that it's not bob who has to change reference frame but at alice it's alice will have to do this there is where the asymmetry is so you can argue if you want that in order for alice to do that she actually has to accelerate but we don't have any sort of discussion of how the acceleration actually went about it's really the change in reference frame which is crucial in this discussion and causes a symmetry between bob and alice you