The Twin Paradox

1. The Twin Paradox

2. A quick note to the ‘reader’ • This is intended as a supplement to my workshop on special relativity at EinsteinPlus 2012 • I’ve tried to make it ‘stand alone’, but in the process it became rather didactic and lecturey, as pointed out by the excellent Roberta Tevlin, who was kind enough to look it over (all mistakes remain my own). • I have gone back and tried to ask more and tell less… but since I want someone to be able to go through this on their own I couldn’t resist keeping some answers in… So on some pages there are questions, and a little symbol will bob up at the bottom of the page: • Clicking the symbol should take you to a ‘hidden’ page that has the answers or other comments. Clicking elsewhere (or using the arrow keys, etc.) should navigate normally! I hope you find this helpful! 

3. Title &Intro(you are here) Description of ‘Paradox’ Click on any box to go to that part, or just click anywhere else to continue to the next slide! Because there are a number of choices you can make as you go through this presentation, I thought it might be helpful to give you an outline of the different parts right away. Intro to Spacetime Diagrams Outline: Qualitative: mapping the paradox Calculations:computing the times From the Travelling Twin’s view The twin paradox &The doppler effect End

4. The Twin Paradox • One of the hardest things to get used to about relativity is the way that time can be different for different observers. • This includes not just how quickly time passes, but also what different observers call “now” • Let’s take a little time to look at what is behind the ‘twin paradox’… which isn’t really a paradox at all, just an example of how we carry our everyday ideas of time into our understanding. Even when we are trying not to! Summary of Twin Paradox Skip the Summary

5. Introduction to the Twin Paradox • In our study of special relativity we have learned that moving clocks ‘run slow’. One tick of your clock One tick of movingclock light light 3.0m > 3.0m Clock moves Light must travel further in moving clock. But light has the same speed relative to all observers, so one tick of the moving clock takes longer than one tick of the stationary one (as measured in the stationary frame)

6. A long trip • If we have two identical twins, one on earth and one in a spaceship which is moving at a speed close to light speed (relative to the earth), what will the stay-at-home twin say about the travelling twin’s clock? • Suppose that the travelling twin’s clock is running at half the rate of the stay-at-home twin. If the trip takes, say, 24 years on the stay-at-home twin’s clock, how long will it take on the travelling twin’s? tship v  tearth Hey sib! Better fix your clock!

7. Different times • We can see that a long trip will take a very different amount of time according to the two twins. • When the twin returns they will be significantly younger than their identical twin! • This isn’t actually the oddest combination… how about a daughter who is much older than her mother? • What do possible situations like this say about our ideas of age and time?

8. The issue • So far this is weird… but it isn’t a paradox. There is nothing contradictory about this except language. • But here’s the rub: Motion is relative. What would a round trip, as the spaceship goes to another planet and returns, look like to the stay-at-home twin? Imagine or sketch the motion of the ship as seen by the twin staying on earth. 

9. From another viewpoint • What would this same round trip look like from the point of view of the twin in the spaceship? • Remember that they don’t see themselves as moving, it is the earth that goes away and comes back! • Imagine or sketch the motion of the ship as seen by the twin who is travelling on the ship 

10. Who’s younger? • What does the twin on the ship (the “travelling twin”) say about the Earth’s motion? • Whose clock does the travelling twin see as running slow? • Which twin should be younger according to the travelling twin? ? 

11. The In-Between • We know that during the trip out and during the trip back both the travelling twin and the stay-at-home twin see the other twin as moving near light speed. • What will they say about one another’s clocks? • What will the travelling twin experience at the turn-around point (what would it feel like on the ship)? • What will the stay-at-home twin experience at the turn-around point (what would it feel like on the earth?) • What is different? 

12. What you need to know: • For this explanation to make sense you need to understand a few things about spacetime diagrams. • There is another powerpoint about this, which you can look at. I’ll give a quick summary here or you can skip that and go straight to the explanation. Intro to spacetime diagrams Cut to the chase! Mapping the twin paradox

13. Space, time, and spacetime • One of the key ideas which emerges from special relativity is the fact that space and time are not separate things, but components of one thing, spacetime. • Thus we can measure time in metres, or distance in seconds. • And different observers can have their time and space axes pointed in different directions (which is responsible for all the ‘strange’ effects of special relativity)

14. Spacetime diagrams • Spacetime diagrams show this 4D spacetime with 2 (or sometimes 3) dimensions by showing only one direction in space (sometimes 2), and using the other direction for time. ct c ct c y x x 1D space, 1D time 2D space, 1D time

15. Spacetime diagrams are like traditional position-time diagrams BUT time goes vertically by convention. So as time passes things are ‘copied up’: 3) 3) Same point in space at different times Different points in space at different times 2) 2) 1) 1) Standing Still Running Notice that for a moving observer the world-line is slanted. The path in spacetime is called a “world-line” 3) time time time time 3) 2) 2) 1) 1) space space space space

16. The time and space axes for a moving observer tilt in toward the light speed line (45 if time is converted to the same units as space by multiplying by c) This is the moving observer’s time axis… it represents the location of the observer at different moments in time. This is the moving observer’s space axis… it represents the “now” of the observer. ct' ct c B In terms of the moving observer’s space and time coordinates, what is the same for the two dots shown on this axis (marked A and B) ? In terms of the moving observer’s space and time coordinates, what is the same for the two dots shown on this axis (marked D and E)? A E x' D x stationary moving 

17. The size and direction of the coordinate axes change, depending on how the one frame moves relative to the other. c rest faster fast slow time (here) How do the time axis (“here”) and the space axis (“now”) change as the relative speed increases?What is the limit as speed gets bigger and bigger?(click to increase speed!) time (here) time (here) time space (now) space (now) space (now) space Changes in rate are due to the changing direction of the time axis… but the changes in what “now” means are also important to understand the resolution of the twin ‘paradox’. 

18. Details • The next few slides show the trip, relative to the stay-at-home frame. • We will use this frame because it remains constant throughout the trip. Later you will have the chance to see the trip from the travelling twin’s view too (the result is the same) • The key idea to keep in mind is that the point where the ship turns around, although brief, is very important.

19. To make the numbers simple we will regard the travelling twin as travelling at 0.866c during the trip (=2) to a planet 10.4 ly away (this distance was chosen so that the trip time to destination = 12 years in earth frame). Ship Earth Destination Planet The time and space axes of the stay-at-home frame are in black. The axes of the travelling frame are in blue.

20. ctplanet Starting out ctship xship(now for ship) Notice that right away the ship and the earth would describe very different times as “the same time on the destination planet as the time the ship left” Light This line shows the velocity of the rocket (its world-line) This line shows the space axis of the ship (its now-line) Which of these points in the history of the destination planet would someone on earth say is at “the same time” as the ship leaves earth? Which is “the same time” as the ship leaves earth in the frame of the ship? Here the travelling twin leaves the earth in the ship, already travelling at 0.866c. xplanet(now for planets) Planet Earth Planet Relativity Ship(v) 

21. Light ctplanet Half way ctship xship (now for ship) How much time has passed on earth at this point, from the point of view of the travelling twin? How does this compare to the time that has passed on earth, from the point of view of the stay-at-home twin? The travelling twin is now half way to the destination planet. xplanet(now for planets) Planet Earth Planet Relativity Ship(v) 

22. Light Arriving at the Destination ctship xship (now for ship) ctearth Which point in the earth’s history corresponds to the time of the ship’s arrival in the earth’s frame? Which point corresponds to the arrival in the ship’s frame? • The ship has now reached its destination. The travelling twin must now slow down and stop. How do the times for the trip compare in the two frames? xplanet(now for planets) Planet Earth Planet Relativity Ship(v) 

23. Light xship (now for ship) xship (now for ship) Arriving at the Destination ctearth ctship Now, as the ship slows down to turn around, watch what happens to the earth time that corresponds to the ship’s NOW. (click to begin) Light Light xship (now for ship) ctship ctship Ship(v=0) xplanet(now for planets) xplanet(now for planets) xplanet(now for planets) Ship(v) Planet Earth Planet Relativity Planet Earth Planet Relativity Planet Earth Planet Relativity Ship(v)

24. xship (now for ship) Return ctearth Light ctship Light ctship Now the travelling twin must begin the trip back. Light Light ctship After you click, notice how the travelling twin’s “now” continues to sweep across the world-line of the stay-at-home twin. (click to begin trip back!) xship (now for ship) xship (now for ship) Ship(v) xplanet(now for planets) xplanet(now for planets) xplanet(now for planets) Ship(v) Planet Earth Planet Relativity Planet Earth Planet Relativity Planet Earth Planet Relativity Ship(v=0)

25. ctplanet … and back again! ctplanet ctship ctship Finally the trip back, with the usual rotation factors. Light Light Light ctship (click to begin trip back!) xship (now for ship) xship (now for ship) xship (now for ship) Ship(v) Ship(v) xplanet(now for planets) xplanet(now for planets) xplanet(now for planets) Ship(v) Planet Earth Planet Relativity Planet Earth Planet Earth Planet Relativity Planet Relativity

26. ctplanet Trip Out Now with numbers! v=0.866c=2 ctship How much time does the trip to the planet take according to the stay-at-home twin(as seen from earth’s now)? Distance = 10.4 ly xplanet(now for planets) Planet Earth Planet Relativity 

27. ctplanet Trip Out Now with numbers! v=0.866c=2 ctship How much time has passed for travelling twin:(slowed by a factor of )? Time that has passed for stay-at-home twin = 12 years xplanet(now for planets) Planet Earth Planet Relativity 

28. ctplanet Trip Out Now with numbers! v=0.866c=2 To the travelling twin it is the stay-at-home twin who is moving at 0.866c, and so the stay-at-home twin’s clock that is slow:(by a factor of ) ctship Time that has passed for stay-at-home twin = 12 years How much time does the travelling twin say has passed for the stay-at-home twin during the 6 year trip? Time on earth relative to SHIP’S NOW = 3 years Time that has passed for travelling twin = 6 years xplanet(now for planets) Planet Earth Planet Relativity 

29. ctplanet Trip Back is much the same! Time on earth relative to SHIP’S NOW = 3 years ctship Time that has passed for travelling twin = 6 years The return trip is a reverse of the trip out, with the same times all around. Time that has passed for stay-at-home twin = 12 years xplanet(now for planets) Planet Earth Planet Relativity

30. ctplanet For the whole trip What is the total time that has passed for the travelling twin? ctship What is the total time that has passed for stay-at-home twin? The travelling twin sees the time on earth as partly having passed during the trip, and partly “swept over” during the turn around. How much earth-time does each of these correspond to? xplanet(now for planets) Planet Earth Planet Relativity (summary)

31. ctplanet Summary for the whole trip ctship Total time that has passed for travelling twin = 6+6 = 12 years Total Time that has passed for stay-at-home twin = 24 years The travelling twin sees the time on earth as 3+3= 6 years while travelling Plus 18 years swept over during the turn around. 6 + 18 = 24 years on earth. xplanet(now for planets) Planet Earth Planet Relativity

32. So that’s the resolution of the ‘paradox’ • Everyone agrees about how much total time has passed for each twin. • The apparent symmetry between the two trips is broken by the act of changing frames, during which the travelling twin’s ‘now’ “sweeps through” the missing time. Extra: See the trip from the travelling twin’s coordinates too!

33. The change of frames of the travelling twin is not relative, and the views are not symmetric! • The act of turning around makes the view of the stay-at-home twin different from the travelling twin, no matter whose point of view you follow. • When the travelling twin changes frames, the meaning of “now” changes for the traveller, and their coordinates are very different, including their own view of their past motion. • Thus changing frames (accelerating) is not relative. But we knew that… (you can FEEL an acceleration, even in a closed room!)

34. The End The real issue is what the twins are going to do about the asymmetry of number of Birthday Presents!! Unless you want a quick aside on what the twins actually SEE each other’s clocks doing on the trip (not the same as the times they calculate).click this button for the extra notes, anywhere else to end!

35. Signals take time to travel Because signals (or images or whatever) can travel no faster than the speed of light, the times when signals from earth reach the spaceship (or signals from the spaceship reach earth) are not necessarily spaced out just according to the rate time seems to flow. Is received here A light signal from here ct ct

36. To understand what we ‘see’ we have to track the signals • This travel time means that we actually see events when their signals catch up to us (or we intercept them). • For example, we saw that during the ship’s turn around the ship’s ‘now’ sweeps through 18 years of the earth’s time. • But that doesn’t mean that the twin on the ship “sees” 18 years pass on earth – it means that 18 years of earth history that they called ‘future’ they now call ‘past’. But news from that past still has not reached the ship.

37. ‘What you gets is what you sees’ • Let’s track signals to see what you would actually receive in the way of signals from earth if you were the travelling twin. • We’ll assume that the ship sets out on the twin’s birthday, and each twin sends the other a birthday greeting each year.

38. ctplanet What the travelling twin sees: On the trip out the signals have to catch up to the ship. Estimate, from the graph, how much time passes on the ship before the first birthday greeting is received? ctship From the graph, about how many signals a year does the ship encounter as it returns? Ticks mark birthdays xplanet(now for planets) Planet Earth Planet Relativity 

39. ctplanet What the stay-at-home twin sees: ctship Coming back the ship is rapidly following its signals, so they will come in very rapidly. About how many are received per year? The stay-at-home twin also gets only infrequent birthday greetings during the outward part of the trip. How many years apart are the birthday messages? xplanet(now for planets) Planet Earth Planet Relativity 

40. If you do the math on this expansion/compression of time (and frequency) you get exactly the relativistic Doppler effect… which perhaps is not a surprise if we think about it! Calculate the ratio between the frequency of signals sent and received at the relative speed of the two ships. When does the travelling twin get slowed down signals? When do they get signals that are sped up? What about the stay-at-home twin? 

41. ct From the point of view of the stay-at-home twin the ship sends 6 signals while moving away from the earth. Going the other way the ship sends 6 signals, but now they are received 1/3.73 years apart. How many years will pass on earth before all those signals are received? (Include at least 1 decimal place in your results) ctship Given that the ratio of frequencies is 3.73, how many years will pass on earth before all those signals are received? (Include at least 1 decimal place in your results) Earth How much time passes on earth during this whole process? (What is the total time taken to get all the birthday messages?) x Earth Planet 

42. From the point of view of the travelling twin the signals from earth are spaced out as the earth moves away. As the earth approaches the ship again signals are received much more often. How many signals are received by the ship during this part of the voyage? (Include at least 1 decimal place in your results) Given that the ratio of frequencies is 3.73, how many birthday greetings from earth are received by the ship as the earth moves away? (Include at least 1 decimal place in your results) ct ctship Ship How many birthday greetings are received by the ship in total? (What is the total number of birthday messages the ship receives?) x Earth Planet 

43. This can also be seen in the travelling twin coordinates. We looked at what messages the travelling twin receives, but we still used the earth coordinates while doing so! You might want to look at this using the actual (changing) ship coordinates. Warning: it’s a little messy, because we have to switch frames half way through. But you can see what the messages are really like from the travelling twin’s perspective. Your call! No thank’s… I’m satisfied. Skip it! Show me all the gory details!

44. The 1st part of the trip in Ship coordinates. Notice that during the first 6 years for the ship the earth moves away from the ship, but not all signals sent are received by the ship. How many signals will the ship receive, given that the ratio of frequencies is 3.73? 

45. Given that the ratio of signals is 3.73, how many signals will be received in the next 6 ship years? What is the total number of signals received by the travelling twin? What is the total number of signals sent by the travelling twin? NOW here is where the shift of frame happens! The ship changes frame. In this frame what WAS the present on earth is now the past…The next signal to be received is the second birthday wish… how many years ago (relative to this new frame) was the signal sent? 

46. When they finally meet the twin on earth will be celebrating the 24th birthday since the travelling twin left, while the travelling twin will be celebrating their 12th! The twins can still celebrate together, but they are no longer the same age! We’ve seen that if we count the messages we get just what our analysis using the spacetime diagrams requires: The travelling twin sends 12 birthday messages and gets 24, and the stay-at-home twin sends 24 ang gets 12. They are aged just the amount we calculated. This time really…The End (And many happy returns!)