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. Hey! That one was just an example!! ;) But your keenness does earn you a reward! Should you find that you have questions that are not answered… or answers to questions I should have asked but didn’t… or anything else… you can drop me a line at: PhilipF@sphericalcows.net We now return you to your regularly scheduled power point!

4. Outline: Title &Intro (you are here) Qualitative: mapping the paradox End Calculations: computing the times Intro to Spacetime Diagrams The twin paradox & The doppler effect Description of ‘Paradox’ From the Travelling Twin’s view 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. Click on any box to go to that part, or just click anywhere else to continue to the next slide!

5. 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! Skip the Summary Summary of Twin Paradox

6. Introduction to the Twin Paradox In our study of special relativity we have learned that moving clocks ‘run slow’. One tick of your clockOne tick of moving clock 3.0m Clock moves > 3.0m light 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)

7. 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? Hey sib! Better fix your clock!  t earth t ship v

8. 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?

9. 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. 

10. The issue The ship travels to the destination planet, then turns and comes back… this is what we usually view as “THE” motion.

11. 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 

12. From another viewpoint From the point of view of the twin on the ship the ship stays in one place (right where the twin is!) while the earth leaves… and then reverses and comes back!

13. 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? 

14. Who’s younger? Since the travelling twin sees the Earth as moving, they will see the stay-at-home twin’s clock running slow, not theirs. So shouldn’t it be the stay-at-home twin who is younger? There is no quick answer here! This question is what the whole power point is about!

15. 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? 

16. The In-Between We know that during the trip out both the travelling twin and the stay-at-home twin see the other twin’s clocks running slow. And that the same thing is true during the trip back. So we might expect that if something weird is happening it must be during the moments BETWEEN the trip out and the trip back. That is indeed the case!

17. TURNING AROUND If the ship turns around very fast then the travelling twin will feel some very strong forces as the ship reaches its destination! (or the destination reaches the ship, from the travelling twin’s point of view!) But the stay-at-home twin doesn’t feel anything at all, even when the travelling twin sees the earth reverse and come back! There is something very different about the frame(s) of the two twins. And that’s what we’re exploring here!

18. 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

19. 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)

20. 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. x ct c 1D space, 1D time x ct c y 2D space, 1D time

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

22. 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) ? x ct c stationary ct' x' moving 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 space axis… it represents the “now” of the observer. 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 B D E This is the moving observer’s time axis… it represents the location of the observer at different moments in time. 

23. x ct c stationary ct' x' moving 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) The moving observer’s space axis represents the “now” of the observer (all the same time for that observer). The dots on this axis are all at the same time relative to this observer. So D&E happen at the same instant (for the moving observer) A B D E The moving observer’s time axis represents the location of the observer at different moments in time (all the same place for that observer). The dots on this axis are all at the same place relative to this observer. So A and B are the same place (at different times) for this observer

24. The size and direction of the coordinate axes change, depending on how the one frame moves relative to the other. time rest space (now) time (here) slow c space time (here) space (now) fast time (here) space (now) faster 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!) 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’. 

25. The faster the one frame moves relative to the other, the more the axes for the moving frame converge toward the light speed line. time rest space (now) time (here) slow c space time (here) space (now) fast time (here) space (now) faster Notice how both the time axis (“here”) and the space axis (“now”) shift and stretch as the relative speed increases. The diagonal line (light speed) is the middle line for all frames. It is the ultimate limit as speed increases (since the axes will meet).

26. 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.

27. 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.

28. Ship (v) x ship (now for ship) ct ship Planet Earth Planet Relativity x planet (now for planets) ct plan et Light Starting out Here the travelling twin leaves the earth in the ship, already travelling at 0.866c. 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” 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? 

29. Ship (v) x ship (now for ship) ct ship Planet Earth Planet Relativity x planet (now for planets) ct plan et Light Starting out The twin on the ship would claim this point in the planet’s history is the same time as when they left earth. The stay-at-home twin would claim that this point in the planet’s history is the same time as when the ship left earth. The travelling twin in the ship and the stay-at-home twin on earth see different events in the destination planets history as “the same time” as the ship sets out.

30. Ship (v) Planet Earth Planet Relativity x planet (now for planets) ct plan et ct ship Light x ship (now for ship) Half way The travelling twin is now half way to the destination planet. 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? 

31. Ship (v) Planet Earth Planet Relativity x planet (now for planets) ct plan et ct ship Light x ship (now for ship) Half way Comparing using the travelling twin’s “now” not much time has passed on earth. The stay-at-home twin determines a different point on its world-line as being ‘at the same time’ as the ship reaching halfway. It perceives a much longer time as having passed.

32. Ship (v) Planet Earth Planet Relativity x planet (now for planets) Light ct ship x ship (now for ship) Arriving at the Destination The ship has now reached its destination. The travelling twin must now slow down and stop. 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? How do the times for the trip compare in the two frames?  ct earth

33. Ship (v) Planet Earth Planet Relativity x planet (now for planets) Light ct ship x ship (now for ship) Arriving at the Destination Relative to the ship’s frame this much time has passed on earth during the trip. How do the times for the trip compare in the two frames? ct earth

34. Ship (v) Planet Earth Planet Relativity x planet (now for planets) ct earth Light ct ship x ship (now for ship) Ship (v) Planet Earth Planet Relativity x planet (now for planets) ct ship Light x ship (now for ship) Planet Earth Planet Relativity x planet (now for planets) Light ct ship x ship (now for ship) Ship (v=0) Arriving at the Destination 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)

35. Light Ship (v=0) Ship (v) Planet Earth Planet Relativity x planet (now for planets) Light ct ship x ship (now for ship) Ship (v) Planet Earth Planet Relativity x planet (now for planets) Light ct ship x ship (now for ship) Planet Earth Planet Relativity x planet (now for planets) Light ct ship x ship (now for ship) Return Now the travelling twin must begin the trip back. 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!) ct earth

36. Ship (v) Planet Earth Planet Relativity x planet (now for planets) ct plane t Light ct ship x ship (now for ship) … and back again! Ship (v) Planet Earth Planet Relativity x planet (now for planets) Light ct ship x ship (now for ship) Ship (v) Planet Earth Planet Relativity x planet (now for planets) ct plane t Light ct ship x ship (now for ship) Finally the trip back, with the usual rotation factors. (click to begin trip back!)

37. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship Trip Out Now with numbers! v=0.866c  =2 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 

38. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship Trip Out Now with numbers! v=0.866c  =2 Time that passed for stay-at- home twin (as seen from earth’s now): Time that has passed for stay-at- home twin = 12 years

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

40. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship Trip Out Now with numbers! v=0.866c  =2 Time that has passed for stay-at- home twin = 12 years How much time has passed for travelling twin: (slowed by a factor of  )? Time that has passed for travelling twin = 6 years

41. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship Trip Out Now with numbers! v=0.866c  =2 Time that has passed for stay-at- home twin = 12 years 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  ) Time that has passed for travelling twin = 6 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

42. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship Trip Out Now with numbers! v=0.866c  =2 Time that has passed for stay-at- home twin = 12 years To the travelling twin it is the stay-at-home twin who is moving at 0.866c, and so their clock is slow: (by a factor of  ) Time that has passed for travelling twin = 6 years Time on earth relative to SHIP’S NOW = 3 years Notice that the earth and the ship disagree about how much time has passed on the earth during the trip. This is because the ship’s “now” and the earth’s “now” are very different. The earth and the ship do not agree as to the time on earth that is at “the same time” as the ship’s arrival at its destination!

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

44. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship For the whole trip What is the total time that has passed for the travelling twin? 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? (summary)

45. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship Summary for the whole trip Total Time that has passed for stay-at-home twin = 24 years Total time that has passed for travelling twin = 6+6 = 12 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.

46. 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!

47. The trip in the coordinates of the ‘stay at home’ twin. Watch how the positions and coordinates behave. What changes? What stays the same? Click to Start Next

48. The ‘turn around’. What changes? What stays the same? Click to Start Next

49. The trip back in the coordinates of the stay-at-home twin. Click to Start Next Do you wonder why we bothered dividing this up? Wait for the next view!

50. The 1 st part of the trip in Ship coordinates. Here the Earth leaves the ship and the planet comes to it. Notice how (except for the values of the coordinates) this is very similar to the previous. Click to Start Next

51. The ‘turn around’. Watch closely. How is this different from the view of the stay-at-home twin? What changes and what is the same? Why is this so different? Click to Start Next

52. The earth returns to the ship in the view of the travelling twin. Notice how the first part of the trip looks completely different. Was that true for the stay-at-home twin? Click to Start Next

53. In this frame the earth has been moving toward the ship’s present position the whole time, at 0.866c. It will arrive after 24 years on earth = 48 years in this frame (since the moving earth’s clocks run slow by a factor of  = 2 24 y 48 y Relative to this frame the earth is moving toward the ship’s position, but the in the first part of the trip the ship was moving toward this position faster. The ship travelled 41.6 ly in 42 years, so relative to this frame it was going at 0.990c. This is exactly what relativistic velocity addition gives. 42 y 41.6 ly Ship on trip outward Earth’s motion seen in this frame

54. 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!)

55. The real issue is what the twins are going to do about the asymmetry of number of Birthday Presents!! The End 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!

56. Extra: What would the twins really ‘see’? We are sometimes rather loose with the way we talk about things in discussing the different ‘frames’ in relativity. We will say things like “the stay-at-home twin sees the travelling twin’s clock running slow.” But what we mean is that “relative to the stay-at-home twin the travelling twin’s clock is running slow… and if the stay-at-home twin works out how quickly the travelling clock is running, or waits to see what is reported when the ship arrives, they will find the time is slowed” You can’t totally blame us for the shortcut – but it’s misleading!

57. 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. ct A light signal from here Is received here

58. 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.

59. ‘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.

60. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship 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? From the graph, about how many signals a year does the ship encounter as it returns? Ticks mark birthdays 

61. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship What the travelling twin sees: The returning ship encounters just over 22 birthday greetings in 6 years, or about 3.7 a year! Notice that on the trip out the signals have to catch up to the ship, so only 1 is received along the way (and one hasn’t quite gotten there when the ship arrives)…. The messages are more than 3 years apart (about 3.7 years if you measure carefully)

62. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship What the stay-at-home twin sees: 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? Coming back the ship is rapidly following its signals, so they will come in very rapidly. About how many are received per year? 

63. Planet Earth Planet Relativity x planet (now for planets) ct planet ct ship What the stay-at-home twin sees: The stay-at-home twin gets just 6 birthday greetings in the first 22 years of waiting… more than 3 years apart (3.7 years apart in fact). Coming back the ship is rapidly following its signals, so they will come in very rapidly. All 6 are received by the stay-at- home twin during less than 2 years as the travelling twin returns… more than 3 a year!

64. 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? 

65. The stay-at-home twin gets slowed down messages as the ship travels away and then faster messages while the ship travels back toward the Earth The travelling twin gets slowed down messages as the Earth travels away (6 years) and then faster messages while the Earth travels back toward the ship (another 6 years)

66. 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) From the point of view of the stay-at-home twin the ship sends 6 signals while moving away from the earth. Earth Planet x ct ct shi p 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) How much time passes on earth during this whole process? (What is the total time taken to get all the birthday messages?) 

67. From the point of view of the stay-at-home twin the ship sends 6 signals while moving away from the earth. EarthPlanet x ct ct shi p Given that the ratio of frequencies is 3.73, a total of 6  3.73 = 22.4 years pass on earth while waiting for those messages to arrive. Going the other way the ship sends 6 signals, but now they are received 1/3.73 years apart. So 6/3.73 = 1.6 years for those signals to arrive. So all the birthday messages take a total of 22.4y + 1.6 y = 24.0 years to arrive. So 24 years pass for the stay-at-home twin, who gets 12 birthday greetings from the travelling twin! 22.4 y 1.6 y 24.0y Earth

68. From the point of view of the travelling twin the signals from earth are spaced out as the earth moves away. Ship EarthPlanet x ct ct shi p 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) 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) How many birthday greetings are received by the ship in total? (What is the total number of birthday messages the ship receives?) 

69. From the point of view of the travelling twin the signals from earth are spaced out as the earth moves away. Ship EarthPlanet x ct ct shi p Given that the ratio of frequencies is 3.73, there is time to receive only 6/3.73 = 1.6 signals As the earth approaches the ship again signals are received much more often… at a rate of 3.73 a year during the next 6 years = 3.73/y  6y = 22.4 signals So during the 12 years that pass for the ship a total of 24 birthday messages are received… which agrees with our calculations! 1.6 messages 22.4 messages

70. 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! Show me all the gory details! No thank’s… I’m satisfied. Skip it! This can also be seen in the travelling twin coordinates.

71. The 1 st 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? 

72. The birthday wishes from the stay-at-home twin are received by the travelling twin every 3.73 years, so a total of 6.0y/3.73y = 1.61 signals are received.

73. 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? 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? 

74. The second birthday wish was sent 38 years ago (relative to the new frame) but is just now reaching the ship’s position. The remaining signals will arrive by the time the earth reaches the ship. Thus we see that the travelling twin gets a total of 1.6 + 22.4 = 24 birthday greetings from the stay- at-home twin. The travelling twin sent 6+6 = 12 birthday greetings. So, in the next 6 years a total of 6  3.73 = 22.4 signals are received (most of these sent long before but only just now reaching the ship).

75. 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!) When they finally meet the twin on earth will be celebrating the 24 th birthday since the travelling twin left, while the travelling twin will be celebrating their 12 th ! The twins can still celebrate together, but they are no longer the same age!