-------------------------------------- THE TWIN PARADOX IN SPECIAL RELATIVITY -------------------------------------- The twin paradox arises when one twin travels at a high speed to some faraway place and returns. Let's say that one twin, Alex, stays at home on earth, while the other, Betty, goes to the star Betelgeuse and returns 20 years later (Alex time), traveling at 0.87c, so that the shrinking factor is 2. The paradox goes like this: o From Alex's point of view, Betty has been traveling at 0.87c, so she should only have aged 10 years, and Alex should be older. o From Betty's point of view, Alex (and the earth) have been moving at 0.87c, so Alex should only have aged half as much as she did. o How can both of them be right that the other ages more slowly? Here's a cartoon of how each twin sees it: Alex's a cartoon: =================================================== Alex Year 0: (Earth) (Betelgeuse) Betty--> --------------------------------------------------- Alex Year 05: (Earth) (Betelgeuse) Betty--> --------------------------------------------------- Alex Year 10: (Earth) (Betelgeuse) <--Betty--> --------------------------------------------------- Alex Year 15: (Earth) (Betelgeuse) <--Betty --------------------------------------------------- Alex Year 20: (Earth) (Betelgeuse) <--Betty =================================================== Betty's cartoon: =================================================== <-- Alex Year 0: <--(Earth) <--(Betelgeuse) Betty --------------------------------------------------- <-- Alex Year 2.5 <--(Earth) <--(Betelgeuse) Betty --------------------------------------------------- <-- Alex--> Year 5: <--Earth--> <--(Betelgeuse)--> Betty --------------------------------------------------- Alex--> Year 7.5 (Earth)--> (Betelgeuse)--> Betty --------------------------------------------------- Alex--> Year 10: (Earth)--> (Betelgeuse)--> Betty =================================================== The answer to the paradox lies in the ACCELERATION that Betty must apply. Unless she accelerates near Betelgeuse, she won't be able to return to earth, where age comparisons can be made (see footnote at end). ---------------------------------------- Let's say that each twin sends the other daily radio messages, reporting what time it is, and how things are going. Also, let's assume that Betty really brakes hard at Betelgeuse, so she turns around quickly (it still works for lower accelerations, but with less dramatic effects). Both Alex and Betty are equipped with Acme Amazing Doppler Correctors (AADC), which allow them to correct received transmissions for the doppler effects. Here is the story from Alex's point of view: Outward journey takes 10 years, Betty's radio messages are drawlingly slow (despite AADC) and only come once every 2 days. She says 5 years have elapsed. As Betty turns around, her speech slows to practically nothing (she is accelerationg, or in a deep gravitational field, so her clocks appear to have slowed way down). On her return trip, Betty's messages are again their usual slow drawl, and although it takes 10 years again, she says it only took 5 years. Back at home, she is indeed only 10 years older, despite Alex's having aged 20 years, and she speaks quite normally now that she's home again. The story from Betty's point of view is different: The outward journey takes 5 years (Betelgeuse is 8.7 Ly away, but this distance appears to be shrunk by a factor of 2, since earth and Betelgeuse are moving by at 0.87c). During this time, Alex's radio messages are annoyingly slow and drawling. He says that only 2.5 years have elapsed. Upon reaching Betelgeuse, Betty fires her retro-rockets, and suffers from many nosebleeds during the months needed to turn around. During these few months, she feels like she is in a colossal gravitational field (about 30x earth's gravity), and since Alex seems to be way up "high" in that field, his clocks appear to be moving very fast. Alex's radio messages now race in at an amazing rate, and she can hardly understand his mickey-mouse gibberish. But she can make out that he thinks that 15 years have elapsed during the few months that she was turning around. Her return trip is much like the outward journey. Alex is back to his irritating drawl (after AADC), and again claims to have aged only 2.5 years during the 5 year journey. Back at home, Betty is relieved to see that Alex is talking quite normally again, but he looks surprisingly old. Both agree that Alex has aged 20 years to Betty's 10. ---------------------------------------- The stories told above are as Alex and Betty would have worked them out after using AADC and thinking things through. Their actual message logs look quite different, due to the finite speed of light. Because of this, the messages actually get stretched out and doppler shifted on the way out, and squashed together on the return. For example, here is Alex's real log of received messages: Alex's log of received messages: Year 0: Betty's off to her great adventure. Once she's up to speed, she zips past Earth one last time, and sends me a message, Sounds pretty slow after AADC, she must be making fun of my staying home with that tedious drawl. Years 1-5: Betty sends messages every two days (after promising daily messages). All messages are drawlingly slow, as well -- how tiresome! At the end of year 5, the latest message says she has been gone for over 1 year, and is only 1/4 the way to Betelgeuse. Of course, I understand that it takes a while for messages to get back to me. Years 6-10: More drawling messages from Betty, saying that she is outward bound. By year 10, messages say Betty is about half way to Betelgeuse. She probably has got there by now, but of course her messages have to wend their way back here at c. Years 11-18.7: Still more drawling messages from Betty. Still outward bound. But at year 18.7, I finally receive a message saying that she has reached Betelgeuse, and is firing her retro-rockets to come home. Doing some quick calculations of message transmission times, I'd say she sent this message just before year 10, as I suspected, but she says it only took 5 years. Around 18.7: I hear very few messages from her turn around. The few messages that I get are very slow indeed, and I can barely hear her. Around 18.8-20.0: Betty says she has turned around, and I'm getting messages every hour. She talks very fast like Mickey Mouse, and it's hard to understand. Of course, the reason things are so fast is that her messages are Doppler-shifted on her return trip. Curiously, when I correct for this with AADC, her messages are drawling, not normal. Year 20: I saw Betty zip past Earth again! As she zipped past, she sent me a drawling message saying that she'll slow down now, and that she has aged only 10 years on the trip. Fancy that! Year 20: She's stopped and come home now, and at last is speaking normally. She says that she never drawls, and on the contrary has found my slow speech quite irritating herself. Similarly, Betty's log will have most of Alex's messages piled up at the end of her trip. ---------------------------------------- Bottom Line: o It is Betty's ACCELERATION, which she could actually feel, which made her age less than Alex. If you construct a more symmetric example, where Alex goes to Antares (roughly the opposite direction) while Betty goes to Betelgeuse, and both meet again at earth to compare notes with Charles, all will agree Alex and Betty are now the same age, but that Charles has aged more than either of them. ========================================================================== Footnote: To make proper age comparisons, you need to be nearby -- close enough so that you can send signals back and forth with age information. You might wonder whether Betty could make age comparisons by talking to Alex's friend Andrew, living on a space station near Betelgeuse. Andrew and Alex correspond frequently, and they have agreed on how to synchronize their clocks. Shouldn't Betty, then, find that Andrew is 2.5 years younger than she is when she reaches Betelgeuse, even though Andrew thinks that Betty is 5 years younger than he is? This is where Einstein's asynchronous clock effect comes in. Under special relativity, Betty and Andrew both think that Andrew is older by 5 years when Betty reaches Betelgeuse. Remember that Andrew's clocks look to Betty like they are set far in advance of Alex's, since Andrew's clock is "trailing" by a long way as Alex and Andrew zip past her. This means that although Andrew (and Alex) appeared to age more slowly, Andrew had a big head start in the aging process. In general relativity, it is Betty's acceleration that leads to these curious time differences that Betty sees between Alex's and Andrew's clocks. Andrew's clocks got their apparent head start (as seen by Betty) when Betty first accelerated at the start of her adventure. Now, again, while Betty fires her retro-rockets, both Betty and Andrew agree that this takes a few months, since they exchange frequent messages to check this. But Betty is feeling this colossal gravitational field, with Andrew at the same depth as she is, while Alex is way up high. In this case, general relativity effects will make Betty think that Alex is aging very rapidly, while Andrew is not. By the time she finishes re-accelerating towards earth, Andrew's apparent head-start is completely erased, and it now looks like Alex's clocks are set ahead. Imagine Betty watching a live video of Alex and Andrew's clocks during her journey: o Before she starts, both are winding along at the normal rate. o While Betty is accelerating, Andrew's clocks speed up dramatically, and Andrew ages much more than Alex, who roughly keeps pace with Betty. o Once she is coasting along, both Andrew and Alex's clocks are advancing very slowly (after AADC and other corrections) at the same rate. Without AADC, of course, Andrew's clocks seem to advance more quickly, but that's because she is running into Andrew's signals, and away from Alex's signals. o When she reaches Andrew, Betty thinks the boys have only aged half as much as she has during the trip, but Andrew's head start makes him much older than Alex. o As Betty decelerates to slow down, however, Alex's clocks speed up, and by the time she decelerates completely, Alex's clocks have caught up with Andrew's (and both are older than she is). o Now that she accelerates for the return trip, Alex's clocks keep going at a high speed, while Andrew's roughly keep pace with Betty's. o On Betty's return trip, the boys are again aging slowly, but now Alex is quite a bit older than Andrew. o On her return to earth, Alex has aged 20 years, and Andrew seems to be between her and Alex in age. o As she decelerates to land, however, Alex's clock speeds up to match hers, but Andrew's clocks start racing ahead. o When Betty comes to rest on Earth, both Alex and Andrew have aged 20 years, while she has aged only 10. Imagine that you are in a relativistic spaceship facing away from the rocket nozzle. When you fire up the rocket to accelerate (or to decelerate after turning around) all clocks in front of you seem to speed up dramatically, at a rate that increases with their distance from you, while clocks behind you appear to slow down correspondingly. Interestingly, things in front also become bluer and brighter, while things behind you become faint and red. Bring your sun-screen lotion and keep looking backwards.