Twins Paradox:

From P. Davies, 'About Time':

Betty is going to leave Earth in the year 2000 and travel by rocket ship to a star eight light-years away (as measured in Earth's frame of reference) at a speed of 240,000 kilometers per second. To keep the sums simple, I shall neglect the periods the ship spends accelerating and braking (i.e., treat these periods as instantaneous), and also assume Betty doesn't spend any time sightseeing when she reaches the star. To achieve 80 percent of the speed of light in a negligible time implies an enormous acceleration, which would be fatal to a real human, but this is incidental to the argument. I could easily include a more realistic treatment of the acceleration, but at the price of making the arithmetic more complicated; the overall conclusions would be unaffected.

First let me compute the total duration of the journey as predicted by Einstein for each twin. At 80 percent of the speed of light. it takes ten years to travel eight light-years, so Ann, on Earth, will find that Betty returns in Earth year 2020. Betty, on return, agrees that it is Earth year 2020, but insists that only twelve years have elapsed for her, and her rocket clock - a standard atomic clock carefully synchronized before takeoff with Betty's identical clock on Earth - confirms this assertion: it reads 2012.

Now suppose we equip our twins with powerful telescopes so that they can watch each other's clocks throughout the journey and see for themselves what is going on. Ann's Earth clock ticks steadily on, and Betty looks back at it through her telescope as she speeds away into space. According to Einstein, Betty should see Ann's clock running at 60 percent of the rate of her own clock. In other words, during one hour of rocket time, Betty is supposed to see the Earth clock advance only thirty-six minutes. In fact, she sees it going even slower than this. The reason concerns an extra effect, not directly connected with relativity, that is usually left out of discussions of the twins paradox. It is vital to include the extra effect if you want to make sense of what the twins actually see.

Let me explain what causes this extra slowing. When Betty looks back I at Earth, she does not see it as it is at that instant, but as it was when the light left Earth some time before. The time taken for light to travel from Earth to the rocket will steadily increase as the rocket gets farther out in space. Thus Betty will see events on Earth progressively more delayed, because of the need for the light to traverse an ever-widening gap between Earth and rocket. For example, after one hour's flight as measured from Earth, Betty is 0.8 light-hours (48 light-minutes) away, so she sees what was happening on Earth forty-eight minutes earlier, for the light, which conveys the images of Earth to Betty, to reach her at that point in the journey. In particular, Ann's clock would appear to Betty-I'm referring to its actual visual appearance-to be slow anyway, irrespective of the theory of relativity. After two hours' flight, the Earth clock would appear to Betty to lag even more behind. This "ordinary" slowing down of clocks. and events generally, as seen by a moving observer, is called the "Doppler effect," named for a Swedish physicist who first used it to describe a property of sound waves. By adding the Doppler effect to the time-dilation effect, you get the combined slowdown factor.

Ann will also see Betty's rocket clock slowed by the Doppler effect, because light from the rocket takes longer and longer to get back to Earth. She will in addition see Betty's clock slowed by the time-dilation effect. By symmetry, the combined slowdown factor of the other clock should be the same for both of them.

Let me now compute the combined slowdown factor, first from Ann's point of view, then Betty's. To do so, I shall focus on the great event of Betty's arrival at the star. The outward journey takes ten years as measured on Earth. However, Ann will not actually see the rocket reach the star in the year 2010, because by this stage Betty is eight light-years away. Since it will take light a further eight years to get back to Earth, it will not be until the year 2018 that Ann gets to witness visually Betty's arrival at the star.

What is the time of the arrival event as registered on Betty's clock? Einstein's formula tells us that Betty's clock runs at 0.6 the rate of the clock on Earth, so ten years of Earth time implies six years in the rocket. The rocket clock therefore stands at six years on Betty's arrival at the star. So, when Ann gets to witness this arrival in 2018, the rocket clock says 2006. Thus, as far as the visual appearance of the rocket clock is concerned, Ann sees only six years having elapsed in her eighteen years-i.e., Betty's rocket clock has been running at one-third the rate of Ann's Earth clock. Now, Ann is perfectly capable of untangling the time-dilation and Doppler effects, and computing the "actual" rate of Betty's clock, having factored out the effect of the light delay. She will find the answer to be 0.6, in accordance with Einstein's formula. Thus Ann deduces (but does not actually see) that throughout Betty's outward journey Betty's clock was running at thirty-six minutes to Ann's hour.

From Betty's perspective, things are the other way about. She agrees, of course, that her rocket clock stands at 2006 when she arrives at the star, but what does she see the Earth clock registering at that moment? We know that in the Earth's frame of reference the arrival event occurs at 2010, but, because the star is eight light-years away, the light that actually reaches the rocket at that moment will be from eight years previously-i.e., 2002. So Betty will look back at Earth, on arrival at the star, and see the Earth clock registering 2002. Her clock says 2006. Therefore as far as the actual appearance of the Earth clock is concerned, it records two years having elapsed for Betty's six years. Thus Betty concludes that the Earth clock has been running at one-third the rate of her own rocket clock for the outward part of the journey. This is the same factor that Ann perceived Betty's clock to be slowed by, so the situation is indeed perfectly symmetric. Again, Betty can untangle the Doppler effect from the time-dilation effect and deduce that Ann's clock has "really" been running at 0.6 the rate of her own.

Without delay, Betty embarks on the return journey. Because Betty is approaching rather than receding from Earth the light-delay (i.e., Doppler) effect now works in opposition to the time-dilation effect. The former causes events to appear speeded, although time dilation still works to tell slow them down. Let's put the numbers in. First, what does Ann see as Betty speeds back towards Earth? Since we are agreed that Betty returns to Earth in the year 2020, and Ann actually sees Betty reach the star in 2018, the return part of the journey will appear to Ann, viewing the approach of the rocket from Earth, to be compressed into just two years of Earth time. We have already determined that, when, in 2018, Ann sees Betty's clock at the halfway point, it registers 2006, and that when Betty returns to Earth it will register 2012. So, for the two Earth years during which Ann sees the rocket traveling back, she will witness the rocket clock progress through the remaining six years. In other words, on the return leg of the journey Ann sees Betty's clock running three times faster than her own, Earthbound, clock. This is a key point: during the return journey the rocket clock appears from Earth to be speeded up, not slowed down. The Doppler effect beats the time-dilation effect. Again, Ann can untangle the time-dilation and light-delay effects and deduce that the rocket clock is "really" running at 0.6 the rate of her clock-i.e., although the rocket clock looks to Ann to be speeded up, she deduces that it is "really" running slow at exactly the same reduced rate (0.6) as it was on the outward journey. So, although the visual appearance of the rocket clock is quite different for the two legs of the journey, the time-dilation factor of 0.6 remains the same throughout.

Finally, let me examine the return journey as observed by Betty, in the rocket. She has experienced six years for the outward trip, and she experiences another six years for the return, reaching Earth in 2012 as registered on her own clock. During the return journey, however, Betty also observes the clock on Earth. She saw it (actually, visually) standing at 2002 at the moment she reached the star. We know she will get home I in 2020, so Betty will see the Earth clock progress through eighteen years during the six years aboard the rocket. Thus the Earth clock appears to the same factor as that by which Ann saw Betty's clock speeded up- there is complete symmetry on the return part of the journey too. Betty can again factor out the light-delay effect and deduce that the Earth clock is "really" running slow-at 0.6 of the rate of her rocket clock.

The crucial point to be extracted from all this is that during the periods when the rocket is traveling at a fixed speed Ann deduces that Betty's clock is running slow and Betty deduces that Ann's clock is running slow. On the outward part of the journey, each actually sees the other's clock running (even more) slowly, but on the return part of the journey each sees the other's clock speeded up. The deductions and experiences all fit together consistently, and refute the claim that there is any paradox attached to the statement that "each clock runs slow relative to the other.

For those readers who have waded through this arithmetic, it contains a hidden conclusion about distances. If you use the fact that in Betty's frame of reference Earth recedes at 0.8 of the speed of light, and the journey to the star takes just six rocket years, then the distance to the star as measured by Betty must be 0.8 x 6 = 4.8 light-years. Thus, although Ann measures the star to be eight light-years away, Betty measures the distance to the star to be only 4.8 light-years. The distance is shrunk by the same factor (0.6) as that by which time is dilated.

Excerpt from the Encyclopedia Britannica without permission.