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free to decide that they are standing still, and the other guy is whizzing around the universe at high speed relative to them. And so it is for the astronaut twin, who is free to say that she is standing perfectly still in her space rocket, watching the earth fly away at high speed. For her, it is therefore the earthbound twin who ages more slowly. Who is right? Can it really be that each of the twins ages more slowly relative to the other? Well it has to be like that—that is what the theory says. There is no paradox yet, because any problems you might be having in believing that each twin observes the other to be aging more slowly are not real problems. They are due to the fact that you are clinging to the idea of universal time. But time is not universal; that much we have learned, and that means there is no contradiction at all. Now comes the apparent paradox: What happens if the astronaut twin returns back to Earth sometime in the future and meets up with her earthbound twin? Obviously they cannot both be younger than the other. What is going on? Is one of them actually older than the other? If so, who?

FIGURE 8

The answer can be found in our understanding of spacetime. In Figure 8 we show the paths through spacetime taken by the twins, as measured using clocks and rulers at rest relative to the earth. The earthbound twin stays on the earth and consequently her path snakes along the time axis. In other words, almost all of her allocated speed through spacetime is expended traveling through time. Her astronaut twin, on the other hand, heads off at close to light speed. Returning to the motorcyclist analogy, that means she charges off in a “northeasterly” direction, using up as much of her spacetime speed as she can to push through space at close to the cosmic speed limit. On the spacetime diagram shown in Figure 8, that means she travels close to 45 degrees. At some point, however, she needs to turn around and come back to the earth. The picture shows that we are supposing that she heads back again at close to light speed but this time in a “northwesterly” direction. Obviously the twins take different paths through spacetime, even though they started and finished at the same point.

Now just like distances in space, the length of two different paths in spacetime can be different. To reiterate, although everyone must agree on the length of any particular path through spacetime, the lengths of different paths need not be the same. This is really no different from saying that the distance from Chamonix to Courmayeur depends upon whether you went through the Mont Blanc tunnel or hiked over the Alps. Of course, walking over a mountain means you travel a longer distance than tunneling through it. In our discussion of the motorcyclist speeding over the spacetime landscape, we established that the time measured on the motorcyclist’s wristwatch provides a direct way to measure the spacetime distance he traveled: we just need to multiply the elapsed time by c to get the spacetime distance. We can turn this statement on its head and say that once we know the spacetime distance traveled by each of the twins, then we can figure out the time that passes according to each. That is, we can think of each twin as a voyager through spacetime with their wristwatches measuring the spacetime distance that they travel.

Now comes the key idea. Look again at the formula for distances in spacetime, s2 = (ct)2 - x2. The spacetime distance is biggest if we can follow a path that has x = 0. Any other path must be shorter because we have to subtract the (always positive) x2 contribution. But the earthbound twin snakes along the time direction with x close to zero, so her path must be the longest possible path. Actually, that is just another way of saying what we already know: that the earthbound twin is traveling as fast as possible through time and so it is she who ages the most.

Our explanation so far has been presented from the viewpoint of the earthbound twin. To fully satisfy ourselves that there is no paradox, we should see how