Why Does E=mc2_ - Brian Cox [39]
how far that distance is, because distances in spacetime are universal and not a matter for debate. That means we can ask the motorcyclist how far he thinks he has traveled over the spacetime landscape and the answer he gives will be the right answer. Now, the motorcyclist can choose to calculate distances in spacetime relative to himself, and from this point of view he has not moved in space. It is just like the person sitting on the airplane in Chapter 1 who doesn’t stray from her airplane seat and who therefore states that she has not moved. She may have moved relative to someone else—for example, someone standing on the ground watching the plane fly by—but that is not the point. So from our motorcyclist’s point of view, he has not moved in space and yet 1 second in time has passed. He can therefore use the spacetime distance equation s2 = (ct)2 - x2 with x = 0 (because he hasn’t moved in space) and t = 1 second to figure out how far in spacetime he has actually traveled: The answer is a distance equal to c multiplied by 1 second. So the motorcyclist tells us that he is traveling a distance of c (multiplied by 1 second) for every second that passes on his watch, and that is just another way of saying that his speed through spacetime is equal to c. If you have been following closely, then you might object that the passage of 1 second was measured on the motorcyclist’s wristwatch and that a different amount of time will pass according to someone else who is moving relative to the motorcyclist. That is true enough, but there is something special about the motorcyclist’s watch, because the motorcyclist does not move relative to himself (a trivial statement).We are therefore free to put x = 0 in the distance equation and so the time that passes on his wristwatch is a direct way to measure the spacetime distance s. This is a nice result: The time that passes on the motorcyclist’s watch is equal to the spacetime distance traveled divided by c. In a sense, his watch is a device for measuring distances in spacetime. Since both the spacetime distance and c are agreed upon by everyone, it follows that the motorcyclist has unwittingly used his watch to measure something that everyone can agree upon. The spacetime speed c that he deduces is therefore also a quantity that everyone can agree upon.
So the speed through spacetime is a universal upon which everyone agrees. This newfound way of thinking about how things move through spacetime can help us get a different handle on why moving clocks run slow. In this spacetime way of thinking, a moving clock uses up some of its fixed quota of spacetime speed because of its motion through space and that leaves less for its motion through time. In other words, a moving clock doesn’t move so fast through time as a stationary one, which is just another way of saying that it ticks more slowly. In contrast, a clock sitting at rest whizzes along in the time direction at the speed c with no motion through space. It therefore ticks along as fast as is possible.
Armed with spacetime, we are ready to contemplate one of the wonderful puzzles of Special Relativity: the Twins Paradox. Earlier in the book we showed that Einstein’s theory allows us to contemplate the possibility of traveling to distant places in the universe. Speeding within a whisker of the speed of light, we imagined journeying off to the Andromeda galaxy within a human lifetime regardless of the fact that it takes light nearly 3 million years to make the journey. There is a paradox lurking here that we previously glossed over. Imagine twins, one of whom trains to be an astronaut and heads off on humanity’s first mission to Andromeda, leaving her twin back home on Earth. The astronaut twin is moving at high speed relative to the earth and consequently her life slows down relative to her twin on Earth. But we have just spent a significant fraction of this book arguing that there is no such thing as absolute motion. In other words, the answer to the question “Who is doing the moving?” is “Whoever you want.” Anybody and everybody is