Why Does E=mc2_ - Brian Cox [44]
concept: Just as a vector can point “north,” we now have the notion of a vector that points “in the time direction.” As is the norm when we talk about spacetime, this is not something we can picture in our mind’s eye, but that is our problem, not nature’s. The spacetime landscape analogy of the last chapter might help you build a mental picture (at least of a simplified spacetime with only one dimension of space). Four-dimensional vectors will be characterized by four numbers. The archetypal vector is the one that connects two points in spacetime. Two examples are illustrated in Figure 9. That one of the vectors in Figure 9 points exactly in the time direction and that both just happen to start out from the same place is only for our convenience. Generally speaking, you should think of any two points in spacetime with an arrow joining them. Vectors like these are not entirely abstract things. Your going to bed at 10 p.m. and subsequent awakening at 8 a.m. defines an arrow linking two events in spacetime; it is “10 hours multiplied by c long” and it points entirely in the time direction. Moreover, we have actually been using vectors in spacetime throughout the book but haven’t used the terminology before. For example, we met a very important vector in our discussion of the intrepid motorcyclist, journeying over the undulating landscape of spacetime with his throttle stuck. We worked out that the motorcyclist always travels at a speed c through spacetime, and the only choice he can make is the direction in which he points his motorcycle (although he doesn’t even have complete freedom of direction, because he is restricted to staying within a bearing of 45 degrees of north). We can represent his motion with a vector of fixed length c, which points in the direction in which he is traveling over the spacetime landscape. This vector has a name. It is called the spacetime velocity vector. To use the correct terminology, we would say that the velocity vector always has length c and is restricted to point within the future lightcone. The lightcone is a fancy name for the area contained within the two 45-degree lines that are so important in protecting causality. We can completely describe any vector in spacetime by specifying how much of it points in the time direction and how much of it points in the space direction.
By now, we are familiar with the statement that the distances in time and space between events are measured differently by observers moving at different speeds relative to each other, but they must change in such a way that the spacetime distance always remains the same. Because of the strange Minkowski geometry, this means that the tip of the vector can move around on a hyperbola that lies in the future lightcone. To be absolutely concrete, if the two events are “going to bed at 10 p.m.” and “waking up at 8 a.m.,” then an observer in the bed concludes that the spacetime distance vector points up his time axis, as illustrated in Figure 9, and its length is simply the time elapsed on his watch (10 hours) multiplied by c. Someone flying past at high speed would be free to interpret the person in bed as doing the moving. She would then have to add in a bit of space movement as well when she viewed the person in bed, and that moves the tip of the vector off her time axis. Because the arrow’s length cannot change, it must stay on the hyperbola. The second, tilted arrow in Figure 9 illustrates the point. As you can see, the amount of the vector pointing in the time direction has increased and this means that the fast-moving observer concludes that more time passes between the two events (i.e., more than 10 hours elapses on her watch). This is yet another way to picture the strange effect of time dilation.
So much, for now at least, for vectors (we will need the velocity spacetime vector again in a moment). The next few paragraphs relate to the second crucial piece of the E = mc2 jigsaw. Imagine you are a physicist trying to figure out how the universe works. You are comfortable with the idea of vectors and on occasion