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υt. In other words, distance and time can be interchanged using something that has the currency of a speed. Let us therefore introduce a calibrating speed; call it c. We can then measure time in meters provided we take any time interval and multiply it by our calibrating speed. At this point in our reasoning c really can be any old speed and we have not committed ourselves at all as to its actual value. Actually, this trick of interchanging time and distance is very common in astronomy, where the distance to stars and galaxies is often measured in light-years, which is the distance light travels in one year. This doesn’t seem so strange because we are used to it, but it really is a distance measured in years, which is a unit of time. In the astronomy case, the calibrating speed is the speed of light.

FIGURE 4

This is progress; we now have time and distance intervals in the same currency. For example, they could both be given in meters, or miles or light-years or whatever. Figure 4 illustrates two events in spacetime, denoted by little crosses. The bottom line is that we want a rule for figuring out how far apart the two events are in spacetime. Looking at the figure, we want to know the length of the hypotenuse given the lengths of the other two sides. To be a little more precise, we shall label the length of the base of the triangle as x while the height is ct. It means that the two events are a distance x apart in space and a distance ct apart in time. Our goal, then, is to answer the question “what is the hypotenuse, s, in terms of x and ct?” Making contact with our earlier example x = 10 meters is the distance in space from bed to kitchen table, and t = 1 hour is the distance in time. So far, since c was arbitrary, ct can be anything and we appear to be treading water. We shall press onward nonetheless.

We have to decide on a means of measuring the length of the hypotenuse, the distance between two events in spacetime. Should we choose Euclidean space, in which case we can use Pythagoras’ theorem, or something more complicated? Perhaps our space should be curved like the surface of the earth, or maybe some other more complicated shape. There are in fact an infinite number of ways that we might imagine calculating distances. We’ll proceed in the way that physicists often do and we will make a guess. Our guess will be guided by a very important and useful principle called Occam’s razor, named after the English thinker William of Occam, who lived at the turn of the fourteenth century. The idea is simple to state but surprisingly difficult to implement in everyday life. It might be summarized as “don’t overcomplicate things.” Occam stated it as “plurality must never be posited without necessity,” which does beg the question: Why didn’t he pay more attention to his own rule when constructing sentences? However it is stated, Occam’s razor is very powerful, even brutal, when applied to reasoning about the natural world. It really says that the simplest hypothesis should be tried first, and only if this fails should we add complication bit by bit until the hypothesis fits the experimental evidence. In our case, the simplest way to construct a distance is to assume that at least the space part of our spacetime should be Euclidean; in other words, space is flat. This means that the familiar way of working out the distance in space between objects in the room in which we are seated reading this book is carried over into our new framework intact. What could be simpler? The question, then, is how we should add time. Another simplifying assumption is that our spacetime is unchanging and the same everywhere. These are important assumptions. In fact, Einstein did eventually relax them and doing so allowed him to contemplate the mind- (and space-) bending possibility that spacetime could be constantly changed by the presence of matter and energy. It led to his general theory of relativity, which is to this day our best theory of gravity. We will meet general relativity in the final chapter, but for the moment we can ignore