Why Does E=mc2_ - Brian Cox [28]
Other events occur far enough away from each other in space and time that they could not have any possible influence on each other. Remarkably, the ordering of these can be reversed. Einstein’s theory exploits a loophole that allows the order of events to be switched provided that doing so makes absolutely no difference to the workings of the universe. We shall explain what we mean by “far enough away” later on. For now, we have introduced the concept of causality as an axiom that we shall use to build our theory of spacetime. The success of the theory in predicting the outcome of experiments will of course be the ultimate arbiter. As an aside, Nostradamus did get one prediction right. While suffering from a particularly acute bout of gout, he apparently told his secretary, “You will not find me alive at sunrise.” The next morning he was found dead on the floor.
What has causality got to do with spacetime and, in particular, distances in spacetime? Well, we will soon discover that insisting on a causal universe constrains the structure of spacetime to such an extent that we are left with no choice in the matter. There will be only one way in which we can merge space and time together to manufacture spacetime while simultaneously preserving the causal order of things. Any other way would violate causality and allow us to do fantastical things like going back in time to prevent our own birth or, in Nostradamus’s case, perhaps avoiding a lifestyle that made him susceptible to gout.
FIGURE 3
Time now to return to the challenge of developing the concept of distance in spacetime. To get warmed up we will set time to one side for the moment and think about the idea of distance in ordinary three-dimensional space, a concept with which we are all familiar. Suppose we try to measure the shortest distance between two cities on a flat map of the earth. As will be very familiar to anyone who has flown on a long-haul flight and watched her progress on the map on the aircraft entertainment system, the shortest distance between any two points on the earth’s surface appears as a curve. This line is known as a great circle. Figure 3 shows a map of the earth, and drawn on it is a line that corresponds to the shortest distance between Manchester and New York. On a globe, this line can be understood but at first glance it is a surprise to see a curved line representing the shortest distance between two points. This occurs because the earth’s surface is not flat, but curved. To be specific, the earth is a sphere. The curved nature of the earth’s surface is also the reason why, on some flat maps, Greenland looks much bigger than Australia, when in reality it is much smaller. The message is clear: Straight lines represent the shortest distance between two points only in flat space. The geometry of flat space is often called Euclidean geometry. What Euclid didn’t know at the time, however, and in fact it did not become clear until the nineteenth century, was that his geometry of flat space is only a specific example of a whole family of different possible geometries, each of which are mathematically consistent and some of which can be used to describe nature. A very good example is the surface of the earth, which is curved and therefore described using a geometry that is non-Euclidean. Specifically, the shortest distance between two points is not a Euclidean straight line.
There are other familiar Euclidean properties that are not obeyed on the surface of the earth. For example, the interior angles of a triangle no longer add up to 180 degrees, and lines that are parallel and point north-south at the equator cross at the poles. If Euclid is no use anymore, we need to figure out how to calculate distances in a curved space, such as on the earth’s surface. One way would be to work directly