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Generally speaking, gravity changes from place to place. Its pull is stronger the closer to the center of the earth you are, although there isn’t that much difference between sea level and the top of Mount Everest. It is much weaker on the moon, because the moon is less massive than the earth. Likewise, the gravitational pull of the sun is much stronger than that of the earth. But wherever you happen to be in our solar system, the force of gravity will not vary too much within your immediate locality. Imagine standing on the ground. The gravity at your feet will be slightly stronger than the gravity at your head but it will be a very small difference. It will be smaller for a short person and bigger for a tall person. You might imagine a tiny ant. The difference in the gravitational pull on its feet compared to its head will be smaller still. Let’s travel the well-worn pathways of the thought experiment one more time and imagine smaller and smaller things, all the way down to a tiny “elevator.” So small is our elevator that the gravity can be assumed to be the same everywhere inside it. The tiny elevator is populated by even tinier physicists whose job it is to carry out scientific experiments within their elevator. Now we can imagine that the little elevator is in free fall. In this case, none of the tiny physicists would ever utter the word “gravity.” A description of the world in terms of observations made by this group of tiny falling physicists has the astonishing virtue that gravity simply does not exist. Nobody would utter the word “gravity” in their tiny squeaky voices because there is no observation that could be made within the elevators that would indicate that there was such a thing. But hang on a second! Clearly something makes the earth orbit the sun. Is this just some clever sleight of hand or are we onto something important?

Let’s leave gravity and spacetime for a moment and return to the analogy of the curved surface of the earth. A pilot planning a trip from Manchester to New York clearly needs to recognize that the earth’s surface is curved. In contrast, when moving between your dining room and your kitchen you can safely ignore the curvature of the earth and assume that the surface is flat. In other words, the geometry is (very nearly) Euclidean. This is ultimately why it took awhile for humans to discover that the earth is not flat but spherical; the radius of curvature is very much bigger than the day-to-day distances that we are used to dealing with. Let’s imagine chopping up the earth’s surface into lots of little square patches, as illustrated in Figure 25. Each patch is pretty near flat, and the smaller we make the patches, the nearer to flat each one is. On each patch, Euclid’s geometry holds sway: Parallel lines don’t cross and Pythagoras’ theorem works. The curvature of the surface becomes evident only when we try to cover large areas of the earth’s surface with our Euclidean patches. We need lots of little patches sewn together to faithfully construct the curved surface of the sphere.

Now let’s return to our little elevator in free fall and imagine it is accompanied by many other little elevators, one at each point in spacetime, in fact. The spacetime inside each is approximately the same everywhere, and the approximation gets better as the elevators get smaller. Now, recall that in Chapter 4 we were very careful to point out our assumption that spacetime should be “unchanging and the same everywhere,” and this was critical in allowing us to construct Minkowski’s spacetime distance formula. Since the spacetime within each tiny elevator is also “unchanging and the same everywhere,” it therefore follows that we can use Minkowski’s distance formula inside each individual little elevator.

FIGURE 25

Hopefully, the analogy with the sphere is beginning to emerge. For “flat patch on the earth’s surface,” read “falling elevator in spacetime,” and for “curved surface of the earth,” read “curved spacetime.