• Choose a category
• All
• Classic-Fiction

By Root 913 0

” In fact, physicists often refer to Minkowski spacetime as “flat spacetime” for this very reason. Minkowski spacetime plays the role of flat Euclidean space in the analogy. In this book, we’ve reserved the use of the word “flat” for Euclidean geometry, and the minus sign in the Minkowskian version of Pythagoras’ theorem motivated us to use the term “curved.” Sometimes the use of language is not as straightforward as we might like it to be! So the assembly of little elevators is to spacetime as the assembly of little patches is to the sphere. In each little elevator, gravity has been banished, but we could imagine sewing all the little Minkowski patches together to form a curved spacetime in exactly the same way that we constructed the curved surface of the earth from flat Euclidean patches. If there were no gravity, then we could get by with one big elevator within which the geometry is that of Minkowski. So what we have just learned is that if there is gravity around, we can make it go away but only at the expense of making spacetime curved. What a remarkable conclusion.

Turn this around, and it looks like we have discovered that the force of gravity is actually nothing more than a signal to us that spacetime itself is curved. Is this really true, and what causes the curving? Since gravity is found in the vicinity of matter, we might conclude that spacetime is warped in the vicinity of matter and, since E = mc2, energy. The amount of warping is something we have so far said nothing at all about. And we don’t intend to say very much because it is, to use a well-worn physics phrase, nontrivial. In 1915, Einstein wrote down an equation that was able to quantify exactly how much warping there should be in the presence of matter and energy. His equation improves upon Newton’s age-old law of gravity in that it is automatically in accord with the special theory of relativity (Newton’s law is not). Of course, it gives very similar results to Newton’s theory for most cases we encounter in everyday life, but it does expose Newton’s theory as an approximation. To illustrate the different ways of thinking about gravity, let’s see how Newton and Einstein would describe the way in which the earth orbits the sun. Newton would say something like this: “The earth is pulled toward the sun by the force of gravity, and that pull prevents it from flying off into space, constraining it instead to move in a big circle.”13 It is similar to whirling a ball on a string around your head. The ball will follow a circular path because the tension in the string prevents it from doing otherwise. If you cut the string, the ball would head off in a straight line. Likewise, if you suddenly turned off the sun’s gravity, Newton would say that the earth would then head off into outer space in a straight line. Einstein’s description is quite different and goes like this: “The sun is a massive object and as such it distorts spacetime in its vicinity. The earth is moving freely through spacetime but the warping of spacetime makes the earth go in circles.”

To see how an apparent force might be nothing more than a consequence of geometry, we can consider two friends walking on the earth’s surface. They are told to begin at the equator and to walk due north parallel to each other in perfect straight lines, which they dutifully do. After a while, they will notice that they are coming closer together and, if they carry on walking for long enough, they will bump into each other at the North Pole. Having established that neither of them cheated and wandered off course, they may well conclude that a force acted between them that pulled them together as they walked northward. This is one way to think about things, but there is of course another explanation: The surface of the earth is curved. The earth is doing much the same thing as it moves around the sun.

To get a better feel for what we are talking about, let’s return to one of our intrepid walkers on the surface of the earth. As before, he is told always to walk in a straight line. Locally, that is an instruction