[Physics FAQ] - [Copyright]
By Don Koks, 2021.
Einstein told us that the curvature of spacetime tells matter how to move. You can see that very graphically from the following demonstration. Find a balloon or a ball that you can draw on, and a felt-tipped pen; then draw what is detailed below. I first saw this presentation given by the physicist Bill Unruh in a talk he gave many years ago.
The ball or balloon will represent a curved spacetime. Don't worry that it is a closed surface, since we are not going to use its entire surface: we are not discussing a closed spacetime here! It's just a convenient curved surface that you can draw on. Because it's a two-dimensional surface, and we require one of those dimensions to be time, we are restricted to discussing only one dimension of space. Call that dimension up-down.
Draw two axes on the ball. The up-down axis is along a meridian, and the time axis is along the equator. Now draw two "small circles" (as opposed to "great circles") around the ball, each at a fixed latitude on the ball: say, plus and minus 40°. These circles represent the world lines of two men standing on Earth's surface at opposite points on our one-dimensional Earth. (Remember: the ball is not Earth; the ball is a piece of curved spacetime.) The ball's meridian, the up-down direction, between the small circles represents Earth's body in the up-down direction. The ball's equator is the world line of Earth's centre, and each direction away from that equator along a meridian represents the direction of up on each side of Earth.
Look at each small circle: it is not a geodesic because it curves away from being a great circle. This curving away is caused by the supporting force of the ground on each man. General relativity postulates that if a hole were suddenly to open up in the ground under one of the men so that he was no longer supported by Earth's body and was thus free to move, his world line would become a geodesic. Draw that, and you'll see that his world line heads toward the ball's equator, meaning downwards. That is, he falls down!
Now go back to the man at rest on Earth's surface, with his world line following a small circle of constant latitude on the ball: say, the +40° circle. Remember, up-down is along a meridian on the ball, and the time axis is along the ball's equator. Have the man jump up, so that his world line departs a little bit from the ball's constant-latitude small circle. What happens? The ground no longer provides a support force, and again the man must now follow a geodesic on the ball. Draw that (a real man making a real jump will only make a small deviation away from the small circle), and you'll see that this geodesic departs from the small circle for a few centimetres, and then meets up with it again. That is, whereas the geodesic takes the shortest path on the ball between the events of the man leaving Earth's surface and his landing again, the small circle takes a longer path between these two events. The result is that from the viewpoint of the ground, the man goes up into the air, comes to a stop, then falls back down. It is just what we know really happens when we jump in the air.
Newton said that when we jump up and fall back down, we had a complicated motion caused by a gravity force, while the ground did nothing. Einstein took a different view. He said "The man who jumps up has as simple a motion as possible: he just sails through spacetime on a geodesic. It's the ground beneath him that has the complicated motion, because its world line [the small circle] is curving in spacetime".
Remember: don't follow any of these curves right around the ball, because that would be doing something with a closed spacetime, and we're not talking about a closed spacetime. A closed spacetime would imply that at some moment in the future, we'd return to a moment in the distant past. And there is no reason to think that our universe is like that.
[Physics FAQ] - [Copyright]