In Newton’s classical physics, space tells bodies how to move. A body given an initial shove will follow a straight line in space unless deflected by some other force. Einstein thought that this “one-way” influence was peculiar. If space acted on bodies, should not bodies act on space? The germ of Einstein’s later theories of space and time, his general theory of relativity, is that this circle is completed.

Central idea of the theory of general relativity: Spacetime tells bodies how to move, and bodies tell spacetime how to curve.
But what is a curved space or curved spacetime? If space is emptiness, or even a kind of nothingness, how could that curve? What would space bend into?

Physicists begin to answer these questions by insisting that only observable entities may play a role in physics. Since space is invisible and we see no lines of latitude and longitude etched across the night sky, the straight lines through space can only be discovered by following bodies around and observing their paths. Thus physicists define a straight line through space as the path traversed by a body moving without interference of any kind, that is, moving inertially.

Suppose, however, that two bodies are shoved along two adjacent, parallel paths in the same direction, but later bump into each other. Newton would say that some force must have pushed them together. If they were moving inertially without disturbance they would have followed two parallel paths, and since parallel lines never intersect, the bodies would never collide. Einstein disagreed, and suggested another possibility. Suppose that the bodies bump into each other because space itself is curved. Suppose that space is crisscrossed in all directions by lines that inertial bodies follow, but that the fabric of space can be bent, twisted and curved. Then bodies coasting along “parallel” paths may indeed bump into each other if the paths converge.

It does sound strange to say that “straight lines curve” or that “parallel lines intersect”, but this is just a superficial oddity of physicist’s language. They adhere to the definition above even when parallel, straight lines intersect, and prefer simply to say that they have enlarged our notion of what “straight” and “parallel” mean.

Einstein’s concept of curved space is often illustrated by the rubber sheet analogy. Strictly speaking, it is spacetime that is curved, but this analogy helps us picture a curved space. Imagine a large sheet of supple rubber stretched out and secured along its edges, perhaps like the surface of a large drum in an orchestra. A heavy lead weight placed in the centre of the sheet will sink down into a smooth, circular well. This gives us a picture of what it means to say that “mass causes space to curve”. More interestingly, suppose that children begin shooting marbles back and forth along the rubber sheet. Suppose a marble is shot straight but not directly at the lead weight. Instead of following a “straight line” its path will bend towards the lead weight. Either it continues on away from the weight after this deflection, or it will spiral down into the well. There is no significant attraction between the lead weight and the marble. The lead weight affects the marble by distorting the spatial surface that it travels along.

 This nicely shows how Newton’s gravitational forces are replaced in Einstein’s theories by curved spaces. For Newton, bodies move towards each other when they exert attractive forces. For Einstein, they influence each other indirectly by affecting the space between them. Each body curves the space in its environment; when other bodies coast along straight lines they veer toward the centre of curvature. Some say that, in general relativity, gravitational forces are geometrized away.

Suppose further that the lead weight on the rubber sheet was moved rapidly up and down. If the sheet were large enough and flexible enough, ripples of upward and downward motion would spread out across the sheet. Perhaps they would look like the circular waves in the surface of a pond caused by a falling stone. Similarly, Einstein’s theory predicts that moving masses will cause travelling distortions in the fabric of space: gravity waves.

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