Warped Passages - Lisa Randall [51]
The Graceful Curves of the Universe
The equivalence principle says that the force of gravity is indistinguishable from constant acceleration. I’m glad you made it to this point, because I need to confess that I simplified, and the two aren’t entirely indistinguishable after all. How could they be? If gravity were equivalent to acceleration, it would not be possible for people in opposite hemispheres to simultaneously fall to Earth. After all, the Earth cannot accelerate in two directions at once. Gravitational pull in the different directions felt in America and China, for example, cannot possibly be accounted for by a single acceleration.
The resolution of this paradox is that the equivalence principle asserts only that gravity can be replaced by acceleration locally. At different places in space, the acceleration that would replace gravity according to the principle would generally be in different directions. The answer to our problem with Chinese/American relations is that American gravity is equivalent to an acceleration in a different direction from the acceleration that would reproduce Chinese gravity.
Figure 37. The “Einstein Cross” is formed when multiple images of a bright, distant quasar are formed by light bending in different directions as it passes by a massive foreground galaxy.
This critical insight led Einstein to a complete reformulation of the theory of gravity. He no longer saw gravity as a force that acts directly on an object. Instead, he described it as a distortion of the geometry of spacetime that reflects the different accelerations required to cancel gravity in different places. Spacetime is no longer a parenthetical background to an event—it is an active player. With Einstein’s theory of general relativity, the force of gravity is understood in terms of the curvature of spacetime, which in turn is determined by the matter and energy that are present. Let’s now consider the notion of the curvature of spacetime, on which Einstein’s revolutionary theory rests.
Curved Space and Curved Spacetime
A mathematical theory must be internally consistent but, unlike a scientific theory, it has no obligation to correspond to an external physical reality. True, mathematicians have often drawn inspiration from what they see in the world around them. Mathematical objects such as cubes and natural numbers do have real-world counterparts. But mathematicians extend their assumptions about these familiar concepts to objects whose physical reality is less certain, such as tesseracts (hypercubes in four-dimensional space) and quaternions (an exotic number system).
Euclid wrote his five fundamental postulates of geometry in the third century BC. From these assumptions a beautiful logical structure developed, one that you might have had a taste of in high school. But later mathematicians found themselves having trouble with the fifth postulate, the one known as the parallel postulate. This postulate states that, given a line and a point outside that line, there is one and only one line that can be drawn through the point that is parallel to the initial line.
For two millennia after Euclid formulated his postulates, mathematicians argued about whether this fifth postulate was actually independent or merely a logical consequence of the other four. Could there be a system of geometry for which all but the last postulate was true? If no such system of geometry existed, the fifth postulate would not be independent,