The Noneuclidean Nature Of Our Universe

The previous section was not only a fun digression into a particularly simple model of a curved geometry; it also contains some of the key features of Einstein's eventual unification of gravity with special relativity, and identification of gravity as a kind of curvature. This was the celebrated general theory of relativity. One of the chapters in Einstein's 1920 book Relativity: The Special and General Theory is titled "The Space-Time Continuum of the General Theory of Relativity Is not a Euclidean Continuum." The general theory of relativity holds that space and time are inextricably linked, both parts of a single four-dimensional geometrical space, which can be (indeed, must be, in the presence of matter) curved and whose curvature gives rise to the influence we commonly know and experience as gravity.

For weak gravitational fields or empty space, the general theory of relativity reduces to the special relativity just discussed.

This is because the geometry of the four-dimensional spacetime is set up in such a way that, locally near each space-time point (or "event") there exists a coordinate system (x, t) similar to the in-ertial frames we have been using. Also in this local neighborhood of a space-time point, the metric or distance function, can be put into the form that we found above for the invariant relativistic interval, thus completing the statement that locally, or in situations where the gravitational field can be regarded as weak, general relativity reduces to special relativity.

Using general relativity, Einstein predicted several phenomena that were subsequently found to be accurate by experiments— phenomena that cannot be explained either using Newton's theory or by special relativity. We now describe an example of one such phenomenon. As noted in Chapter 3 (Figure 3.3), the perihelion of a planet's orbit is the unique point on its orbital trajectory when it is closest to the Sun. Let us fix our attention on a single planet, say, Mercury. If you draw a ray from the center of the Sun to Mercury's perihelion, it turns out that even in a coordinate system where the center of the Sun is fixed, the ray from the center of the Sun to the perihelion of Mercury changes ever so slightly as a function of time. The precise nature of this change is called precession; roughly speaking, this means that the tip of the ray we have drawn follows a circular path, although the length of the ray does not change. This phenomenon, called the perihelion precession of Mercury, was first explained properly by Einstein using general relativity.

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