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Unit 8

Geometries Beyond Euclid

8.7 General Relativity


We saw earlier that an ant on a globe can find the curvature of his world by drawing circles of differing sizes and seeing how the value of pi changes. The globe is a surface of constant positive curvature, which simply means that no matter where our ant decides to begin drawing circles, he'll find that pi changes in the same way.


However, what if the surface that the ant draws on were not so uniform? Let's say that, just like the real surface of Earth, the ant's surface has hills and valleys. Dimensionally, this would mean that some parts of his world are more curved than others. Riemann studied surfaces like this and came up with a generalized way to deal with such surfaces and manifolds—ones with so-called, non-constant curvature.

While the details of this method are beyond our scope, there are a couple of key features worth noting. The first condition is that all the hills and valleys must be smooth—in other words, there is a way to get from point to point without encountering any cliffs or walls. As long as this condition is met, Riemann said that it is possible to use the local geometry to find the lengths of curves. So, our ant would basically draw circles, as it did before, but it would see that pi changes in different ways, depending on the specific placements of the circles.

Pi Changes

Using this method, the ant can measure the intrinsic curvature of the different regions of its world. We find this idea of a non-constantly-curved surface easy to visualize because we have a vantage point from the third dimension.

There is no reason, however, to think that only 2-D manifolds can be curved. In fact, it would be surprising if that were true, because we have seen that math concepts often generalize to higher dimensions. What would curvature in a 3-D manifold look like? This question puts us in a bit of a bind, for, just like the ant, we have no higher-dimensional viewpoint from which we can observe this curved space. We are stuck in it! Also, we must consider that our space is not of uniform curvature. That is, the ant's surface can have mountains and valleys, so our space might have the 3-D equivalent. Thus, there might be pockets of our space that are more curved than others, and there might be places that are more or less flat.

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Einstein shed light on these questions with his General Theory of Relativity, which he developed in 1915-1916. He used the notion that some areas of a surface are more curved than others. All that is required is that, if we look closely enough at the surface, it appears to be flat.

Locally Flat

Likewise, if you look at a planar curve closely enough, it will appear to be a straight line. A straight line is, of course, the geodesic of a Euclidean plane. So, there are regions, however small, of any line that will appear as a geodesic of the flat plane. A geodesic in a 3-D manifold will also be a line. If that manifold has curvature, then the geodesic line will be curved also. Nonetheless, just as with the planar curve, we can look at this curve in our 3-D manifold closely enough for it to seem straight.


What would a geodesic in spacetime look like? Following the definition, it would be the minimal-length connection between one time-and-place and another time-and-place. Another way to think about a minimal-length connection is to think of the path of least resistance. A geodesic can then be considered to be the path that requires the least amount of energy. If we place an object at an arbitrary point in spacetime, whatever it does naturally can be considered to be the geodesic of that spacetime. If there happens to be a massive object, such as a planet, nearby, we would expect our test object to "fall" toward the planet. This suggests that the state of free fall is a minimal-length connection "in action" in spacetime. In other words, falling is like following a geodesic.

Einstein noticed that an object in free fall "feels" no force of gravity. (Supposedly, this was after interviewing a painter who had fallen off of a house.) This is analogous to an ant looking very closely at a curved surface and seeing it to be flat—it doesn't see the curvature. Likewise, if one thinks of gravity as the curvature of our 4-D manifold of spacetime, then being in free fall, in which the effects of gravity are not noticed, is like looking closely at the curved surface. In other words, it is a perspective from which geodesics appear to be flat (straight lines). Free fall is just a straight-line geodesic through spacetime.

Using reasoning such as this as a starting point, Einstein re-interpreted gravity to be a result of the geometry of a curved spacetime. He said that it is not a force in the Newtonian sense; rather, it is the effect of living in a curved manifold. So, in other words, whereas the ant must draw circles to experience the curvature of his world, we experience the curvature of our world though gravity.

This interpretation of gravity has been experimentally verified in numerous ways. The first such verification occurred during a solar eclipse in 1919. Arthur Eddington found that certain stars, which, according to calculations, were behind the sun at the time, were visible during the eclipse.


The only way to interpret this phenomenon is to say that the light of the hidden stars "bent" around the sun, like so:


This was an early verification that light, which should normally travel in straight lines, actually travels in a curved path in the presence of a massive object. (Light paths are also geodesics of spacetime; we cannot conceive of something that would find a more efficient path from point A to point B.) This implied that the geodesics in the vicinity of the sun are curved, the cause being the mass of the sun.

Since this initial verification, other experiments also have shown the predictions of general relativity to be accurate. Einstein's theory that massive objects cause the spacetime in their vicinity to warp and that we experience this effect as gravity was a breakthrough in our understanding of physics. It was also a beautiful example of a mathematical idea that at first had little real-world application (i.e., the Riemannian geometry of non-constant curvature) turning out to be at the heart of one of the most fundamental phenomena in our human experience, gravity.

The General Theory of Relativity describes how mass causes the geometry of spacetime to curve locally. One can extend this thinking from considering the masses and motions of planets around a star to stars around a galaxy, galaxies around each other, clusters of galaxies around other clusters, and eventually to the large-scale curvature of the universe itself. All the mass in the universe must surely curve spacetime into some shape, and most probably that shape exhibits non-constant curvature.

We know how mass causes local curvature of spacetime, but would spacetime be flat were the mass not present? If so, would all of spacetime be flat without mass, or would it just appear to be flat because we are like the ant looking too closely at his surface to see any curvature? One of the most intriguing questions in both mathematics and physics is the question of the underlying geometry of reality.

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Next: 8.8 Geometrization Conjecture


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