Our space is curved
Now comes the main question. Is it true? That is, is the actual physical three-dimensional space we live in curved? Once we have enough imagination to realize the possibility that space might be curved, the human mind naturally gets curious about whether the real world is curved or not. People have made direct geometrical measurements to try to find out, and haven’t found any deviations. On the other hand, by arguments about gravitation, Einstein discovered that space is curved, and we’d like to tell you what Einstein’s law is for the amount of curvature, and also tell you a little bit about how he found out about it. Einstein said that space is curved and that matter is the source of the curvature. (Matter is also the source of gravitation, so gravity is related to the curvature—but that will come later in the chapter.)  Let us suppose, to make things a little easier, that the matter is distributed continuously with some density, which may vary, however, as much as you want from place to place.3 The rule that Einstein gave for the curvature is the following: If there is a region of space with matter in it and we take a sphere small enough that the density $\rho$ of matter inside it is effectively constant, then the radius excess for the sphere is proportional to the mass inside the sphere. Using the definition of excess radius, we have \begin{equation} \label{Eq:II:42:3} \text{Radius excess}= r_{\text{meas}}-\sqrt{\frac{A}{4\pi}} =\frac{G}{3c^2}\cdot M. \end{equation}  Here, $G$ is the gravitational constant (of Newton’s theory), $c$ is the velocity of light, and $M=4\pi\rho r^3/3$ is the mass of the matter inside the sphere. This is Einstein’s law for the mean curvature of space.  Suppose we take the earth as an example and forget that the density varies from point to point—so we won’t have to do any integrals. Suppose we were to measure the surface of the earth very carefully, and then dig a hole to the center and measure the radius. From the surface area we could calculate the predicted radius we would get from setting the area equal to $4\pi r^2$. When we compared the predicted radius with the actual radius, we would find that the actual radius exceeded the predicted radius by the amount given in Eq. (42.3). The constant $G/3c^2$ is about $2.5\times10^{-29}$ cm per gram, so for each gram of material the measured radius is off by $2.5\times10^{-29}$ cm. Putting in the mass of the earth, which is about $6\times10^{27}$ grams, it turns out that the earth has $1.5$ millimeters more radius than it should have for its surface area.4 Doing the same calculation for the sun, you find that the sun’s radius is one-half a kilometer too long. You should note that the law says that the average curvature above the surface area of the earth is zero. But that does not mean that all the components of the curvature are zero. There may still be—and, in fact, there is—some curvature above the earth. For a circle in a plane there will be an excess radius of one sign for some orientations and of the opposite sign for other orientations. It just turns out that the average over a sphere is zero when there is no mass inside it. Incidentally, it turns out that there is a relation between the various components of the curvature and the variation of the average curvature from place to place. So if you know the average curvature everywhere, you can figure out the details of the curvature components at each place. The average curvature inside the earth varies with altitude, and this means that some curvature components are nonzero both inside the earth and outside. It is that curvature that we see as a gravitational force. Suppose we have a bug on a plane, and suppose that the “plane” has little pimples in the surface. Wherever there is a pimple the bug would conclude that his space had little local regions of curvature. We have the same thing in three dimensions. Wherever there is a lump of matter, our three-dimensional space has a local curvature—a kind of three-dimensional pimple. If we make a lot of bumps on a plane there might be an overall curvature besides all the pimples—the surface might become like a ball. It would be interesting to know whether our space has a net average curvature as well as the local pimples due to the lumps of matter like the earth and the sun. The astrophysicists have been trying to answer that question by making measurements of galaxies at very large distances. For example, if the number of galaxies we see in a spherical shell at a large distance is different from what we would expect from our knowledge of the radius of the shell, we would have a measure of the excess radius of a tremendously large sphere. From such measurements it is hoped to find out whether our whole universe is flat on the average, or round—whether it is “closed,” like a sphere, or “open” like a plane. You may have heard about the debates that are going on about this subject. There are debates because the astronomical measurements are still completely inconclusive; the experimental data are not precise enough to give a definite answer. Unfortunately, we don’t have the slightest idea about the overall curvature of our universe on a large scale. 