Curvature

When considering the beings living on a sphere it is easy for us to differentiate between the sphere and some plane surface: we actually see the sphere being curved. But when it comes to us, and our curved space, we cannot see it since this would entail our standing outside space and looking down on it. Can we then determine whether space is curved by doing measurements inside it?

To see that this can be done let's go back to the beings on the sphere. Suppose they make a triangle by the following procedure: they go form the equator to the north pole along a great circle (or meridian) of the sphere, at the north pole they turn 90^o to the right and go down another great circle until they get to the equator, then they make another 90^o turn to the right until they get to the starting point. They find that all three lines make 90^o angles with each other, so that the sum of the angles of this triangle is 270^o, knowing that angles in all flat triangles always add up to 180^o they conclude that the world they live on is not a flat one. Pythagoras' theorem only holds on flat surfaces
 
 

Figure 7.18: A path followed by a determined being living on the surface of a sphere; each turn is at right angles to the previous direction, the sum of the angles in this triangle is then 270^o indicating that the surface in which the bug lives is not flat. 

We can do the same thing: by measuring very carefully angles and distances we can determine whether a certain region of space is curved or not. In general the curvature is very slight and so the distances we need to cover to observe it are quite impractical (several light years), still there are some special cases where the curvature of space is observed: if space were flat light would travel in straight lines, but we observe that light does no such thing in regions where the gravitational forces are large; I will discuss this further when we get to the tests of the General Theory of Relativity in the following sections.

The curvature of space is real and is generated by the mass of the bodies in it. Correspondingly the curvature of space determines the trajectories of all bodies moving in it. The Einstein equations are the mathematical embodiment of this idea. Their solutions predict, given the initial positions and velocities of all bodies, their future relative positions and velocities. In the limit where the energies are not too large and when the velocities are significantly below c the predictions of Einstein's equations are indistinguishable from those obtained using Newton's theory. At large speeds and/or energies significant deviations occur, and Einstein's theory, not Newton's, describes the observations.