Intrinsic geometry and curvature

In the previous chapters of this book we have frequently used the concept ‘Riemannian space’ or ‘curved space’. Except in Section 14.4 on the geodesic deviation, it has not yet played any rôle whether we were dealing only with a Minkowski space with complicated curvilinear coordinates or with a genuine curved space. We shall now turn to the question of how to obtain a measure for the deviation of the space from a Minkowski space.

If one uses the word ‘curvature’ for this deviation, one most often has in mind the picture of a two-dimensional surface in a three-dimensional space; that is, one judges the properties of a two-dimensional space (the surface) from the standpoint of a flat space of higher dimensionality. This way of looking at things is certainly possible mathematically for a four-dimensional Riemannian space as well – one could regard it as a hypersurface in a ten-dimensional flat space. But this higher-dimensional space has no physical meaning and is no more easy to grasp or comprehend than the four-dimensional Riemannian space. Rather, we shall describe the properties of our space-time by four-dimensional concepts alone – we shall study ‘intrinsic geometry’. In the picture of the two-dimensional surface we must therefore behave like two-dimensional beings, for whom the third dimension is inaccessible both practically and theoretically, and who can base assertions about the geometry of their surface through measurements on the surface alone.