We have pointed out in §10.2 that in general relativity we have to deal with the curvature of spacetime and that tensors provide a natural mathematical language for describing such curvature. We now plan to give an introduction to tensor analysis and then an introduction to general relativity at a technical level. It will be useful for readers to be familiar with the qualitative concepts introduced in §10.2 before studying this chapter.
Since general relativity is a challenging subject, it is helpful to clearly distinguish the purely mathematical topics from the physical concepts of general relativity. So, when we develop tensor analysis in the next section, we shall develop it as a purely mathematical subject without bringing in general relativistic concerns at all. The two-dimensional metrics (10.7), (10.8) and (10.9) introduced in §10.2 will be used as illustrative examples repeatedly to clarify various points. When various formulae of tensor analysis are applied to metrics of dimensions higher than two, the algebra can be horrendous. It is, therefore, advisable to develop a familiarity with tensors by first applying the important results to two-dimensional surfaces.
After introducing the basics of tensor analysis in the next section, we shall start developing the basic concepts of general relativity from §12.3.