Einstein's general theory of relativity gives a mathematical description of space, time and gravitation which is extraordinarily concise, subtle and accurate. It has, however, the appearance of being concise only to those who are already familiar with the mathematical formalism of Riemannian geometry. To someone who is not familiar with that body of mathematical theory – a theory which, though remarkably elegant, is undoubtedly sophisticated, and usually becomes extremely complicated in detailed application – Einstein's General Relativity can seem inaccessible and bewildering in its elaborate structure. But the complication and sophistication lie only in the details of the formalism. Once that mathematics has been mastered, the precise formulation of Einstein's physical theory is, indeed, extremely compact and natural. Although a little of this formalism will be needed here, it will be given in a compact form only that should be reasonably accessible.
The mathematical theory of Riemannian geometry applies to smooth spaces of any (positive whole) number N of dimensions. Such a space M is referred to as an N-manifold, and to be a Riemannian manifold it must be assigned a metric, frequently denoted by g, which assigns a notion of ‘length’ to any smooth curve in M connecting any two points a, b. (See Figure 3.1.) For a strictly Riemannian manifold, this length function is what is called positive definite which means that the length of any such curve is a positive number, except in the degenerate situation when a = b and the curve shrinks to a point, for which the length would be zero.