An Introduction to General Relativity

Cartesian tensors: an invitation to indices

LOCAL DIFFERENTIAL GEOMETRY consists in the first instance of an amplification and refinement of tensorial methods. In particular, the use of an index notation is the key to a great conceptual and geometrical simplification. We begin therefore with a transcription of elementary vector algebra in three dimensions. The ideas will be familiar but the notation new. It will be seen how the index notation gives one insight into the character of relations that otherwise might seem obscure, and at the same time provides a powerful computational tool.

The standard Cartesian coordinates of 3-dimensional space with respect to a fixed origin will be denoted xi (i = 1,2,3) and we shall write A = Ai to indicate that the components of a vector A with respect to this coordinate system are Ai. The magnitude of A is given by A · A = AiAi. Here we use the Einstein summation convention, whereby in a given term of an expression if an index appears twice an automatic summation is performed: no index may appear more than twice in a given term, and any ‘free’ (i.e. non-repeated) index is understood to run over the whole range.