The Affine Connexion in Physical Field Theories

Abstract

1. The loss of connexion by general invariance. The essential physical entities are mathematically described as invariants, vectors, tensors. That comes from the isotropy and homogeneity of space in Euclidean geometry—or from the pseudo-isotropy of space-time, in the Restricted Theory of Relativity. When the latter is replaced by the idea of general invariance, that is, by regarding the three space-co-ordinates and the time only as continuous labels of the world points, which labels may equivalently be replaced by any quadruplet of continuous functions of themselves, the notion of vector, or tensor subsists, but any such entity is now necessarily bound to a given world-point, it is 'a tensor at P'. For example, the displacement-vector, dx_k/, leading from a world-point P with co-ordinates x_k (that is, x₁ x₂, x₃, x₄) to a neighbouring point Q with co-ordinates x_k + dx_k, is the prototype of a contravariant vector A^k at P. If you change the labels (that is to say, if you execute a general transformation of the frame) the A^k transform by definition as the dx_k thus*: