, the square of which is a homogeneous second order function of the coordinate differentials. It follows from this that (apart from certain conditions of reality) Euclidian geometry is valid in any infinitely small region. Hence to every line element (or vector) at a point
is assigned a parallel and equal line element (or vector) through any given infinitesimally adjacent point
' (affine correlation). Riemannian metric determines an affine correlation. Conversely, however, when an affine correlation (law of infinitesimal parallel displacement) is mathematically given, generally no Riemannian metric determination exists from which it can be derived.
The most important concept of Riemannian geometry, "space curvature", on which the gravitational equations are also based, is based exclusively on the "affine correlation". If one is given in a continuum, without first proceeding from a metric, it constitutes a generalization of Riemannian geometry but which still retains the most important derived parameters. By seeking the simplest differential equations which can be obeyed by an affine correlation there is reason to hope that a generalization of the gravitation equations will be found which includes the laws of the electromagnetic field. This hope has in fact been fulfilled although I do not know whether the formal connection so derived can really be regarded as an enrichment of physics as long as it does not yield any new physical connections. In particular a field theory can, to my mind, only be satisfactory when it permits the elementary electrical bodies to be represented as solutions free from singularities.
Moreover it should not be forgotten that a theory relating to the elementary electrical structures is inseparable from the quantum theory problems. So far also relativity theory has proved ineffectual in relation to this most profound physical problem of the present time. Should the form of the general equations some day, by the solution of the quantum problem, undergo a change however profound, even if there is a complete change in the parameters by means of which we represent the elementary process, the relativity principle will not be relinquished and the laws previously derived therefrom will at least retain their significance as limiting laws.
[1]The Lecture was not delivered on the occasion of the Nobel Prize award, and did not, therefore, concern the discovery of the photoelectric effect.
[2]As regards the red shift, the agreement with experience is not yet completely assured, however.