D O C U M E N T 5 2 1 U N I F I E D F I E L D T H E O R Y 4 4 7 521. “Unified Field Theory” [Berlin, 10 May 1929][1] Contents.… (see (1a))[2] For roughly the past year, I have been pursuing a new path toward a unified the- ory of gravitation and electromagnetism.[3] Since these investigations have now ar- rived at a certain preliminary conclusion, in that the derivation of the field equations has been successful in a fairly natural manner, the theory is presented here from the beginning, in such a way that anyone familiar with the general theory of relativity can readily become acquainted with the new theory. This is all the more important because certain concepts and also relations have been abandoned in the course of the investigations, so that a study of the articles that have already appeared would necessitate a needless expenditure of time. In order to arrive at the most complete description, I refer here neither to my earlier works on this subject, nor to the numerous articles by those mathematicians who have already anticipated the formal basis of the theory for the most part. A summary of that literature can be found in Weitzenböck’s article.[4] § 1. Riemannian Metric and Distant Parallelism The fundamental idea on which the whole theory rests is the following.The theory is based on the following consideration: the present general theory of rela- tivity rests upon the formal hypothesis that the four-dimensional space-time con- tinuum has a structure,[5] § 1. The Structure of Space The (n-dimensional) continuum is presumed to have a Riemannian metric. This means that for every line element or every vector there is a measurable number, its magnitude. For every point P on the continuum one can calculate the magnitude of a line element or vector attached to P with respect to a local coordinate system, making use of Pythagoras’s rule: . . . (1) (a summation is to be carried out over the index s), where the are the local com- ponents of the vector and A is its magnitude. Such local components are always de- noted in the following by Roman-letter indices, in contrast to the “usual” compo- nents, which are defined with respect to the general Gaussian coordinate system in the well-known manner. The latter always have Greek-letter indices. [p. 1] [p. 3] A2 A2 s = A s