Significant page of a handwritten scientific manuscript by Albert Einstein, one page, 8.5 x 11, no date but circa 1943. A superb example of Einstein's creative process and working method, exploring the application of bivector fields to the tensor equations of General Relativity. Bivectorial mathematics deals with the possibility of operationally viewing a field as a composite product of two vectors. At the time of this manuscript's composition, Einstein was just beginning his efforts to generalize the relativistic theory of gravitation into a fuller unified system of equations integrating electromagnetic field theory. This project of unification and Unified Field Theory would preoccupy Einstein for the remainder of his life, and the present autograph is accordingly to be understood as one of the earliest extant specimens of Einstein's final approach to Unified Field Theory. Einstein is clearly moving in uncharted waters here, his grasp on the material tentative and the movements subject to reversal and revision—Einstein here striking through an entire sequence of meticulously articulated equations and redrafting sentences even as he is putting pen to paper.
Einstein published two articles on bivector fields in 1944: Bivector Fields I dealt with symmetric field components and Bivector Fields II dealt with antisymmetric components (defined with relation to the affine connection). Submitted simultaneously for publication in 1943, they are Einstein's only Unified Field Theory papers to appear during World War II. As Einstein himself explained in these papers, his prior attempts at Unified Field had failed because he attempted to change only a single basis for the theory (e.g. by adding a higher spatial dimensionality) whereas Einstein came to realize that 'a decisive modification of the fundamental concepts is unavoidable.' Departing therefore from a strict Riemannian metric, Einstein indicates that 'we have found new concepts and relations which from a logical viewpoint seem simple and natural enough to be of interest.' But Einstein cautioned, 'whether we have succeeded in the solution of this physical problem is still uncertain.'
Einstein's two articles on Bivector Fields were published in English in the Annals of Mathematics. The articles were originally written in German and translated by Einstein's assistant Bergmann. The Library of Congress contains the finished manuscripts of the two articles. Only a handful of leaves like the present representing early states of the article are known to have survived, and they are of the greatest rarity in private hands. In fine condition.
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