Einstein, Albert

in German, headed "Fünf dim. Raum Koord. x₁ x₂ x₃ x₄ x₅", a working manuscript written in blue ink, with revisions, deletions and alterations (including the deletion of part of the fourth and final section), containing equations in differential geometry and dimensional calculations, including a version of the geodesic equations relating to how freely-falling matter moves through space-time

1 page, folio (27.7 x 22.5cm), on fine thin paper ("Pendrift Bond"), no place or date, [1938, or earlier], remains of mount on verso, light browning towards edges



...Koordinatenwahl. a) Auf den Geodäten soll x₁...x₄ konstant sein.
b) Auf jeder Geodäte soll sich x₀ so ändern, dass allen dx₀ auf einer Geodäten proportionale geodätische [?] Längen zukommen und dass für eine Periode [delta]x_o = 1 sei.
Dann sind alle Feldgrössen in x₀ periödisch von der Periode 1. Ferner ist [gamma]₀₀ von x₁...x₄ abhängig.
Die "x₀ Linien" sollen gemäss 3) geodätische Linien sein, also die Gleichungen erfüllen...[geodesic equation]

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This is an important working manuscript apparently representing Einstein's notes for a paper entitled "On a Generalization of Kaluza's Theory of Electricity" which he wrote jointly with Peter Bergmann, and which was published in the Annals of Mathematics, vol. 39, no. 3, July 1938, pp.683-701. Although the manuscript differs in many details from the published article (written in English), there are enough correspondences in wording, and also with respect to the equations to the section of the article headed 'The Space Structure' to suggest very strongly a link between it and the 1938 paper. In sum, the manuscript details part of Einstein's attempt to construct a unified theory of electromagnetism, gravitation and quantum mechanics based on a curved five dimensional spacetime with five spacetime coordinates x₁, x₂, x₃, x₄, x₀ and four spatial coordinates, one of which, x₀, is periodic. Through every point it is assumed that there passes a closed geodesic given by x₁, x₂, x₃, x₄ constant. This particular approach is sometimes referred to as "Projective Relativity" and is type of unified theory pioneered by T. Kaluza and later by O. Klein in the 1920s. Kaluza and Klein's ideas play a key part of modern Super String theory and are currently being extensively pursued by theoretical physicists.

After obtaining his doctorate at the German University in Prague in 1936 under the direction of Philipp Frank, Peter Bergmann (1915-2002) collaborated with Einstein, as his research assistant, at the Institute for Advanced Study between 1936 and 1941. In 1942, Bergmann published a textbook on General Relativity, Introduction to the Theory of Relativity, which contained a foreword by Einstein.

We understand that Einstein presented this manuscript to the daughter of Luther P. Eisenhart, Chairman of the Mathematics Department at Princeton University.

We gratefully acknowledge the kind assistance of Professor Gary Gibbons in our cataloguing of this lot.