7 2 0 D O C . 4 5 9 O N K A L U Z A ’ S T H E O R Y : P A R T 1 Published in Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1927): 23–25. Presented 20 January 1927, published 14 March 1927. The first two pages of the manuscript are preserved as an ADft [1 061]. [1]Einstein himself wrote several papers on affine field theory in which he attempted to think through to its end Eddington’s approach. He especially tried to find unified field equations within this approach. His last paper in this direction was Einstein 1925t (Doc. 17). [2]Einstein had argued early on that the reduction of the group of five-dimensional coordinate transformations via the cylinder condition was a severe shortcoming of Kaluza’s approach (see Doc. 365, note 5, for details). [3]This sharpened version of Kaluza’s cylinder condition was first introduced in Klein, O. 1926. Klein had sent the paper to Einstein on 29 August 1926 (Doc. 363). [4]This “sharpened cylinder condition” is not introduced by Kaluza 1921, which explicitly dis- cusses the consequences of a new scalar field that emerges if one merely demands the normal, rather than the sharpened, cylinder condition to be fulfilled. This is acknowledged by Einstein in the foot- note at the end of the paper. Klein, O. 1926, like Einstein, demands the sharpened cylinder condition to be fulfilled. Einstein here gives an explicitly coordinate-independent formulation of the cylinder condition. In modern terminology, he shows that the normal cylinder condition corresponds to the existence of a Killing vector in the direction of the fifth coordinate, while the sharpened cylinder con- dition corresponds to the further demand that the Killing vector is normalized to 1. [5]Einstein identifies the same invariants as does Klein, O. 1926, p. 896. [6]Einstein here points out that the group of five-dimensional coordinate transformations restricted by the sharpened cylinder condition is isomorphic to the product group of four-dimensional coordi- nate transformations and gauge transformations of the electromagnetic four-potential i.e., the symmetry group is the same as that of the normal four-dimensional Einstein-Maxwell equations. Klein, O. 1926 had made the same point. [7]The last sentence of this paragraph replaces the following sentences from the manuscript: ‘‘Dass er damit das Richtige getroffen hat, bezweifle ich. Abgesehen davon, dass sich diese Bedingung mit der ,,verschärften Zylinderbedingung” nicht vertragen will, erscheint es mir unnatürlich, die Kovari- anzbedingung im R5 mit der Zylinderbedingung zu kombinieren. Allgemein kovariant kann man dies so ausdrücken: Wenn man in der ,,Zylinderbedingung” neben den noch einen Vektor der ein- führen muss, ist es nicht berechtigt, zu verlangen, dass in die ,,Feldgleichungen” die nicht einge- hen. An die Stelle dieses Postulates von Kaluzas Theorie möchte ich eine weniger kühne, aber—wie mir scheint—natürlichere Betrachtung setzen.” Einstein had voiced similar criticism regarding the tension between the cylinder condition and the use of field equations that are invariant under five- dimensional coordinate transformations before see Doc. 365, note 5. [8]In a letter to Kaluza of 5 May 1919 (Vol. 9, Doc. 35), Einstein had criticized the “non-covariant” character of the cylinder condition. However, this is not strictly in contrast with his invariant charac- terization of the condition here: even in this formulation, the cylinder condition is not covariant with respect to five-dimensional coordinate transformations, yet it is invariant with respect to the restricted group of transformations that Einstein introduces on the following page. [9]Klein, O. 1926, eq. (2), identifies the same reduced symmetry group. [10]In the draft, the following text is deleted after eq. (5): “Hierauf folgt mit Rücksicht auf (3) also . . . . (6) Aus der zweiten der Gleichungen (5) folgt ferner wegen der Konstanz von mit Rücksicht auf (4) γμν ξα ξα dτ2 dτ2 2dϕdψ dψ2 + + = dϕ dϕ dψ += dτ2 dϕ2 – dτ2 dϕ2 –= γ00