I N T R O D U C T I O N T O V O L U M E 1 5 l i i i tioning with his sons (Doc. 319), but asked to see Klein’s paper two and a half months after he had been first informed of it (Doc. 356). Most likely at Ehrenfest’s suggestion, Klein wrote to Einstein directly in late August 1926 (Doc. 363). He sent Einstein not only the manuscript of his paper (Klein, O. 1926), but also pro- posed how he wanted to develop the theory further. In particular, he explained his idea of assuming a periodicity of the fifth coordinate, averaging over it, and recon- ceptualizing Schrödinger’s wave function as a component of the five-dimensional metric by effectively dropping the sharpened cylinder condition and using the same version of the condition that Kaluza had used seven years earlier. A few months later, Einstein wrote two papers on Kaluza’s theory, Einstein 1927i (Doc. 459) and Einstein 1927j (Doc. 480). In the first paper, Einstein focuses on his old concern regarding the meaning and consequences of the cylinder condi- tion. First, he gives a coordinate-independent formulation of both the original cyl- inder condition (used by Kaluza) and of the sharpened cylinder condition (initially used by Klein). The latter is both more, and less, restrictive than Kaluza’s condi- tion: it is less restrictive in that it only demands that the derivatives of the metric with respect to the fifth coordinate, rather than of all physical quantities, vanish but it is also more restrictive in that it demands that the component be a con- stant, rather than a variable, as permitted by Kaluza. Second, Einstein identifies the invariants of the reduced group of five-dimensional coordinate transformations re- sulting from the sharpened cylinder condition, namely, a four-dimensional sym- metric tensor and a four-dimensional antisymmetric tensor, both of second rank. And third, he argues that thereby “Kaluza’s idea offers a deeper understanding of the fact that besides the symmetric metric tensor ( ) only the antisymmetric ten- sor ( ) of the electromagnetic field (which is derivable from a potential) plays a role” (Doc. 459). In the second paper, Einstein examines the relation between general relativity and Kaluza-Klein theory. He investigates how the five-dimensional metric should be projected into four dimensions so as to recover the original Einstein-Maxwell equations exactly, and notes that Kaluza had only managed to derive them approx- imately. He concludes by stating that in order to do this “in the usual form, one must presume the 0-direction to be spacelike” (Doc. 480). Both Kaluza and Klein had assumed the fifth dimension to be spacelike Einstein believed he had found a con- sistency argument in its favor. It is noteworthy that Einstein did not comment at all on the links that Klein had tried to forge between five-dimensional general relativ- ity and Schrödinger’s wave mechanics. The middle of the second paper consists of a thorough examination of how five- dimensional geodesics are related to four-dimensional ones. This had been g55 gmn φmn