D O C . 5 1 N O T E T O F R I E D M A N N 8 3 Published in Zeitschrift für Physik 16 (1923): 228. Received 31 May 1923. A manuscript is also available ([1 026]). [1]In Friedmann 1922, it was shown that solutions to the gravitational field equations with cosmo- logical term exist, describing a universe with time-dependent spatial curvature. Friedmann’s solution contained Einstein’s so-called cylinder world with constant spatial curvature and De Sitter’s spherical universe as special cases. In a comment on Friedmann’s paper (Einstein 1922p [Vol. 13, Doc. 340]), Einstein had challenged Friedmann’s results by arguing that the relevant premises entail the relation for the mass density ρ and timelike coordinate , which in turn would imply that the spa- tial curvature is also time-independent. In a letter to Einstein of 6 December 1922 (Vol. 13, Doc. 390), Friedmann had refuted Einstein’s criticism by showing that indeed one can only derive the relation , and that his results would stand. [2]Yuri A. Krutkov (1890–1952) was Professor of Physics at the University of Petrograd. He vis- ited Leyden from late March to 25 May 1923 (see Yuri Krutkov to Tatiana Krutkova, 26 March and 1 June 1923 (RuPRAS, 07504090104_3B_00690 and 0759040104_3B_00790 [93 985] and [93 993] see also Frenkel 1971). He was present when Einstein discussed his most recent work with Paul Ehrenfest, Johannes Droste, and a Belgian (perhaps Théophile E. de Donder) on 7 May 1923 (see Yuri Krutkov to Tatiana Krutkova, 7 May 1923 RuPRAS, 094601003900080 [93 990]). Frenkel 2002, p. 7, quotes from Krutkov’s notebooks that “on Monday, 7 May 1923, Einstein and I read Friedmann’s article in Zeitschrift für Physik.” In the letter of 7 May, he proudly announced to his sister that he “defeated Einstein in the debate over A. A. Friedmann’s paper. Petrograd’s honor is saved!” (“ɉɨɛɟɞɢɥ ɗɣɧɲɬɟɣɧɚ ɜ ɫɩɨɪɟ ɨ ɫɬɚɬɶɟ Ⱥ. Ⱥ. Ɏɪɢɞɦɚɧɚ. ɑɟɫɬɶ ɉɝɪɚɞɚ ɫɩɚɫɟɧɚ!”). [3]In the manuscript “aufklärend” replaces the deleted word “interessant.” [4]In the manuscript, the sentence continues with the deleted text: “denen eine physikalische Bedeutung kaum zuzuschreiben sein dürfte.” For further historical discussion of Friedmann’s paper and Einstein’s initial criticism of it, see Frenkel 1988. ∂ρ ∂x4 -------- 0 = x4 ∂ gρ ∂x4 ------------- - 0 =