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

∂x4
------------- - 0 =
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