D O C U M E N T 3 4 0 C O M M E N T O N F R I E D M A N N 4 9 1 Published in Zeitschrift für Physik 11 (1922): 326. A manuscript [1 025] is also available. [1]Friedmann 1922. Friedmann’s goal was to show that it is possible to have a universe whose cur- vature is constant in space but changes in time, and which includes Einstein’s cylindrical and De Sit- ter’s spherical universes as special cases. [2]A: , with . [3]C: for , and (ρ is the density). D3: . The function R is propor- tional to the radius of curvature. [4](8): , with A a constant. [5]Friedmann challenges Einstein’s argument in Doc. 390, and Einstein would later retract the statements made in the current document in Einstein 1923g. For a detailed analysis, see Frenkel 2002. In 1916, Einstein had objected to De Sitter’s solution both because it violates Mach’s principle and because it is nonstatic (see the editorial note, “The Einstein-De Sitter-Weyl-Klein debate,” Vol. 8, p. 356). Friedmann’s solution violates the condition that every solution has to be static, but it does not (in general) violate Mach’s principle. For the reverse case, see Einstein 1922q (Doc. 370), in particu- lar note 5. Rik 1 2 --gikR - – λgik + κTik –= i k , 1 2 3 4) , , , = ( R gikRik = Tik 0= i k , 4 ≠ T44 c2ρg44 = dτ2 R2(x4) c2 ----------------( dx1 2 sin2x1dx2 2 sin2x1sin2x2dx3 2) + + – dx4 2 + = ρ 3A κR2 --------- =