論空間的曲率

Friedmann takes Einstein's field equations of general relativity and asks for cosmological solutions under two simple postulates: that matter is, on the large scale, uniform and at rest (a pressureless ‘dust’), and that space has the same constant curvature everywhere — homogeneous and isotropic. Crucially, he does NOT assume the universe is static. Where Einstein and de Sitter had each found one unchanging model, Friedmann lets the radius of curvature of space be a function of time, R(t), and solves for how it must evolve.

The equations admit a whole family of non-static universes, which Friedmann sorts into types. In the ‘monotonic’ worlds the radius of curvature grows without bound — the universe expands forever, in one case starting from zero size. In another solution the radius oscillates: space swells to a maximum and then contracts again.

Friedmann computed the period of such an oscillating world and noted that the cosmological constant Λ could be positive, negative, or zero, giving different histories. He made no claim that the real universe is one of these models — he had no data on cosmic expansion; he established only that Einstein's equations permit a universe that moves. Observational confirmation (Hubble's redshift–distance law) and the physical ‘primeval atom’ reading came later, from others.

The full derivation — the field equations under the symmetry assumptions, the classification of solutions, and the period of the periodic world — is at the source below, in the German original and in English translation.