’ s Note : EŒect of inhomogeneity on cosmological models

In the Einstein stat ic Universe the gravit ational self-attract ion of homogeneously dist ribut ed matter and the cosmological repulsion implied by the cosmological constant balance each other. Eddington [1] showed in 1930 that this balance is unstable: any departure from it will cause the Universe to either expand or collapse away from the init ial state. The perturbed solut ions are the Friedmann models. Tolman, in the paper reprinted in this issue, showed that there exists one more instability: both the Einstein model and the Friedmann model with posit ive spat ial curvature are unstable against perturbat ions of the spat ially homogeneous density distribut ions (the init ial velocity distribut ion remaining unchanged) . The perturbat ions are provided by the solut ion of Lemâõ tre [2] presented as the preceding part of this series [3]. The instability is triggered by local condensat ions as well as by local rarefact ions in the density distribut ion; the latter would be today called voids. Tolman was well aware of the implicat ions of this result for relat ivist ic cosmology; the paper contains an explicit warning against light -hearted use of the Friedmann models for describing the Universe over large regions of space and long periods of t ime. One is tempted to think that , by saying so, Tolman demonstrated a deeper understanding of cosmology than many modern astrophysicist s do. Had he only been taken seriously... The modern reader can recognize a modest predecessor of the singularity theorems in this paper [the paragraphs after eqs. (25) and (35) ]. Tolman’ s paper is often cited today as the source for the solut ion of Einstein’ s equat ions discussed in it , and the solut ion is usually called the a Tolman modelo . However, there is evidence that this paper is almost