On eddington's problem of the expansion of the universe by condensation

1.The discovery of the general receding motion of the spiral nebulae by Hubble lent importance to the Friedmann-Lemaître solution of Einstein’s field equations and it was promptly suggested that our present universe started from a static condition, and owing to certain unknown causes began expanding and has since been doing so continuously. Eddington* pointed out that the static Einstein universe was unstable and so “exploded” (as Eddington put it) in some past age. Eddington suggested that the reason for explosion was the condensation of matter into stellar bodies out of the nebular mass uniformly filling up the Einstein universe. McCrea† and McVittie, working on this idea, proposed a proof showing that for a single condensation the universe would start contracting, but for more condensations start expanding from the equilibrium state. This proof they have recently withdrawn as being erroneous. Meanwhile, Lemaître§ himself enunciated a theorem stating that condensation itself could not cause expansion or contraction, but it was the stagnation of energy (ultimately amounting to condensation) which disturbed the equilibrium and caused the universe to swell up, but McCrea and McVittie showed that his proof was incorrect. Eddington’s problem thus remains where it was when first proposed. In this note we give a proof which shows that condensations, no matter whatever be their number, would start expansion of the Einstein universe.