Lemaître's hesitating universes

The notion of a hesitating universe was first presented by Lemaître in the Monthly Notices of the Royal Astronomical Society, in 1931, in the paper, The Expanding Universe.

Eddington had suggested that the expansion of a universe in equilibrium may be started by the formation of condensations, but Lemaître points out he finds no effects due to these factors on the equilibrium of the universe. He attributes expansion due to a stagnation process, elaborating:

When there is no condensation, the energy, or at least a notable part of it, may be able to wander freely through the universe. When condensations are formed this free kinetic energy has a chance to be captured by the condensations and then to remain bound to them.

That is what I mean by a "stagnation" of the world - a diminution of the exchanges of energy between distant parts of it.

He introduces the notion of a neutral zone between a particular condensation and the rest of the universe. Crucially, he finds that,

$$\frac{2}{R}\frac{d^2 R}{ds^2} + \frac{1}{R^2}\left( \frac{dR}{ds}\right)^2 + \frac{1}{R^2} = \lambda - \kappa P$$

where $\lambda$ is the cosmological constant and $P$ is the normal pressure on both faces of a material shell in the neutral zone. From this relation, he deduces no dependence on the condensation process but rather only on the normal pressure $P$.

Lemaître goes on to consider a homogeneous universe of variable mass in this set-up. He finds that,

$$4\kappa p R^2 = \left(\frac{dx}{d\tau} \right)^2 - (2x-1)^2 + 1 - 4y$$

where in his notation, $x \sim R^2$, $\tau \sim t$ and $y \sim R^2$. He goes on to conclude, if in a universe in equilibrium, the pressure begins to vary, the radius of the universe varies in the opposite sense, and that therefore stagnation processes induce expansion.

He goes on in section 6 to consider the case of a sudden stagnation, that is, in the case where the diminution of pressure arises suddenly, or more quantitatively that at $t=0$, $p$ drops to zero.