The $\mathrm{\Lambda }$ to zero limit of spacetimes and its physical interpretation

We study the $\mathrm{\Lambda }\to 0$ behaviour of Schwarzschild-de Sitter spacetime and show, according to Geroch’s notion of spacetime limits, that it converges to the Schwarzschild spacetime. We use an embedding into $Ad{S}_3$ to illustrate and quantify this limiting behaviour. We use these quantitative observations to establish a hierarchy of validity between the Einstein-de Sitter equations and the Einstein equations (and therefore in a weak field limit also Newton’s equations), analogous to the quantum-classical limit when taking $\mathrm{\hslash }\to 0$.