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

TL;DR

This paper investigates the limit of Schwarzschild-de Sitter spacetime as the cosmological constant approaches zero, demonstrating convergence to Schwarzschild spacetime and clarifying the physical interpretation of this limit.

Contribution

It provides a rigorous analysis of the $\Lambda o 0$ limit using Geroch's spacetime limits and introduces an embedding into $AdS_3$ to quantify this behavior.

Findings

Schwarzschild-de Sitter spacetime converges to Schwarzschild spacetime as $\Lambda o 0$

The embedding into $AdS_3$ effectively illustrates and measures the limiting process

Establishes a hierarchy of validity between Einstein-de Sitter and Einstein equations

Abstract

We study the $\Lambda \to 0$ 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 $AdS_3$ 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 $\hbar \to 0$.