The Consistent Newtonian Limit of Einstein's Gravity with a Cosmological Constant

Abstract

We derive the `exact' Newtonian limit of general relativity with a positive cosmological constant $Λ$. We point out that in contrast to the case with $Λ= 0 $, the presence of a positive $Λ$ in Einsteins's equations enforces, via the condition $| Φ| \ll 1$, on the potential $Φ$, a range ${\cal R}_{max}(Λ) \gg r \gg {\cal R}_{min} (Λ)$, within which the Newtonian limit is valid. It also leads to the existence of a maximum mass, ${\cal M}_{max}(Λ)$. As a consequence we cannot put the boundary condition for the solution of the Poisson equation at infinity. A boundary condition suitably chosen now at a finite range will then get reflected in the solution of $Φ$ provided the mass distribution is not spherically symmetric.