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 $\Lambda$. We point out that in contrast to the case with $\Lambda =0$, the presence of a positive $\Lambda$ in Einsteins's equations enforces, via the condition $|\Phi |\ll 1$, on the potential $\Phi$, a range ${\mathcal{R}}_{max}(\Lambda )\gg r\gg {\mathcal{R}}_{min}(\Lambda )$, within which the Newtonian limit is valid. It also leads to the existence of a maximum mass, ${\mathcal{M}}_{max}(\Lambda )$. 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 $\Phi$ provided the mass distribution is not spherically symmetric.

Latex, 15 pages, no figures, errors corrected

References in corpus (1)

• Geometry and Destiny

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