On The Equivalence of the FRW Field Equations and those of Newtonian Cosmology

TL;DR

This paper explains why the Friedmann-Robertson-Walker (FRW) field equations are equivalent to Newtonian cosmology by analyzing a limiting process involving conformally rescaled metrics, revealing Newtonian space and time.

Contribution

It introduces a simple limiting argument connecting FRW Einstein equations with Newtonian cosmology, clarifying their equivalence through conformal rescaling and scalar field analysis.

Findings

The Einstein equations can be derived from Newtonian theory via a limiting process.

The conformally invariant scalar field equation helps recover Einstein equations from Newtonian limits.

The limiting procedure relates the Cartan formulation of Newtonian gravity to relativistic cosmology.

Abstract

We present a simple argument to explain why the field equations of the Friedmann-Robertson-Walker metric are equivalent to those of Newtonian cosmology. By passing to the infinite limit of a family of conformally rescaled FRW metrics in suitable coordinates, we reveal Newtonian space and time. The limiting process preserves the Einstein equations and these may be elucidated directly from the Newtonian limit up to the determination of the scalar curvature parameter. Consideration of the conformally invariant scalar field equation on the FRW spacetime is used to recover the Einstein equations efficiently from the Newtonian theory. We proceed to examine the limiting procedure in connection with the Cartan formulation of Newtonian gravity.