Deriving the Schwarzschild solution from a local Newtonian limit

TL;DR

This paper presents an accessible derivation of the Schwarzschild solution using a local Newtonian limit and infalling coordinates, suitable for undergraduate teaching without advanced differential geometry.

Contribution

It introduces a novel derivation method that relies on a simple form of Einstein's equations linked to Newtonian tidal forces, making the Schwarzschild metric more approachable.

Findings

Derivation does not require advanced differential geometry.

Uses infalling coordinates tailored to the problem.

Connects Einstein's equations to Newtonian tidal effects.

Abstract

The Schwarzschild metric is derived in a manner that does not require familiarity with the formalism of differential geometry beyond the ability to interpret a general spacetime metric. As such, the derivation is suitable for an undergraduate course on general relativity. The derivation uses infalling coordinates that are particularly well adapted to the situation, as well as Einstein's equation in the simple form introduced by Baez and Bunn. That version of the vacuum Einstein equations corresponds to requiring a particular local Newtonian limit: that, to first order, the deformation of a "test ball" of freely falling, initially-at-rest test particles is governed by the tidal forces of Newtonian gravity.