The Schwarzschild metric: It's the coordinates, stupid!

TL;DR

This paper emphasizes the importance of coordinate choices in general relativity by illustrating how different coordinate systems affect the complexity of deriving the Schwarzschild solution, highlighting that the geometry can be described in infinitely many ways.

Contribution

It provides a detailed comparison of multiple derivations of the Schwarzschild metric, demonstrating how coordinate choices influence the simplicity and complexity of calculations.

Findings

Different derivations vary in complexity

The Schwarzschild geometry admits infinitely many coordinate representations

Coordinate choice impacts calculation simplicity

Abstract

Every general relativity textbook emphasizes that coordinates have no physical meaning. Nevertheless, a coordinate choice must be made in order to carry out real calculations, and that choice can make the difference between a calculation that is simple and one that is a mess. We give a concrete illustration of the maxim that "coordinates matter" using the exact Schwarzschild solution for a vacuum, static, spherical spacetime. We review the standard textbook derivation, Schwarzschild's original 1916 derivation, and a derivation using the Landau-Lifshitz formulation of the Einstein field equations. The last derivation is much more complicated, has one aspect for which we have been unable to find a solution, and gives an explicit illustration of the fact that the Schwarzschild geometry can be described in infinitely many coordinate systems.