The Schwarzschild metric: It's the coordinates, stupid!
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.
Keywords
Cite
@article{arxiv.1308.0394,
title = {The Schwarzschild metric: It's the coordinates, stupid!},
author = {Pierre Fromholz and Eric Poisson and Clifford M. Will},
journal= {arXiv preprint arXiv:1308.0394},
year = {2014}
}
Comments
15 pages, to be submitted to the American Journal of Physics