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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.

© 2014 American Association of Physics Teachers

Received 09 August 2013
Accepted 04 December 2013

Acknowledgments:
This work was supported in part by the National Science Foundation under Grant Nos. PHY 09-65133, 12-60995 and by the Natural Sciences and Engineering Research Council of Canada. P.F. is grateful to the University of Florida for its hospitality during an internship sponsored by the École Normale. C.M.W. is grateful to the Institut d'Astrophysique de Paris for its hospitality during the completion of this work.

Article outline:

I. INTRODUCTION
II. THE “TEXTBOOK” SOLUTION
III. SCHWARZSCHILD'S SOLUTION OF 1916
IV. SOLUTION USING THE LANDAU-LIFSHITZ FORMULATION
A. The case *C* = 0
B. The case *C* ≠ 0
V. DISCUSSION