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.

Encoding Energy-Density as Geometry

“…In "The Schwarzschild metric: it's the coordinates, stupid", Will et al [9] observe that "Every general relativity text emphasizes that coordinates have no physical meaning… Every student of general relativity is taught that coordinates are irrelevant to physics." "The principle of general covariance, upon which general relativity is built, implies that coordinates are simply labels of space-time events that can be assigned completely arbitrarily.…”

On the Generation of Equivalent ‘Black Hole’ Metrics: A Review

“…The simplest black hole solution to Einstein's field equations is the 'Schwarzschild solution'. A means for generating the required infinite set of equivalent metrics for spherically symmetric Schwarzschild spacetime was recently sought by (Fromholz et al, 2013), without success (Crothers, 2014a). In this case the fundamental issue is the vacuum state, described by the Einstein field equations R µν = 0, for a static gravitational field in the absence of matter (Einstein, 1916).…”

“…Moreover, purported black hole metrics are not restricted to spherical symmetry or the absence of matter, the latter being, according to Einstein, everything except his gravitational field (Einstein, 1916). Accordingly, a means for generating an infinite set of equivalent solutions for black hole universes must not be restricted to spherical symmetry in the absence of matter; the restriction applied by (Fromholz et al, 2013). The sought for metric ground-form must encapsulate all types of supposed black hole universes; otherwise it is incomplete.…”

Newtonian limit and general radial coordinate in the Brans-Dicke theory of gravity

“…Generally, in this case, the Schwarzschild radial coordinate is utilized in the spherical symmetry solution to the Einstein vacuum field equations. However, for convenience, different radial coordinates can be used [7]; the Fock radial coordinate is an example [6].…”