I was reading the Schwarzschild's original paper where he derives the Schwarzschild metric for the first time(The english translated version found in arXiv : On the Gravitational Field of a Mass Point according to Einstein’s Theory).

The derivation is not very complicated and I could understand(?) most of it. However, in the final solution (Pg.6 - Eq.14) the radial coordinate $R$ is not the spatial $r$ of polar coordinate. They are related by - $$R = (r^3 + \alpha^3)^{1/3}$$ This was surprising to me because in our GR class and also in GR textbooks I used (including Wikipedia) - this was never mentioned and the radial coordinate in the metric is used as if it was spatial co-ordinate.

Why is this so? Is the original derivation wrong?


Is it that this $R$ should be considered as the spatial co-ordinate instead of $r$. If so, why?


Yes, the solution by Schwarzschild is the same solution as the one we call the Schwarzschild metric nowadays. The solution for the electrostatic potential of a point charge is valid if you re-express it in spherical instead of Cartesian coordinates. The same is true if you reexpress the Schwarzschild metric in isotropic coordinates, Gullstraind-Painlevé coordinates and so on and so on.

Coordinates do not have any direct meaning in GR, and the skill of beating the physics out of the coordinate expressions is not acquired easily. To do so, you have to consider a range of thought experiments such as various observers measuring distances by various methods such as holding a rope, laser ranging, etc. In this sense, the coordinate $r$ which we usually call the Schwarzschild radius $r$ is the curvature radius of the symmetry sphere (spanned by the angles $\vartheta,\varphi$), but it will not fall out of any simple measurement as some kind of natural "radial distance". So rescaling this coordinate locally actually is not any big issue.

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    $\begingroup$ Aren't you missing the forest behind the trees? :) Schwarzschild's $r$ is the radial coordinate from the origin defined as $r=\sqrt{x^2+y^2+z^2}$. In his solution, this coordinate is zero at the horizon thus meaning that no such a region as "inside the horizon" exists. In the paper link above, see: "Foreword. This fundamental memoir contains the ORIGINAL form of the solution of Schwarzschild’s problem. It is regular in the whole space-time, with the only exception of the origin of the spatial co-ordinates; consequently, it leaves no room for the science fiction of the black holes." $\endgroup$ – safesphere Sep 14 at 19:39
  • $\begingroup$ Yes, the original coordinates used by Schwarzschild do not cover the entire space-time as defined by maximal geodesic extension. But the same is true for the usual coordinates $t,r,\vartheta,\varphi$. $\endgroup$ – Void Sep 16 at 16:12
  • $\begingroup$ No, the usual coordinates do cover the area "inside" the horizon. For example, the usual $r=r_s$ at the horizon while $r=0$ deep inside the horizon. In contrast, the Schwarzschild $r=0$ at the horizon, so the horizon is at the origin and no such area as "inside" the horizon exists. This has nothing to do with the maximally extended solution in the Kruskal coirdinates. $\endgroup$ – safesphere Sep 16 at 16:20
  • $\begingroup$ The solution widely known today as "Schwarzschild" was actually given by Hilbert several months later and contained an error in the definition of $r$. The original solution of Schwarzschild himself did not contain this error and did not produce black holes. Here are the details: sjcrothers.plasmaresources.com/Abrams1989.pdf $\endgroup$ – safesphere Sep 19 at 19:34

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