- #1

Jonathan Scott

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To avoid such infinities, one can instead consider a finite mass and shrink it down. To avoid other complexities, that finite mass can take the form of a thin uniform shell. I believe that by Birkhoff's theorem, it should be possible to devise such a shell that has the same field externally as a spherical mass M. However, inside the shell, space is flat (again by Birkhoff's theorem). The proper area of the inside of a thin shell should be equal to the proper area of the outside, which would then correspond to the Schwarzschild radial coordinate of the surface, so the proper radius of the shell would be equal to its Schwarzschild radial coordinate.

If the shell changes in radius, the internal proper radius changes by the same amount as the Schwarzschild radial coordinate, but the radial proper distance from some point outside it to the surface changes proportionally to the radial component of the metric, [itex]1/\sqrt(1-2GM/rc^2)[/itex]. The exterior and interior parts of the range of the Schwarzschild radial coordinate are therefore physically quite different. However, it seems reasonable to attempt to extrapolate the exterior radial coordinate to the centre, by taking the proper radius of the sphere and assuming it to be scaled by the same radial metric factor as just outside the sphere. That is, if you poked a ruler through a hole in the sphere and measured the distance to the centre, through flat space, it might be reasonable to assume that the radial scale factor to convert that proper distance back to an extrapolated exterior radial coordinate would be the same as immediately outside that surface.

The proper distance to the centre is [itex]r[/itex] so the extrapolated radial coordinate distance to the centre is [itex]r \sqrt(1-2GM/rc^2)[/itex] which is the same as [itex]\sqrt(r (r - 2GM/c^2))[/itex]. Note that this tends to zero as [itex]r[/itex] tends to [itex]2GM/c^2[/itex].

This says that if you have a shrinking shell and extrapolate the distance to its centre in terms of the external radial coordinate, using continuity at the surface, then as the radial coordinate of the sphere approaches the Schwarzschild radius, the extrapolated remaining distance to the centre in terms of the external radial coordinate approaches zero.

Skipping some calculation, if you express the radial coordinate of the shell as [itex]r = R+2GM/c^2[/itex], where [itex]R[/itex] tends to zero as [itex]r[/itex] approaches the Schwarzschild radius, then for small [itex]R[/itex] the extrapolated location of the centre is external Schwarzschild radial coordinate [itex]2GM/c^2 - \sqrt(2GMR/c^2)[/itex].

This seems to suggest that as such a shell shrinks down towards the Schwarzschild radius, the extrapolated distance to the centre (in terms of a continuation of the Schwarzschild coordinate just outside the surface) shrinks down to zero, which suggests that perhaps Schwarzschild's original solution is merely the hypothetical limit of an otherwise physically reasonable situation.

Does this seem valid? It obviously doesn't actually prove anything, but it does suggest that the Schwarzschild radial coordinate is nothing like as simple as it might seem at first glance.