Schwarzschild coordinates for the Schwarzschild black hole solution become very weird near the event horizon because the radial coordinate is based on the proper circumference of a sphere but that has a minimum at the event horizon. This is easy to see in isotropic coordinates, where the Schwarzschild radial coordinate ##r## is related to the isotropic radial coordinate ##r'## as follows:(adsbygoogle = window.adsbygoogle || []).push({});

$$r = r' \left ( 1 + \frac{Gm}{2r'c^2} \right)^2$$

By differentiating with respect to ##r'## it is easy to show that this has a minimum at ##r' = Gm/2rc^2##, which is simply the event horizon in isotropic coordinates, and it then increases towards infinity as the isotropic radial coordinate decreases to zero, corresponding to the singularity.

This effectively says that radial lines become parallel in local space at the event horizon, then space gets wider as one falls past the event horizon, and is infinitely wide by the singularity. This interpretation is not valid on its own as the time coordinate also has problems at the event horizon, although another curiosity is that in isotropic coordinates the time rate seems to go negative rather than imaginary at the event horizon (although in these coordinates an object still takes an infinite time to fall to the event horizon in the first place):

$$d\tau = dt \, \frac{(1-Gm/2r'c^2)}{(1+Gm/2r'c^2)} $$

Various approximations (PPN) or alternative theories of gravity use isotropic metrics where the time and space factors are actually simply reciprocals of one another, for example ##\exp(-Gm/r'c^2)## and ##\exp(+Gm/r'c^2)##. It is usually claimed that this sort of metric cannot lead to black holes, because the factors are non-singular down to ##r=0##, which seems obvious enough. However, this exponential metric and other related metrics still have the property that for small enough ##r##, the proper circumference of a sphere has a minimum value then starts increasing towards infinity:

$$ r = r' \exp\frac{Gm}{r'c^2}$$

$$\frac{dr}{dr'} = \exp\frac{Gm}{r'c^2} + r' \left (- \frac{Gm}{r'^2c^2} \right ) \exp\frac{Gm}{r'c^2}

= \left ( 1 - \frac{Gm}{r'c^2} \right ) \exp\frac{Gm}{r'c^2}$$

The derivative is zero at ##r' = Gm/r'c^2## and this gives the minimum value of ##r##. Note that as the coordinates are isotropic, the radial coordinate behaves in the same way, in that as a spherical surface contracts, an increment in the radial coordinate corresponds to a larger and larger increment in the proper distance in the radial direction, apparently allowing a shrinking mass (regardless of the nature of the internal solution) to occupy more and more space.

A similar result occurs for any radial factor of the form ##r = r' (1+Gm/nr'c^2)^n## where ##n > 1##, although there is no such problem for ##n = 1## or ##n < 1##.

As far as I can see, this means that if you have an object which has starts at the radius of minimum proper circumference and has a fixed mass, it seems that it can expand into a greater amount of space by decreasing its radius, and that the more it does so, the more proper space it has to expand into, although as it does so its time rate will get slower and slower. This suggests that there would be no resistance to prevent it collapsing into a "black dot", except that it's not clear to me where the energy would go in that case!

I know this is a hypothetical metric (although a well-known one), but I'd like a better understanding of what would happen in this case, for contrast with standard "black hole" theory. Can anyone provide further clarification of what would be expected to happen in this case?

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# A Isotropic metric and circumference of sphere

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