The Schwarzschild radial coordinate ##r## is defined in such a way that the proper circumference of a sphere at radial coordinate ##r## is ##2\pi r##. This simplifies some maths but creates some rather odd side-effects, so to get a more physical picture I like to use isotropic coordinates instead, where the spatial metric scale factor is the same in all directions. The isotropic radial coordinate ##R## is related to the Schwarzschild radial coordinate ##r## as follows:(adsbygoogle = window.adsbygoogle || []).push({});

$$ r = R (1 + Gm/2Rc^2)^2 $$

or, turning this round (and choosing one of many possible forms)

$$ R = \frac{r - Gm/c^2 + r \sqrt{1 - 2Gm/rc^2}}{2} $$

In isotropic coordinates, unlike in Schwarzschild coordinates, the spatial part of the Schwarzschild metric has a factor of ##(1+Gm/2Rc^2)^2## in all directions and appears to be perfectly well-behaved when approaching the event horizon, where ##R = Gm/2c^2## and proper distance at the event horizon is simply 4 times coordinate distance in any direction.

If we calculate the proper circumference of a sphere via ##r## as it shrinks towards the event horizon, we find that as we get near the event horizon, it stops decreasing (so if we imagine it made of a ring of rulers, the rulers are now shrinking proportionally to the coordinate radius). What is more, if we pass the event horizon in isotropic coordinates the proper circumference starts increasing again, as the rulers are now shrinking more rapidly than proportionally to the radius, which means that the Schwarzschild radial coordinate defined in terms of that circumference would actually now be increasing again!

As can be seen from the equations relating ##r## and ##R##, when ##0 < r < 2Gm/c^2## the expression for ##R## would involve the square root of a negative number, so it could not have a real value. However, the isotropic expression for the space factor in the metric goes past the event horizon without anything special happening (although of course the time factor goes to zero there).

This presumably means that this metric (at least with real coordinates) can only be a valid solution of the Einstein field equations in one of the two coordinate systems and not both. Can anyone point me to the specific way in which one of these coordinate systems does not give a valid solution inside the event horizon? The fact that the isotropic solution seems to behave nicely at the event horizon and implies that the Schwarzschild radial coordinate starts to increase again seems to make sense physically, but is clearly incompatible with the usual assumptions about black holes. Is it simply that no isotropic coordinate system is possible inside the event horizon, and if so, why?

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# A Event horizon vicinity in isotropic coordinates

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