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Posted by Bruce on November 29, 2001 00:16:11 UTC

Because we can't use the Schwarzchild metric to evaluate spacetime at r < 2M (there is a coordinate singularity at r = 2M and there are no shells between r = 2M and r = 0) then we can find another solution which evaluates remote coordinates from a proper frame. This solution can be seen in its multiplied out form in the second revised edition of Rindler W. 1977 Essential Relativity (Springer-Verlag) pg 124

The way I learned it was from Taylor and Wheeler's "Exploring Black Holes" where Professor Taylor calls this the 'rain' metric. This metric is good for any r (outside or inside the event horizon). It doesn't have a coordinate singularity at r = 2M.

dt2 = (1-2M/r)dt2rain - (2M/r)1/2dtraindr-dr2-r2df2

You can solve this for dr/dtrain and take an integral of that expression and remove the coordinate singularity at r = 0. From this you can find the proper time intervals and proper velocity for any falling object between any set of r's including r = 0 and r = 2M. It turns out that the average proper velocity of a freely falling object between r = 2M and r = 0 is 3/2 c. If you do a transformation to a local frame where the freely falling astronaut (object) measures the local coordinate speed of light the answer is still c. So using this metric you can show that any freely falling astronaut (object) will cross r = 2M and reach r = 0 in a finite amount of wristwatch time while always measuring the local coordinate speed of light to be c. You can analyse wrt to relativity the spacetime inside the event horizon.

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