Now I feel silly, or at least baffled. I was freaking out at the integrand for proper distance, but when you integrate it, it is cool. Looking at proper distance inside an event horizon, the integrand is √[r/(2m-r)] .  In the E.H. limit, r–> 2m- .  The denom has a pole, gut integrating eases it !!!   In this region, the numerator is changing much more slowly… If this analysis holds, the result at the EH is also zero, my, my !!!!  Also at the origin, there is zero residue, dear, dear.  I AM HEARING WORDS FROM MY TEXTBOOK RATTLING LOUDLY IN MY BRAIN: “Proper intervals are invariant but not integrable. Coordinate intervals are integrable, but  not invariant.”  GO FIGGER.  (from the text on GR, by Adler, Bazin, and Schiffer: PROPER INTERVALS ARE INVARIANT, BUT  NOT INTEGRABLE. COORDINATE INTERVALS ARE INTEGRABLE, BUT NOT INVARIANT.”   Yessssssssss.


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