THIS is not so much, and we are surprised to be learning how many might be “around”, haaaaaaaaaaa. I gave the solution of the Schwarzchild metric in proper distance. Tomorrow, I won’t be tired !!! the complete integration form has two terms. The first is √r(r-1), in Schwarz. units. This has AT MOST a first-order remainder, or ‘error term’ in any calc outside the very near-field, which you can see expanding the polynomial under the root. This term, the first-order deviation from ‘r’, is just 1/2 a Schwarz. radius.
The second term is a very different story, because tho logarithms change slowly, THEY DO CHANGE. This has all been a subtle study of not throwing out the baby with the bathwater !!! Hang out with my numbers on one BH, I’ll key the post. (Out of the Ballpark, E-11 lightyears, for one 3-sun BH)