>>42
http://www.homestead.com/flowstate/files/socrates.jpg
>>15
You did not answer the question I asked. I did not ask what
r is as a function of the mass of the black hole. I asked what
r is as a function of the Schwarzschild coordinate
R. In any case, the fact that your answer is right (for the wrong question) is purely a coincidence between Newtonian gravity and GR.
Because space is curved, the physical distance between
R and 2
R is
not R. Integrating the metric yields
= Integral[1/sqrt(1-
R/r),dr,
R,2
R]
= UglyFunction(2
R) - UglyFunction(
R)
=
R(sqrt(2) + Ln(sqrt(
R)(1 + sqrt(2))))-
RLn(
R)/2
=
R(sqrt(2) + Ln(1 + sqrt(2))) >
R
Attempting the same integral for 0
R to 1
R yields an imaginary distance because time and space flopped in some bizarre way that I still have trouble figuring out. All objects falling into a black hole take a finite proper time to reach the singularity, and travel at a finite speed, so it seems conceivable that there is a finite distance between the singularity and the event horizon, but I'm not sure what coordinates to integrate over. I've puzzled over this for quite a while, but I can't even gurantee that a solution exists. I'm hoping that somebody on /sci/ knows more GR than I do...