• The Flamm paraboloid gives the lengths of circumferences (the circles in the diagram) by the formula 
      C = 2πr  for r ≥ R_sch  ,
    where C is a proper distance, but r is NOT a proper distance: it is special ancillary coordinate called the Schwarschild coordinate radius.

  • The radial proper distance is given in differential form by 
                      dr                    dx                  x dx
       ds = --------------- = R_sch ----------- = R_sch ------------
            sqrt(1-R_sch/r)         sqrt(1-1/x)         sqrt(x**2-x)  

             for r ≥ R_sch or x = r/R_sch ≥ 1
    (Wikipedia: Schwarzschild metric: Formulation).

  • The finite radial proper distance measured from the Schwarzschild radius formula R_sch = 2GM/c**2 (i.e., the event horizon) is given by 
      s/R_sch = sqrt(x**2-x) + (1/2)*ln[2x-1+2*sqrt(x**2-x)]   exactly

          = 0   for x = 1
  
          = x*sqrt(1-1/x) + (1/2)*ln(x) 

              + (1/2)*ln(2) +(1/2)*ln[(1-1/(2x))+sqrt(1-1/x)]   exactly

          = x + (1/2)*ln(x) + [ln(2) - 1/2] - (3/8)*(1/x)   to order 1/x

          = x + (1/2)*ln(x)   to order ln(x)
 
          = x   to order x,
    where we have used table integrals (e.g., Wikipedia: List of integrals of irrational functions: Integrals involving R=sqrt(ax**2 + bx + c); HL-8).

    The formula shows that s grows faster than x when x is NOT much greater than 1.

    Note relative difference of x and s/R_Sch decreases as ln(x)/x as x→∞ which means r turns into proper distance as as ln(x)/x as x→∞.

  • File: Black hole file: black_hole_schwarzschild_flamm_paraboloid_1bb.html.