Caption: The Flamm paraboloid is a tool for visualizing the geometry of spacetime around a Schwarzschild black hole outside of the event horizon (i.e., at radius R ≥ Schwarzschild radius R_sch = 2GM/c**2). The circle at the bottom of the Flamm paraboloid represents the event horizon and has radius R_sch and circumference C = 2πR_sch.

Features:

1. The 2-dimensional surface of the Flamm paraboloid represents curved (3-dimensional) space. The 2-dimensional surface surface representation is needed since we have difficulty visualizing curved space.

2. Measurements of lengths on the Flamm paraboloid are correct proper distances (i.e., distances measured a one instant in time with a ruler). Note that really only curved space is being illustrated by the Flamm paraboloid, NOT general spacetime.

3. What the image shows is that it can be SHORTER to go around on the largest circle to get to the far side than to dive straight inward to the shortest circle (which is has the Schwarzschild radius R_sch = 2GM/c**2 = (2.9532 km)*(M/M_☉) = (19.741 AU)*[M/(10**9*M_☉)] and is the event horizon) and go around that circle and climb out again to get to the far side.

   Whether it is SHORTER or NOT depends on how far out you are at the start:

   1. Too far out, it is it LONGER.
   2. But if you start on the second inner-most circle, it is diagrammatically clear (with enough staring) that it is SHORTER.

4. 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.

5. 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).

6. 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→∞.

7. Although Flamm paraboloids are often shown to visualize the geometry of spacetime around Schwarzschild black holes, black holes in general, and spherically symmetric mass distributions in general, they are NOT really very good for illustrating motion in the gravity wells of these astronomical objects.

   Test particles move along geodesics in spacetime: i.e., they move through space in time. So they do NOT move on paths on a single Flamm paraboloid (see Wikipedia: Schwarzschild metric: Flamm paraboloid).

   So Flamm paraboloids or pseudo Flamm paraboloids used in illustrations of gravitational lensing do NOT prove that gravitational lensing should occur in any direct sense, but merely illustrate an ingredient in gravitational lensing.

   For a discussion of gravitational lensing near Schwarzschild black holes, see Bisnovatyi-Kogan et al. 2019, "Shadow of black holes at local and cosmological distances" and Bisnovatyi-Kogan & Tsupko 2019, "Gravitational Lensing in presence of Plasma: Strong Lens Systems, Black Hole Lensing and Shadow".