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

- Too far out, it is it
- 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→∞.

- 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".

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:

Image link: Wikimedia Commons: File:Flamm.jpg.

Local file: local link: black_hole_schwarzschild_flamm_paraboloid.html.

File: Black hole file: black_hole_schwarzschild_flamm_paraboloid.html.