Gravitational energy travels from the central mass: that explains the radial stretching in the Schwarzschild metric

We have been claiming that the radial Schwarzschild metric of a spherical mass M is stretched because moving a test mass m radially ships energy from the gravity field of M over a long distance.

             ● <------------  r  -------------> •

            M                                         m

What is that long distance? The obvious candidate is that the energy is shipped from the center of M to our small test mass.

Let us calculate the extra inertia which m gains for a radial movement. The force of gravity is

       F = G m M / r².

The work that we gain when we move the test mass radially a distance s toward the central mass is

       W = s G m M / r².

That work corresponds to a mass W / c² and that mass is shipped over a distance r. The mass displacement is

       d' = s G m M / c² * 1 / r.

The mass displacement of the test mass m over a distance s is

       d = s m.

The extra inertia in a radial movement is

       d' / d = G M / c² * 1 / r

                 = 1/2 r_s / r,

where r_s is the Schwarzschild radius of M.

https://en.wikipedia.org/wiki/Schwarzschild_metric

The radial Schwarzschild metric is

       1 / sqrt(1 - r_s / r)

       = 1  +  1/2 r_s / r,

when r_s is small.

Everything travels to the radial direction a little bit slower, including light, because the inertia is slightly larger than to the horizontal direction. We take as an axiom:

If light travels to a certain direction by a factor f < 1 slower, then distances in that direction appear to be stretched by a factor 1 / f > 1.