Hi Vladimir
Perhaps I can clarify.
Classical gravitational potential can be linked to general relativity
through escape velocity. Einstein was pleased, as he should have been,
to find that the Schwarzschild solution to his field equations gave a
radial escape velocity exactly matching the classical calculation.
Unfortunately, it is the equality of escape velocities that betrays
general relativity's incompletely relativistic origins.
It is important to note that attention in the paper is restricted
to this problem of determining a radial escape velocity This allows
a metric to be associated with the revised potential energy, calculated
by a multiplicative process to match the expected relativistic redshift.
There is an assumption that only gravity (or an equivalent acceleration)
is being considered.
The two processes to be compared can be summarized as follows:
(A) Classical potential energy -> classical escape velocity -> GR metric
(B) Einstein field equations -> Schwarzschild solution -> GR metric
vs
(a) Revised potential energy -> revised escape velocity -> exponential metric
(b) Einstein-???????? equations -> ?????????? solution -> exponential metric
The essay presents (a) to be compared with (A). It begs the question, "what
is (b)?".
By the way, Hamilton and Lisle's River Model of Black Holes (ref 7) gives
an excellent (and entertaining) explanation of the rather obscure but very useful Gullstrand-Painleve coordinates, although I am arguing against black holes.
Hope this helps
Colin