Ok so now that I know how to post it, here's the description in Latex generated images,
That last statement is saying that v behaves the same as the Newtonian electrostatics electric potential. To be clear it is directly proportional to the electric potential. Meaning that this is the solution to Einstein's field equations for a static electric field configuration. As discussed his log is base e, not 10 and composing the two and using my mentioned preferences for conventions results in the composition of these as
The first order expansion of the
yields
In a Newtonian limit this has to relate to the gravitational potential
by
So if we write
where it is understood that
is proportional to the electric potential, then this is the exact solution with it in units of gravitational potential and yields a Newtonian limit gravitational acceleration field that is proportional to the electric potential.
Now consider for a moment the solution for a nonrotating but charged black hole the Reissner-Nordström solution,
I think it doesn't let me copy more images than that to a post so to be continued...