Bladesinger-Converting mass (M) to units of length is a common practice, just as converting time to units of length ct. It allows you to get rid of constant such as G and c when doing physics. I don't think this was what you were asking me though. I became familar with the equation r(ouch) and many more from my study using Edwin Taylor's and John Archibald Wheeler's introduction to General Relativity 'Exploring Black Holes'. Developing this equation was the main portion of exercise 6, chapter 2 'Astronaut stretching according to Newton'. The exercise opens up "As you pluge radially inward toward the center of a black hole, you are not physically stress free and comfortable! True, you detect no overall 'force of gravity' accelerating you inward. But you feel a tidal force pulling your feet and head apart and additional forces squeezing your middle inward from the sides like a high quality corset. When do these tidal forces become uncomfortable? We cannot yet answer this question using General Relativity, but Newton is available for consultation, so let's ask him." The problem asks the student to find the Newtonian expression for g in conventional units and then convert to units of length and solve for Earth.
g = M/r^2
g = .00444m / (6.4x10^6m)^2 = 1x10^-16m^-1

From there it asks you to "Take the derivative with respect to r of the local acceleration g to obtain an expression dg/dr in terms of M and r."

dg/dr = |2M|/r^3
dg = 2Mdr/r^3
r(ouch) = [2Mdr/dg]^1/3

This is a great book with a specific intention to make General relativity available to a wider range of folks. It covers everthing in General Relativity which can be covered using the Schwarzchild and Kruzkal-Zerkes metrics, algebra and elementary calculus. This is the presentation which was made by Professor Taylor when he accepted the Oersted Medal for his contribution to education. I'll put it in a post below this.

• Professor Edwin Taylor's and Wheeler's vision. - bruce - November 30, 2000 - 21:18 UTC