If anyone wants to tackle some messy algebra and some derivatives to help check what I'm about to do, well here's the deal.

My E=mc^2 reasoning, using the Schwarzchild formula for the coordinate speed of light yields this:

(v/c)^2 = u*(1-u^2), where u = (1 - R/r)^2, R being the Schwarzschild radius. If you read the thread where I first posted this, you can see the details of where I got the various "max speed" factors. Now, this is v(r), giving v as a function of the r coordinate, not v(t). The above formula I posted in this thread gives us the acceleration, and involves v(t) and r(t).

That sucker is non-linear and second order, a mess to solve. I have no idea if an analytic solution is possible. But you could plug that into a numerical routine and see what it looked like. But I wondered if the two might be equivalent, and the way to see that is to convert it to v(r) form.

Now, we can write dv/dt = dv/(dr/v) = vdv/dr, and convert the above acceleration to an equation for v(r). When we do that, we get,

v dv/dr = -GM/r^2 * [ 1 + 2R/r - 3 *(v/c)^2 ]

which now gives an equation for v(r).

So the question is to take the first v/c expression and see if it agrees with this last one. That is some messy algebra. I have been working with the first in v(u) and du/dr terms and have gone through two pages of tiny little scribbling, then discovered I made a sign error half way through that threw everything off.

First, to get rid of GM, note that R = 2GM/c^2 or GM = c^2/2R. That will get everything in terms of R and c^2.

We need to get v dv/dr from the first equation and see if that equals the second.

-Richard