"Oh so close, yet oh so far..." That might be another way to describe trying to dip your fishing hook into the event horizon, but it's how my
E=mc^2 v(r) compares to the above dv/dt expression I found.

This gets messy, and it took me a lot of scribbling and going over several times to condense it to following, but here's how it goes, and please check my work if anyone is interested.

My E=mc^2 reasoning yielded the following expression for v(r) for a Schwarzschild free-fall starting out from infinity with zero initial velocity. And this (supposedly) would be from a frame at infinity.

v^2(r) = c^2 * u(1 - u^2), where u = (1 - R/r)^2, R being the Schwarzschild radius.

Now, differentiate the above with respect to u, and get the following:

2v dv/du = c^2 * (1 - 3u^2).

We're after dv/dr, so we need to multiply both sides by du/dr. That comes out to be du/dr = 2(1 - R/r)*(R/r^2). This cancels the factor of 2 and we have:

v dv/dr = dv/dt = Rc^2/r^2 * [1-3u^2]*(1 - R/r)

If this were correct, this would be an equation for dv/dt that didn't include v itself, just r, which would be nicer. However, we want to compare that with the above dv/dt equation I found. We want a (v/c)^2 in that sucker. We can get that by a little trick of back substituting for u in terms of v. From the u equation, u^2 = 1 - (v/c)^2/u. Substitute that in for u^2, and we get:

dv/dt = Rc^2/r^2 * [1 - 3(1 - v^2/uc^2)]*(1 - R/r)

Now, 1 - R/r = srt(u). Multiply that out in brackets and rearrange and one gets:

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

Now, recognize that Rc^2 = 2GM and pull out a minus sign and we have

dv/dt = -GM/r^2 * [ 4 - 4R/r -6(v/c)^2/(1 + R/r)^2 ]

See what I mean about "Oh so close, yet oh so far"?
The GM/r^2 fell out nicely, but the mess in brackets didn't work out right. It's darn close, we got a constant, 2R/r and 3 (v/c)^2, but it's mulitplied by another factor of 2, the sign is wrong on R/r.

The difference, what we would have to add to the above in brackets to get the first dv/dt formula is this:

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

Doesn't look familiar. So something is awry, but it's something small I imagine, and I don't have a clue what it could be.

-Richard