Ken,

I'm still not exactly sastified. Note the "launch velocity" at which the acceleration will be zero increases as r decreases. That would seem to indicate it would still increase speed, so long as v/c never got above the value determined by 1 + 2R/r. And that may the difference between this and my E=mc^2 thing.

The above is a second order, non-linear differential equation in r(t), and I have little confidence that I can easily find a analytic solution. But what I think I will try is to convert that from r(t) to v(r) and see how that compares to my E=mc^2 v(r) equation. If that's easily possible, that is. If you're half-way interested, I wouldn't mind if you (and anyone else) would play with the equation.

And something else about the above Schwarzschild acceleration. I don't know for sure if this is not *proper time* rather the stationary observer's time. The source I got the above from is not clear about that. It doesn't matter for r >> R anyway and low v/c, and so wouldn't be of concern for calculating something like Mercury's orbit. But it would matter for high v/c and near R.

You've always got to be careful about that in GR, and in many cases the proper coordinates are desired as that would be what would be locally measured (which is what you've pointed out many times in many threads.)

-Richard