The relationship between the General Relativistic derivation of the Schwarzschild radius and the Newtonian derivation is kind of interesting.
Here is an interesting fact: If you have a point-mass of mass ##m## whose velocity is directed radially away from a large spherically symmetric mass ##M##, then you have the following equation of motion for the object's position ##r##: (in the limit as ##m \ll M##):
EOM: ##E = \frac{1}{2} m (\dot{r})^2 - \frac{GmM}{r}##
where ##E## is a constant of the motion. (Note: the presence of ##m## on the right-hand side is irrelevant, but is just thrown in so that it looks like a Newtonian energy equation. The equations of motion are independent of ##m##, if you fix ##E/m##).
This equation is
exact, not an approximation, in two very different cases:
- Newtonian physics, where ##E## is interpreted as the total Newtonian total energy ##E_N = \frac{1}{2} mv^2 + U##, and ##\dot{r}## means the derivative with respect to coordinate time ##t##.
- General Relativity, where ##\dot{r}## means the derivative with respect to proper time ##\tau##, and ##r## is the Schwarzschild radial coordinate.
In the General Relativity case, the interpretation of ##E## is a little complicated. It's actually:
##E_{rel} = \frac{c^2}{2m} ((p_0)^2 - m^2)##
where ##p_0 = \frac{dt}{d\tau}## the ##t## component of the 4-momentum ##p^\mu = m \frac{dx^\mu}{d\tau}##. ##p_0## is conserved in any spacetime in which the metric is independent of ##t##.
Without knowing any of the details about solutions to Einstein's field equations, one should guess that
EOM is approximately true, when ##\dot{r} \ll c## and when ##r## is large. That's because we know that Newtonian gravity is approximately true (in the appropriate coordinate system) when gravitational fields are weak and when objects are moving slowly compared to the speed of light (so that ##\frac{dr}{d\tau} \approx \frac{dr}{dt}##).
What's complicated to show is that
EOM is
exact when using Schwarzschild coordinates, even when ##\dot{r}## is comparable to the speed of light, and when gravitational fields are strong.
But given
EOM, the Schwarzschild radius is easily derivable. To escape to infinity, the object must have ##E \geq 0##. So the limiting case is when ##E = 0##, which implies that: ##\dot{r} = \sqrt{\frac{2GM}{r}}##. Since ##\dot{r} < c##, it follows that escape is only possible if ##r > 2GM/c^2##.
So the odd and wonderful fact about the Schwarzschild solution is that a very careful, rigorous derivation gives exactly the same answer as an approximate, heuristic argument. It's a little like how Bohr's heuristic quantization procedure gives the same answer for hydrogen's energy levels as Schrodinger's equation.
