By using Newtonian gravity, we can equate the kinetic and potential energy of a test mass in order to obtain the escape velocity of an object from a large mass $M$:
$$\frac{1}{2} m v^2 = \frac{GMm}{r}$$
therefore:
$$v_{esc}^2 = \frac{2GM}{r}$$
We can then obtain the Schwarzschild radius $r_s$ by setting $v_{esc}$ to the speed of light $c$:
$$r_s = \frac{2GM}{c^2}$$
Is there a reason why Newtonian gravity gives the correct value of $r_s$? Some people seem to say it is just coincidence, others that the $\frac{G}{c^2}$ needs to be there to make the dimensionality correct, but the correct factor of 2 is just a coincidence. I have read this short paper which seems to conclude that there is a reason this prediction is correct, although I admit my knowledge of General relativity is quite basic and I have a hard time following the argument.
