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Deriving the Schwarzschild radius equation

The force between two masses is given by Newton’s law of universal gravitation:

$$F=\frac{Gm_{1}m_{2}}{r^{2}}$$

The total work done on moving an object with mass $m$, from a distance $R$ to $∞$ from a second body with mass $M$ is:

$$W=\int_{R}^{∞}G\frac{mM}{R^{2}}{dr}=\frac{GmM}{R}$$

For the object to have enough kinetic energy to do this:

$$\frac{1}{2}mv^{2}=\frac{GmM}{R}$$

This can be rearranged to give the escape velocity for any large body, for example the Earth, but the definition of a black hole is that the escape velocity is at least $c$, so in the example above $v=c$. In this case we can transform the equation to see what radius this mass $M£ would have to be squeezed into to make the object a black hole, which is known as the Schwarzschild radius:

$$R=\frac{GM}{c^{2}}$$

When objects move close to the speed of light, and masses warp space as much as black holes do, classical physics doesn’t work very well, so in fact, to correctly derive an equaiton for the Schwarzschild radius we should use relativity, but this equation is approximately correct, hence it is normally written as:

$$R\approx\frac{GM}{c^{2}}$$