# What is the so-called Schwarzschild-Radius of a Black Hole?

Apr 20, 2018

#### Explanation:

In astronomy and astrophysics is defined as the distance (from the center) of a spherical distributed mass to the event horizon of a Schwarzschild black hole. Schwarzschild radius is proportional to the object's mass. (some parameters are here).

Schwarzschild radius was introduced in 1916 when Karl Schwarzschild calculated the exact solution for the gravitational field outside a non-rotating, spherically symmetric star. However it can be derived without the complex mathematical formalism of general relativity, using classical approach.

We are calculated how much mass is needed, that the escaping velocity will be the speed of light $v = c$

Let's write the kinetic and potential energy:
${E}_{c} = \frac{1}{2} m {v}^{2} = \frac{1}{2} m {c}^{2}$
${E}_{g} = m g \left(r\right) r = m \frac{G M}{r} ^ 2 r = m \frac{G M}{r}$
$c$-speed of light in vacuum
$G$-Newton's gravitational constant
$r$-radius
$m$-mass of testing object
$M$-black hole mass

${E}_{c} = {E}_{g}$
$\frac{1}{2} m {c}^{2} = m \frac{G M}{r}$ $\implies {r}_{s} = \frac{2 G M}{c} ^ 2$