Question

The Schwarzschild radius of a black hole depends on its mass m, the speed of light
c, and the gravitational constant G, which has units of m3/(kg · s2). Use dimensional analysis
to write down an expression for the Schwarzschild radius in terms of these three quantities.
Explain your reasoning. (Hint: The Schwarzschild radius has dimension of length)

Answer 1

Let's take the gravitational constant $[G] = \dfrac{m^3}{kg\cdot s^2}$ and try to get rid of $m^2$ in the numerator and $kg\cdot s^2$ in the denominator.

First, let's notice that $[c^2] = m^2/s^2$, thus, we can exclude these units by putting $c^2$ into the denominator of the final expression. Then, only $kg$ has left. To get rid of it we should put $[m] = kg$ into the numerator of the final expression.

Finally, obtain:

$$R = G\dfrac{m}{c^2}$$

Checking dimension, get:

$$[R] = \dfrac{m^3}{kg\cdot s^2}\cdot \dfrac{kg\cdot s^2}{m^2} = m$$

Answer. $R = G\dfrac{m}{c^2}$.