Calculate the Schwarzschild radius for the sun and Earth.
Strategy We use the masses of the sun and Earth in Equation (15.6) to find the Schwarzschild radius for each.
r_{ S }=\frac{2 G M}{c^{2}}=\left(1.5 \times 10^{-27} m / kg \right) M (15.6)
Chapter 15
Q. 15.2
Step-by-Step
Verified Solution
If we substitute the mass of the sun and Earth into Equation (15.6), we obtain
\begin{aligned}r_{ S }(\text { sun }) &=\left(1.5 \times 10^{-27} m / kg \right)\left(2.0 \times 10^{30} kg \right)=3.0 km \\r_{ S }(\text { Earth }) &=\left(1.5 \times 10^{-27} m / kg \right)\left(6.0 \times 10^{24} kg \right)=9.0 mm\end{aligned}
If the Earth were a black hole, the dark spot would be very tiny indeed! Our current understanding is that neither the Earth nor the sun will ever become a black hole.