Question

The average density of a white dwarf of radius 10⁹ cm is 10⁶ gm/cm³ . Is general theory of relativity needed to study the dynamics of this star? What happens if the star shrinks to a radius thousand times smaller? 

Answer 1

The mass of a white dwarf is

$\displaystyle M = \rho \cdot V = \rho \cdot \frac{4 \pi R^3}{3} = 10^6 \cdot 4.2 \cdot 10^{27} = 4.2 \cdot 10^{33} g = 4.2 \cdot 10^{30} kg$

The Schwarzschild radius is

$\displaystyle r_g = \frac{2GM}{c^2} = \frac{2 \cdot 6.67 \cdot 10^{-11} \cdot 4.2 \cdot 10^{30} }{9 \cdot 10^{16}} =6.23 \cdot 10^{3} \, m = 6.23 \cdot 10^{5} \, cm \approx 10^6 \, cm$

$\displaystyle\frac{r_g}{R} \approx \frac{10^6}{10^9} = 10^{-3} = 0.1\%$ , therefore the effects of general relativity play a negligible role.

If the star shrinks to $R' = 10^6\, cm$ then

$\displaystyle\frac{r_g}{R'} \approx \frac{6.23 \cdot 10^5}{10^6} = 0.623$. In this case the general theory of relativity is needed to study the dynamics of this star.