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 general theory of relativity for such an object becomes important when its physical radius $R$ becomes comparable to its Schwarzschild radius, which is given by the formula

$$r_s = \frac{2 G M}{c^2} \, ,$$

where $G$ is the Newton's gravitational constant, $M$ is the mass of the star, and $c$ is the speed of light. Estimating the mass as $M = 4 \pi \rho R^3 / 3$, where $\rho$ is the average density, we conclude that the importance of the general theory of relativity is determined by the magnitude of the ratio

$$\frac{r_s}{R} = \frac{8 \pi G \rho R^2}{3 c^2} \, .$$

It remains to substitute the values $G = 6.67 \times 10^{-11}\,\text{m}^3 / \text{kg s}^2$, $\rho = 10^6\, \text{g} / \text{cm}^3 = 10^9\, \text{kg} / \text{m}^3$, $R = 10^9\, \text{cm} = 10^7\, \text{m}$, and $c = 3 \times 10^8\, \text{m} / \text{s}$. We obtain

$$\frac{r_s}{R} \approx 6 \times 10^{-4} \, .$$

The smallness of this ratio means that the effects of general relativity are insignificant. If the star shrinks to a radius thousand times smaller, its mass and, therefore, its Shwarzschild radius remain the same. Hence, the ratio $r_s/R$ increases thousand times and reaches the value of about 0.6. In this case, the general theory of relativity will be needed to study the dynamics of the star.