Calculating the Schwarzschild Radius | AI Learning Guide

We can calculate the critical radius where escape velocity equals the speed of light using the formula $r_s = 2GM/c^2$. Let us try this in the code editor. Enter the mass of the Earth. You will find the Schwarzschild radius is only about 9 millimeters. Now enter the mass of the Sun. The result is roughly 3 kilometers. These numbers illustrate the extreme density required to form a black hole.

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Schwarzschild Radius Formula

Setting the Newtonian escape velocity equal to the speed of light ($v_e = c$): \[ \frac{1}{2}mv^2 = \frac{GMm}{r} \implies v = \sqrt{\frac{2GM}{r}} \] Setting $v = c$: \[ r_s = \frac{2GM}{c^2} \]

Earth's Schwarzschild Radius

For Earth ($M \approx 5.97 \times 10^{24}$ kg), $r_s \approx 8.87$ mm. Earth would need to be compressed to the size of a marble to become a black hole.

Sun's Schwarzschild Radius

For the Sun ($M \approx 1.99 \times 10^{30}$ kg), $r_s \approx 2.95$ km. The Sun would need to be compressed to a sphere less than 6 km in diameter.