Schwarzschild Radius

Schwarzschild Radius

In 1916, Karl Schwarzschild found the first exact solution to Einstein's field equations — the geometry of spacetime surrounding a spherically symmetric, non-rotating mass $M$. The most famous prediction of this solution is the Schwarzschild radius $r_s$, the radius at which the escape velocity equals the speed of light:

$r_s = \frac{2GM}{c^2}$

where $G = 6.674 \times 10^{-11}$ m³ kg⁻¹ s⁻² is Newton's gravitational constant and $c = 299{,}792{,}458$ m/s.

If a mass $M$ is compressed inside a sphere of radius $r_s$, spacetime curves so severely that nothing — not even light — can escape. That boundary is the event horizon of a black hole.

Typical Values

The Sun would need to be squeezed to a ball less than 3 km across to become a black hole. Earth's Schwarzschild radius is smaller than a penny.

Compactness

A useful dimensionless measure is the compactness — the ratio of the Schwarzschild radius to the actual radius:

$\mathcal{C} = \frac{r_s}{R} = \frac{2GM}{c^2 R}$

For the Sun, $\mathcal{C} \approx 4.2 \times 10^{-6}$ (very Newtonian). For a neutron star, $\mathcal{C} \sim 0.4$.

Your Task

Implement three functions. All physical constants must be defined inside each function body.

• schwarzschild_radius(M) — returns $r_s = 2GM/c^2$ in metres
• schwarzschild_mass(r_s) — inverse: given $r_s$, returns $M = r_s c^2 / (2G)$
• compactness(M, R) — returns $r_s / R = 2GM / (c^2 R)$