Black hole — the Schwarzschild radius

What just happened?

You doubled the mass, and the event horizon grew exactly twofold. The Schwarzschild radius is directly (linearly) proportional to mass — there is no power or root involved.

Schwarzschild radius:

— where $M$ is the mass of the object, $G$ — the gravitational constant, $c$ — the speed of light. It is the radius of the ball into which the mass $M$ must be compressed for it to become a black hole.

What is the event horizon?

It is the boundary around a black hole with radius $R_s$. Nothing escapes from beneath it — neither matter nor light. It is not a physical surface but a “point of no return”: every path outward leads back inside.

Why does even light not escape?

The escape velocity grows with mass and falls with distance. At the event horizon it reaches the value $c$ — the speed of light. Closer to the centre it would have to exceed $c$, and nothing moves faster than light. That is why a photon aimed “outward” falls inward anyway.

How small would Earth or the Sun be?

If you compressed the entire Earth into a black hole, its horizon would have a radius of ≈ 9 mm — smaller than a walnut. The Sun would fit into a ball of radius ≈ 3 km. Black holes are not “large” — they are extremely dense.

Why this matters for your exam:

The formula $R_s = 2GM/c^2$ is a classic astrophysics problem: plug in the mass, compute the radius, compare it with the size of the object. The trick: $R_s$ grows linearly with mass — 10× more mass means a 10× larger horizon, no more, no less.

Remember: the event horizon is the boundary from beneath which nothing escapes, and its radius $R_s = 2GM/c^2$ grows linearly with mass: $R_s \propto M$.