Schwarzschild Radius of a Black Hole

Lesson notes

Overview

Compress any mass inside this radius and not even light can escape. This topic is rated advanced and is number 39 of 98 in Earth & Space. Deeper models, formal results and harder reasoning.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

The Schwarzschild radius, r_s = 2GM/c², is the radius to which a mass must be compressed to form a black hole, since the escape velocity at that radius equals the speed of light. For the Sun it is about 3 km and for the Earth about 9 mm. The surface at r_s is the event horizon, beyond which nothing can return. The radius grows in direct proportion to mass, so a supermassive black hole of a billion solar masses is about the size of the solar system.

Reading the relationship

The expression "r_s = 2GM/c²" is a compact summary of the idea. It uses the symbols r_s, G, M, c; name what each one stands for before you start. Notice that it contains a squared term, so doubling that quantity changes the result by a factor of four, and a ratio, so increasing the denominator makes the result smaller. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Try to explain why the pattern holds, not just what it is. Look for limiting cases and what happens at the edges.

Common pitfalls

Earth and space systems work on very different scales of time and distance. Check which scale a number refers to before comparing it with another.

How it connects

It builds on “Parallax and Stellar Distance”, the topic before it in Earth & Space. Once you are comfortable, continue with “Exoplanet Transit Light Curve”.

Why it matters

Earth and space science helps us understand our planet, its climate and the wider universe.