The Schwarzschild Radius: How Small a Black Hole Really Is
The Schwarzschild radius R_s = 2GM/c² is the size any mass must be crushed to before it becomes a black hole. The Sun gives 3 km, Earth 9 mm.
The Schwarzschild Radius: How Small a Black Hole Really Is
There is one number that decides whether a lump of matter is an ordinary object or a black hole, and you can write it on the back of a napkin. It is the Schwarzschild radius, named after Karl Schwarzschild, who solved Einstein's field equations for a point mass in 1916 while serving on the Russian front. Squeeze any mass inside its Schwarzschild radius and its gravity becomes so strong that not even light can climb back out. Stay above it, and the object is just a star, a planet, or a person.
The formula is short enough to memorize:
R_s = 2GM/c²
Here M is the mass, G is the gravitational constant (about 6.674 × 10⁻¹¹), and c is the speed of light (about 3 × 10⁸ metres per second). The radius you get back is the size of the event horizon: the one-way boundary around a non-rotating black hole.
What the formula is actually telling you
The cleanest way to see where R_s = 2GM/c² comes from is to ask a Newtonian question and let the answer surprise you. The escape velocity from the surface of a body of mass M at radius r is the square root of 2GM/r. Now demand that the escape velocity equal the speed of light, because at the event horizon light itself cannot escape. Set c equal to the square root of 2GM/r, square both sides, and solve for r. You land on r = 2GM/c².
That this naive calculation gives the right answer is a small miracle. The full derivation from the Schwarzschild metric in general relativity is far more involved, yet it produces the identical expression. The Newtonian shortcut has no business being correct, but it is, which is why it shows up in every introductory astrophysics course. If you want to feel the connection in your hands, run the same mass through an escape velocity calculator and watch the speed climb toward c as the radius shrinks toward R_s.
A worked example: the Sun and the Earth
Numbers make this concrete. Take the Sun, with a mass of 1.989 × 10³⁰ kg.
Plug it in: R_s = (2 × 6.674 × 10⁻¹¹ × 1.989 × 10³⁰) / (9 × 10¹⁶), which works out to roughly 2,954 metres, about 3 km. So if you could crush the entire Sun — all of it — into a sphere about 3 km in radius, it would become a black hole. The real Sun is about 696,000 km in radius, so it sits about 230,000 times larger than its own event horizon. It is in no danger of collapsing.
Now the Earth, mass 5.972 × 10²⁴ kg. The same arithmetic gives a Schwarzschild radius of about 8.9 mm — smaller than a marble. The entire planet, every ocean and mountain, would have to fit inside something the size of a grape to trap light. And a 70 kg person? Their Schwarzschild radius is around 10⁻²⁵ metres, far smaller than a single proton.
The reason these numbers feel absurdly tiny is the denominator. You are dividing by c², which is about 9 × 10¹⁶. That enormous divisor crushes everything down. You can run any of these yourself with the Schwarzschild radius calculator, which has presets for the Earth, the Sun, and Sagittarius A* — the 4.3-million-solar-mass black hole at the centre of our galaxy.
Every object has one, but almost nothing reaches it
A point that trips people up: every mass has a Schwarzschild radius. Your coffee mug has one. The Moon has one. That does not make any of them black holes. The Schwarzschild radius is a threshold, not a property you can see. An object only becomes a black hole if its actual physical size gets pushed inside that threshold.
This is why nature needs extreme events to make black holes. A star supports itself against gravity by the outward pressure of fusion. When a massive star runs out of fuel, the core collapses, and if it is heavy enough — above roughly three solar masses of remnant — nothing can halt the fall. The core plunges through its own Schwarzschild radius and an event horizon snaps into existence. For everything lighter, some pressure wins: white dwarfs hold up on electron degeneracy, neutron stars on neutron degeneracy. The Sun will end as a white dwarf about the size of the Earth, never getting anywhere close to its 3 km horizon.
Where it scales beautifully — and where it bends
One feature of R_s = 2GM/c² that makes it pleasant to work with: the radius is directly proportional to mass. Double the mass, double the horizon. A ten-solar-mass black hole has a horizon ten times wider than a one-solar-mass one. That linearity is why supermassive black holes get so large. Sagittarius A*, at 4.3 million solar masses, has a Schwarzschild radius of about 12.7 million km — bigger than the orbit of Mercury — yet it is still describable by the same one-line formula.
Two caveats worth keeping straight. First, R_s is a radius, not a diameter. The full width of the event horizon is twice this value, so one solar mass means a 3 km radius but a horizon about 6 km across. Reporting the radius as if it were the width is the single most common error I see. Second, the formula assumes a non-rotating, uncharged black hole. Real black holes spin, and a spinning black hole's horizon is described by the more complicated Kerr metric. The Schwarzschild radius stays the standard reference number, but it is the simple, idealized case.
My own moment with the numbers
I'll admit the first time I actually computed these by hand, I assumed I had dropped a factor somewhere. The Earth's horizon coming out under a centimetre felt like a typo. I redid it three times, switching units, before I accepted that a body 12,742 km across has an event horizon you could lose between your fingers. What finally made it click was lining up three masses back to back — a person, the Earth, the Sun — and watching the radius leap across thirty orders of magnitude while the formula stayed the same. The math was never wrong; my intuition was. That gap between what the equation says and what feels reasonable is, to me, the whole appeal of this corner of physics.
Putting it to use
If you are checking a homework answer, building a classroom demo, or just satisfying curiosity about how small a black hole the Sun would make, the workflow is the same: enter a mass, read the radius, and remember that the object only collapses if its real size sinks below that line. Toggle between kilograms and solar masses so the units match your source — mixing them up is the second most common mistake, right behind confusing radius with diameter. For the messier conversions and back-of-envelope checks that come up alongside this, an escape velocity calculator covers the closely related gravity question that the Schwarzschild radius quietly grows out of.
Black holes sound exotic, but the line between "ordinary matter" and "trapped light" is just one short equation. R_s = 2GM/c². Memorize it, and you can size the event horizon of anything in the universe.
Made by Toolora · Updated 2026-06-13