What is the Schwarzschild Radius?
The Schwarzschild radius (also called the event horizon radius) is the critical radius at which an object becomes a black hole. If all the mass of an object were compressed within this radius, the escape velocity at the surface would equal the speed of light — meaning nothing, not even light, could escape. The Schwarzschild Radius Calculator lets you compute this radius and the surface gravity at the event horizon from just the mass of the black hole.
What is the Event Horizon?
The event horizon is the boundary around a black hole beyond which no information or matter can escape. For a non-rotating (Schwarzschild) black hole, the event horizon is a perfect sphere whose radius is the Schwarzschild radius. It was first described by the German physicist Karl Schwarzschild in 1916, shortly after Einstein published his general theory of relativity.
Outside the event horizon, objects can orbit the black hole just like planets orbit a star. Inside the event horizon, all paths — even those of light — lead inevitably toward the singularity at the centre.
The Black Hole Equation and How to Use This Calculator
The calculator uses two key equations from general relativity and classical gravity:
Schwarzschild Radius Formula
The Schwarzschild radius r is calculated as:
- r = 2GM / c²
Where:
- G — Gravitational constant = 6.67430 × 10⁻¹¹ N·m²·kg⁻²
- M — Mass of the black hole (in kg, solar masses, tonnes, pounds, or slugs)
- c — Speed of light in vacuum ≈ 2.998 × 10⁸ m/s
Surface Gravity Formula
The gravitational acceleration at the event horizon is:
- g = GM / r²
This is the same formula used for gravitational acceleration on any massive body, with the distance set to the Schwarzschild radius. The result gives the surface gravity in m/s² (metric) or ft/s² (American units).
How to Use the Calculator
- Enter the mass of the black hole in the input field.
- Select the mass unit from the dropdown: solar masses (M☉), kilograms (kg), metric tonnes (t), pounds (lb), or slugs.
- Click Calculate.
- The results show the Schwarzschild radius in meters, kilometres, astronomical units (AU), light-years, feet, and miles — plus the surface gravity in m/s² and ft/s².
Unit Systems Supported
Mass Input Units
- Solar masses (M☉) — The most common unit in astrophysics. 1 M☉ = 1.989 × 10³⁰ kg.
- Kilograms (kg) — The SI unit of mass (metric system).
- Metric tonnes (t) — 1 t = 1,000 kg (metric system).
- Pounds (lb) — American / imperial unit. 1 lb ≈ 0.4536 kg.
- Slugs — The American / imperial unit of mass used in physics. 1 slug ≈ 14.594 kg.
Radius Output Units
- Meters (m) and kilometres (km) — metric system.
- Astronomical Units (AU) — 1 AU ≈ 1.496 × 10¹¹ m (the average Earth–Sun distance).
- Light-years (ly) — 1 ly ≈ 9.461 × 10¹⁵ m.
- Feet (ft) and miles (mi) — American / imperial units.
Worked Examples
Schwarzschild Radius of a 10 Solar Mass Black Hole
For a stellar-mass black hole with M = 10 M☉:
- M = 10 × 1.989 × 10³⁰ kg = 1.989 × 10³¹ kg
- r = 2 × 6.674 × 10⁻¹¹ × 1.989 × 10³¹ / (2.998 × 10⁸)² ≈ 29,540 m ≈ 29.54 km
- In American units: ≈ 96,900 ft ≈ 18.4 miles
Earth's Schwarzschild Radius
Earth's mass is about 5.972 × 10²⁴ kg. Its Schwarzschild radius is:
- r ≈ 8.87 mm (0.35 inches)
This means the Earth would need to be compressed to a ball about 9 mm in diameter to become a black hole — over 700 million times smaller than its current size!
Supermassive Black Hole (Milky Way Centre)
The black hole at the centre of the Milky Way, Sagittarius A*, has a mass of about 4 million solar masses:
- r ≈ 11.8 million km ≈ 7.3 million miles ≈ 0.079 AU
FAQs
How do I calculate the Schwarzschild radius?
Use the formula r = 2GM/c²: multiply the mass by the gravitational constant G (6.67430 × 10⁻¹¹) and by 2, then divide by the square of the speed of light (c² ≈ 8.988 × 10¹⁶ m²/s²). For a 1 solar mass object this gives r ≈ 2,954 m ≈ 2.95 km.
What is Earth's Schwarzschild radius?
Approximately 8.87 mm — less than a centimetre. Earth would have to be squeezed to this tiny radius to become a black hole.
What is the Schwarzschild radius of the Sun?
About 2.95 km (1.83 miles). The Sun would need to be compressed into a sphere roughly 3 km in radius to collapse into a black hole.
Does every object have a Schwarzschild radius?
Yes — any mass has a theoretical Schwarzschild radius. However, most objects (planets, stars, people) are far larger than their Schwarzschild radius, so they are not black holes. Only objects compressed below their Schwarzschild radius become black holes.
What happens at the event horizon?
For a distant observer, nothing special appears to happen at the event horizon — but from the infalling object's frame it is reached in finite proper time. Once inside, all spacetime paths lead to the singularity; even light cannot escape to the outside.