Schwarzschild Radius Calculator

Compute the event horizon radius for any mass. Understand black hole scales, from microscopic to supermassive.

Formula: Rs = 2GM / c²

G = 6.67430×10⁻¹¹ m³ kg⁻¹ s⁻²   |   c = 299,792,458 m/s

Sun Earth Jupiter Human (70 kg) Sgr A* (4.1e6 M☉) Mountain

⚫ Understanding the Schwarzschild Radius

The Schwarzschild radius (also known as the gravitational radius) is the distance from the center of a non-rotating black hole to the event horizon. It was first derived by Karl Schwarzschild in 1916 from Einstein's equations of general relativity, just months after the theory was published.

Key concept: At the Schwarzschild radius, the escape velocity equals the speed of light. Nothing, not even light, can escape from within this radius. This defines the point of no return.

? Derivation and Validity

Although general relativity is needed for a complete description, a surprising Newtonian approach gives the same formula: set escape velocity vesc = √(2GM/R) = c, then solve for R = 2GM/c². This coincidence is a happy accident; the full general relativistic derivation (Schwarzschild metric) confirms the result exactly.

? Scales and Examples (verified)

  • Sun (M☉ = 1.989×10³⁰ kg): Rs ≈ 2.95 km — the Sun would need to be compressed to about 3 km to become a black hole.
  • Earth (M⊕ = 5.972×10²⁴ kg): Rs ≈ 8.87 mm — about the size of a peanut.
  • Human (70 kg): Rs ≈ 1.04×10⁻²⁵ m — far smaller than a proton (10⁻¹⁵ m).
  • Supermassive black hole (Sgr A* at Milky Way center): M ≈ 4.1×10⁶ M☉ → Rs ≈ 12 million km (≈ 0.08 AU). This fits inside Mercury's orbit.
  • M87* (first black hole imaged): M ≈ 6.5×10⁹ M☉ → Rs ≈ 19 billion km (≈ 127 AU), larger than the Solar System.

? Physical Implications

  • Linearity with mass: Rs ∝ M — double the mass, double the radius. This means supermassive black holes can have huge event horizons but extremely low average density.
  • Density: Average density ρ ∝ M / Rs³ ∝ 1/M². For a solar mass black hole, density ≈ 2×10¹⁹ kg/m³ (nuclear density). For a supermassive black hole like M87*, density can be less than air!
  • Time dilation: Clocks near the event horizon run extremely slow relative to distant observers. At the horizon, time appears frozen for an outside observer.

?️ Types of Black Holes and the Schwarzschild Radius

The Schwarzschild radius strictly applies to a non-rotating, uncharged black hole (Schwarzschild black hole). For rotating (Kerr) black holes, the event horizon is smaller and depends on spin, but the Schwarzschild radius still provides the characteristic scale. The formula also applies to the "gravitational radius" in more complex metrics.

? Observational Evidence

  • Event Horizon Telescope (EHT): Images of M87* (2019) and Sgr A* (2022) show a dark shadow with a size consistent with the predicted Schwarzschild radius, accounting for black hole spin and relativistic effects.
  • Gravitational waves: Detections from LIGO/Virgo of merging black holes provide mass and spin estimates, confirming the existence of objects with radii close to their Schwarzschild radii.

? Historical Note

Karl Schwarzschild derived his solution while serving on the Russian front during World War I. He sent the paper to Einstein, who presented it on his behalf. Schwarzschild died the same year from a rare autoimmune disease, at age 42. His work remains foundational to black hole physics.

❓ Frequently Asked Questions

The event horizon is the boundary around a black hole beyond which no information or matter can escape. It is a one-way membrane in spacetime.

For a non-rotating black hole, yes — it's the radius of the event horizon. For rotating black holes, the horizon is oblate and slightly smaller, but the Schwarzschild radius gives the right order of magnitude.

No. Earth's mass is not dense enough to collapse under its own gravity; it would require compression to about 9 mm, which can only happen artificially or in extreme astrophysical events (not for a planet).

It relates to the black hole's surface area (A = 4πRs²), which is proportional to its entropy (Bekenstein-Hawking entropy S = k A / 4ℓP²). Hawking radiation temperature T = ħc³/(8πGMk) is also inversely proportional to mass.

All values verified using CODATA 2018 constants and standard astrophysical masses.