Schwarzschild Radius Calculator
Compute the Schwarzschild radius — the event horizon size of a non-rotating black hole — from its mass via r_s = 2GM/c². The defining length scale of black holes, equal to about 3 km per solar mass.
Last updated: May 2026
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About this calculator
The Schwarzschild radius r_s = 2GM/c² is the radius at which the escape velocity from a spherical body equals the speed of light — the event horizon of a non-rotating (Schwarzschild) black hole. Any mass compressed below this radius forms a black hole. The formula comes from Karl Schwarzschild's 1916 solution to Einstein's field equations of general relativity, which describes the spacetime around a non-rotating spherical mass. Inside r_s nothing — including light — can escape outward; the radial coordinate becomes timelike, and all worldlines lead to the central singularity. Variables: mass in solar masses M☉ = 1.989×10³⁰ kg. The constants used: G = 6.674×10⁻¹¹ N·m²/kg² (gravitational constant), c = 299,792,458 m/s (speed of light). This calculator returns r_s in kilometres. Useful numerical result: r_s ≈ 2.95 km per solar mass. Edge cases: mass must be > 0; very small masses give absurdly tiny r_s (the Sun would need to be compressed to 3 km radius to become a black hole — physically impossible for a star); very large masses give correspondingly large radii (Sagittarius A*, the Milky Way's central supermassive black hole, has mass ~4×10⁶ M☉ and r_s ≈ 1.2×10⁷ km, about 17 solar radii). The formula assumes non-rotating, uncharged black holes; for rotating (Kerr) black holes, the event horizon is r_+ = r_s/2 + √((r_s/2)² − (a/c)²), where a is the spin parameter — this can be smaller than r_s and the structure (ergosphere, ring singularity) is more complex. Astrophysical black holes typically rotate, with a/M between 0 and ~1; r_s remains a useful order-of-magnitude scale even when the actual horizon is smaller.
How to use
Example 1 — Stellar-mass black hole. A 10 solar mass black hole, typical of those formed from massive-star core collapse. Enter Black Hole Mass = 10. r_s = (2 × 6.674×10⁻¹¹ × 10 × 1.989×10³⁰) / (299792458)² / 1000 ≈ 29.5 km. ✓ A ~30 km event horizon — about the size of a small city. The accretion disk and X-ray-emitting hot gas extend out to hundreds or thousands of km, but the horizon itself is tiny. Example 2 — Supermassive black hole. Sagittarius A* at the centre of the Milky Way is ~4 million solar masses. Enter 4000000. r_s = (2 × 6.674×10⁻¹¹ × 4×10⁶ × 1.989×10³⁰) / (299792458)² / 1000 ≈ 1.18×10⁷ km. ✓ About 12 million km, or 17 solar radii, or ~0.08 AU. This is what the Event Horizon Telescope imaged in 2022 — the shadow of the black hole is roughly 2.5×r_s wide on the sky, corresponding to an angular diameter of about 50 microarcseconds from Earth (8000 pc away). The shadow image confirmed both the predicted size and the rough match to general relativity.
Frequently asked questions
What does "event horizon" mean physically?
The event horizon is a one-way membrane in spacetime: signals (matter, light, gravitational waves) can cross it inward but not outward. From far away, an object falling toward the horizon appears to freeze and redshift to invisibility, never quite reaching r_s (because time dilation in the horizon's gravitational field becomes infinite from the external observer's frame). From the falling object's own frame, however, it crosses r_s in finite proper time and continues toward the singularity. Inside r_s the radial coordinate behaves like a time coordinate: there's no "staying put" or "escaping outward"; every worldline ends at r = 0. The horizon is not a physical surface — there's nothing locally special there for a freely falling observer except the no-return property — but it's a global feature of the spacetime geometry. Quantum mechanically, Stephen Hawking showed in 1974 that black holes slowly radiate (Hawking radiation), giving the horizon a tiny temperature and an evaporation timescale much longer than the age of the universe for stellar-mass black holes.
Why is the Schwarzschild radius proportional to mass?
Linear scaling with mass is the simplest non-trivial result: r_s ∝ M. The factor 2GM/c² emerges from solving Einstein's field equations exactly for a non-rotating spherical mass; mathematically it's also where the metric component g_tt = 1 − 2GM/(rc²) crosses zero. A useful sanity check: setting the Newtonian escape velocity √(2GM/r) equal to c gives r = 2GM/c², the same result — the "Newtonian black hole radius" Mitchell and Laplace described in the 1780s coincides numerically with the modern GR result, though their physical interpretation was different. The linear M dependence means r_s for the Sun is 2.95 km, for Earth is 8.87 mm (a marble-sized horizon), for Sagittarius A* is 12 million km. Density to make a black hole scales as 1/M² — so supermassive black holes have far lower average density than stellar-mass ones. SgrA* has mean density less than water; a typical 10 M☉ black hole has density comparable to nuclear matter.
Can anything escape from inside the Schwarzschild radius?
Classically, no — by definition, nothing including light can escape once it crosses the event horizon. Quantum mechanically, however, Stephen Hawking showed in 1974 that black holes emit thermal radiation (Hawking radiation) via virtual particle pair production near the horizon — one falls in (carrying negative energy from the black hole's perspective), the other escapes as real radiation. This causes the black hole to slowly lose mass and eventually evaporate. The radiation has a temperature T = ℏc³/(8πGMk_B), inversely proportional to mass: a solar-mass black hole has T ≈ 60 nanokelvin (utterly undetectable against cosmic background); a small primordial black hole would be much hotter. Evaporation timescale: ~10⁶⁷ years for a solar-mass black hole, longer than the age of the universe by 60+ orders of magnitude. Hawking radiation is theoretically established but has never been observed; analog experiments in fluid and optical systems have produced consistent analog signals.
What are the most common misconceptions about black holes?
The first is that black holes "suck" matter in like a vacuum — they don't. A black hole's gravity is exactly the same as any other mass at distances much greater than r_s; if the Sun were instantly replaced by a 1-solar-mass black hole, Earth's orbit would be unchanged. Matter only falls in if it loses orbital angular momentum (via viscosity in accretion disks, radiation drag, gravitational wave emission). The second is that black holes are infinitely dense — only the singularity at r = 0 has infinite density in GR; the volume inside r_s has a perfectly finite average density that scales as 1/M². The third is that all black holes are non-rotating; in fact, observed astrophysical black holes typically rotate near-maximally (a/M close to 1), and the Kerr metric describes them, not Schwarzschild. The fourth is that the singularity is a physical "point" — it's a coordinate breakdown in classical GR where the theory predicts infinite curvature; quantum gravity is needed to describe what actually happens there. The fifth is that nothing can leave: Hawking radiation does, very slowly.
When should I not use this calculator?
Skip it for rotating (Kerr) or charged (Reissner-Nordström, Kerr-Newman) black holes; their event horizons differ from the Schwarzschild radius and depend on spin a/M and charge Q. Don't use it for compact objects that aren't black holes (white dwarfs, neutron stars); their gravity is intense but their actual radii are much larger than their hypothetical Schwarzschild radii. It's the wrong tool for objects with mass measured in non-solar units — convert to M☉ first, or use the underlying formula r_s = 2GM/c² with SI inputs. Avoid it for black-hole shadow predictions for image interpretation; the observed shadow is ~5×r_s in diameter due to the photon-sphere effects bending light, not directly equal to r_s. Finally, for quantum-gravity or string-theory predictions about black hole microstructure (Planck-scale resolution of the singularity, holographic principle), this classical formula is the wrong tool entirely — those need a complete theory of quantum gravity, which doesn't yet exist.