How to compute the Schwarzschild radius: fundamental general relativity parameter (Einstein 1915, Schwarzschild 1916) defining the event horizon of a black hole (sphere within which escape velocity equals speed of light c, no information can escape). Formula: R_s = 2 * G * M / c², where G = 6.674×10⁻¹¹ N·m²/kg² gravitational constant, c = 2.998×10⁸ m/s speed of light, M mass (kg). Essential tool for relativistic astrophysics, PhD theses USA universities (MIT Boston, Caltech Pasadena, UT Austin), Event Horizon Telescope EHT 2019 first M87* photo, gravitational wave detection LIGO-Virgo 2015 Nobel 2017, Hawking radiation T_H = hbar*c³/(8*pi*k_B*G*M), supernova collapse formation M >3 M_sun. Example Sun 1.989×10³⁰ kg: R_s = 2.95 km (compressed to 6.9 km already black hole). Example Earth 5.972×10²⁴ kg: R_s = 8.87 mm (marble).

General relativity and Schwarzschild formulas

Schwarzschild radius: R_s = 2 * G * M / c² = 2.953 km/(M_sun). Karl Schwarzschild 1916 solved Einstein equations 1915 for vacuum spherically symmetric spacetime metric. Example Sun M=1.989×10³⁰ kg: R_s = 2*6.674e-11*1.989e30/(2.998e8)² = 2953 m = 2.95 km. Real Sun R=696,000 km, would need to be compressed to 1/235,000 of current radius to become black hole.

Escape velocity at R_s: v_esc = sqrt(2GM/r). At r=R_s: v_esc = sqrt(2GM*c²/(2GM)) = c (light speed). So within event horizon light cannot escape, object invisible (black hole). Pre-relativistic concept Mitchell 1783 and Laplace 1796 (dark stars), formalized Einstein-Schwarzschild 1916.

Mean event horizon density: rho = M / V = M / (4/3*pi*R_s³) = 3*c⁶/(32*pi*G³*M²). Decreases with M²! Sun BH: rho_BH=2×10¹⁹ kg/m³ (dense). SMBH M87* 6.5 billion M_sun: rho=0.003 kg/m³ (less dense than air!). Paradox: SMBH formed large are "lighter" volumetrically.

Hawking temperature: T_H = hbar*c³/(8*pi*k_B*G*M) = 1.227e23/M (kg) Kelvin. Stephen Hawking 1974 quantum gravity demonstrated thermal radiation. Sun BH: T_H=6×10⁻⁸ K (tiny, below CMB). Mini-BH primordial 10¹² kg: T_H=10¹¹ K (intense gamma rays). Lifetime evaporation: t_ev = 5.1×10⁻⁶⁷ * M³ / 3.154×10⁷ years.

Time dilation at horizon: gamma = 1/sqrt(1-R_s/r). Approaching horizon (r->R_s+) proper time dilates infinitely as seen by distant observer. Example ISCO orbit (Innermost Stable Circular Orbit) r=3*R_s Schwarzschild: gamma=sqrt(3/2)≈1.22 (22% clock slowing).

Black hole classification: stellar 3-100 M_sun (supernova collapse, Cygnus X-1 21 M_sun), intermediate IMBH 100-10⁵ M_sun (rare, ~150 LIGO candidates), supermassive SMBH 10⁵-10⁹ M_sun (galaxy centers, Sgr A* 4.3M M_sun, M87* 6.5B M_sun first EHT photo 2019), ultra-massive >10⁹ M_sun (TON 618 66B M_sun, Phoenix A 100B).

History and context

Black hole concept dates to John Mitchell (Britain 1783) classical "dark star", Pierre-Simon Laplace (France 1796) "Exposition du systeme du monde" first formula v_esc=c for compact object. Albert Einstein (Germany 1879-1955, USA Princeton 1933) Nobel 1921, published general relativity 1915 "Die Feldgleichungen der Gravitation". Karl Schwarzschild (Germany 1873-1916) Potsdam astronomer, from Russian front WWI solved Einstein equations for spherical symmetry vacuum publishing "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie" (1916), died months later. Rigorous "black hole" term coined John Wheeler 1967 Princeton. Roger Penrose (UK 1931-) Nobel 2020 demonstrated singularity inevitable collapse. Stephen Hawking (UK 1942-2018) Cambridge demonstrated 1974 thermal radiation black holes (no Nobel but Copley Medal 2006). Reinhard Genzel (Germany) and Andrea Ghez (USA) Nobel 2020 observation stellar orbits Sgr A* Galactic Center T=4×10⁶ M_sun. USA: Caltech Pasadena Kip Thorne Nobel 2017 (LIGO theory), MIT Boston Rainer Weiss Nobel 2017 (LIGO design), UC Berkeley Reva Bouman MIT EHT imaging algorithm. LIGO-Virgo collaboration: first gravitational wave GW150914 detection 14 September 2015 from 36+29 M_sun BH merger, Nobel 2017 Weiss-Barish-Thorne. USA LIGO Hanford Washington and Livingston Louisiana Caltech-MIT, KAGRA Japan, EGO Cascina Pisa Italy.

Event Horizon Telescope EHT collaboration 2019 first SMBH M87* photo (Katie Bouman MIT algorithm), 2022 Sgr A* photo. Industry global astrophysics: NASA Goddard Space Flight Center, JPL Pasadena, Caltech IPAC, Smithsonian Astrophysical Observatory Cambridge MA. US space telescopes: Hubble HST since 1990, JWST 2021 IR L2, Chandra X-ray 1999.

Astronomical objects Schwarzschild radii table

• Hydrogen atom: M=1.67e-27 kg, R_s=2.5e-54 m (below Planck length, no BH possible)
• Mini primordial BH (Hawking): M=10¹² kg, R_s=10⁻¹⁵ m (proton radius, T_H=10¹¹ K)
• Human brain: M=1.4 kg, R_s=2.1e-27 m (subatomic, no BH)
• Person 70 kg: R_s=1.04e-25 m (impossible naturally)
• Car 1500 kg: R_s=2.2e-24 m
• Asteroid Ceres (M=9.4e20 kg): R_s=1.4e-6 m=1.4 um
• Moon (M=7.35e22 kg): R_s=1.1e-4 m=0.1 mm
• Earth (M=5.97e24 kg): R_s=8.87e-3 m=8.87 mm (marble)
• Jupiter (M=1.9e27 kg): R_s=2.82 m (basketball court)
• Sun (M=1.989e30 kg): R_s=2953 m=2.95 km (compressed to city)
• Neutron star (M=1.4 M_sun): R_s=4.2 km, real radius 10-12 km, near BH
• Cygnus X-1 (M=21 M_sun): R_s=62 km (stellar BH)
• Sgr A* Galactic Center (4.3M M_sun): R_s=12.7 million km=0.085 AU
• M87* (6.5B M_sun): R_s=19.4B km=130 AU (beyond Pluto orbit 39 AU)
• TON 618 ultra-massive (66B M_sun): R_s=200B km=1300 AU
• Observable Universe (M=10⁵³ kg): R_s=4.5e26 m=46 billion light years

Worked numerical examples

Example 1 - Sun compressible to black hole. M=1.989×10³⁰ kg. R_s = 2*6.674e-11*1.989e30/(2.998e8)² = 2.953 km. Density at R_s: rho=M/(4/3*pi*R_s³)=1.84×10¹⁹ kg/m³ (1840 billion ton/m³, 1000x atomic nuclear density). Hawking temperature: T_H=1.227e23/1.989e30=6.17e-8 K (colder than CMB 2.7 K, no thermalization). Real Sun R=696,000 km, would need compression to 1/235,000 current radius. Chandrasekhar limit 1.4 M_sun white dwarf, beyond Type II supernova or stellar BH. Sun won't become BH, will evolve white dwarf in 8 billion years.

Example 2 - Earth reduced to black hole. M=5.972×10²⁴ kg. R_s = 2*6.674e-11*5.972e24/(2.998e8)² = 8.87×10⁻³ m = 8.87 mm (marble size, hazelnut). Density at R_s: rho=2×10²⁷ kg/m³ (extreme). Marble volume 2.9e-6 m³ containing entire Earth mass. Time dilation: at r=R_s+1mm gamma=1.07 (7% clock slowing). Conceptually: compress Earth into 9 mm marble, infinite surface gravity, event horizon visually absolutely black sphere.

Example 3 - Sagittarius A* Galactic Center. M=4.3×10⁶ M_sun=8.55×10³⁶ kg. R_s = 2*6.674e-11*8.55e36/(2.998e8)² = 1.27×10¹⁰ m = 12.7 million km = 0.085 AU = 1.7 R_sun. Mean density: rho=2×10⁶ kg/m³ (1000x water). Sun-Earth distance 1 AU, Sgr A* R_s size Mercury orbit 0.38 AU. Reinhard Genzel-Andrea Ghez Nobel 2020 monitoring orbits 'S-stars' around Sgr A* observation 25 years proves SMBH existence. EHT 2022 first Sgr A* photo via global VLBI.

Example 4 - M87* EHT 2019 photo. M=6.5×10⁹ M_sun=1.29×10⁴⁰ kg. R_s = 2*6.674e-11*1.29e40/(2.998e8)² = 1.93×10¹³ m = 19.3 billion km = 129 AU (beyond Pluto orbit 39 AU). Mean density: rho=0.003 kg/m³ (less dense than air, 1300 hab beyond Rome). EHT Event Horizon Telescope 2019 (Katie Bouman MIT imaging algorithm) first historical photo supermassive BH event horizon at heart of M87 galaxy (Virgo Cluster, 53.5 million light years). Observational confirmation Einstein 1915 general relativity after 100+ years.

Example 5 - human person hypothetical BH. M=70 kg adult. R_s = 2*6.674e-11*70/(2.998e8)² = 1.04×10⁻²⁵ m. Planck length l_P=1.616e-35 m, R_s_person ~10⁻¹⁰ * l_P still beyond Planck (10 orders magnitude over). Conceptually impossible to collapse person into BH classically: gravity below Planck mass 2.18e-8 kg requires minimum mass natural BH formation (~3 M_sun stellar). Hawking primordial mini-BH 10¹² kg evaporated 10¹⁰ years Universe lifetime, dark matter candidate but exhausted.

Limits and considerations

1. Schwarzschild metric vacuum spherically symmetric, no rotation: spinning BH requires Kerr 1963 metric with exterior R+ and interior R- horizons + ergosphere.
2. Charged BH: Reissner-Nordstrom 1916-1918 metric with charge Q, rare astronomical (astrophysical BHs neutral).
3. Hawking radiation evaporation T_H ~ 1/M: stellar BHs T_H<<CMB, don't evaporate. Mini-primordial BH 10¹² kg evaporate 10¹⁰ yr (Universe).
4. Central singularity r=0: classical concepts fail, requires quantum gravity (loop quantum gravity, string theory).
5. Information paradox Hawking-Susskind 1980-2010: what happens to information falling through horizon? Still debated.
6. Tidal forces: spaghettification for stellar BHs (internal Roche limit), negligible for SMBH large radius.
7. Accretion disk: spiraling matter into BH emits X-rays, Eddington limit luminosity L_E ~ 1.3e31*(M/M_sun) W.
8. Relativistic jets: some BHs (M87, AGN) emit 99.99% c jets polarly collimated, magnetic field origin (Blandford-Znajek 1977).
9. Wormhole/ER bridge: Einstein-Rosen 1935 + Wheeler ER=EPR conjecture 2013, no observational evidence.
10. Naked singularity: Penrose cosmic censorship 1969 hypothesis: all singularities covered horizon (no naked).

Common errors

1. Confusing R_s with physical star R: Sun R=696,000 km, R_s=2.95 km (235,000x smaller).
2. Neglecting speed of light c² denominator: orders magnitude error.
3. Confusing Sun mass 1.989e30 kg with Earth mass 5.97e24 kg (250,000x different).
4. Demanding star collapse into BH if M<3 M_sun: minimum TOV (Tolman-Oppenheimer-Volkoff) limit 2-3 M_sun, below stable neutron star.
5. Calculating stable orbit r<3R_s (ISCO Schwarzschild at 3R_s): photon sphere at 1.5R_s, within fall.
6. Ignoring rotation: Kerr BH has 2 horizons, ergosphere Penrose process extraction.
7. Neglecting time dilation: clock near horizon slows exponentially vs distant.
8. Confusing R_s (event horizon) with singularity (r=0 mathematical point).
9. Calculating Hawking T_H for stellar BH: T_H<<CMB no thermalization, evaporation invisible.
10. Demanding BH "sucks" distant objects: gravity identical point mass at large distance, no anomalous suction.

Numbered applications

1. Relativistic astrophysics: NASA Goddard, Caltech IPAC, MIT Boston, UT Austin PhD research.
2. Event Horizon Telescope EHT: M87* 2019, Sgr A* 2022, next-gen ngEHT 2030.
3. Gravitational wave detection: LIGO USA Hanford-Livingston, KAGRA Japan, GW150914 Nobel 2017.
4. Space telescopes: Hubble HST, JWST 2021 IR, Chandra X-ray 1999, Athena ESA 2030.
5. Primordial cosmology: PBHs primordial black holes Hawking 1974, dark matter candidate.
6. Hawking radiation: Stephen Hawking 1974 thermal radiation, mini-BH 10¹² kg evaporation.
7. Singularity theorems: Roger Penrose 1965 Nobel 2020, cosmic censorship hypothesis.
8. Black hole thermodynamics: Bekenstein-Hawking 1973-1974 entropy S=A/4 c³/(G*hbar).
9. Quantum gravity: loop quantum gravity, string theory black hole microstates.
10. Educational: relativistic astrophysics MIT Caltech Stanford UT Austin Princeton Harvard.

For support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary general relativity. For research consult Schwarzschild K 1916 'Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie', Einstein A 1915 'Die Feldgleichungen der Gravitation', Hawking SW 1974 'Black Hole Explosions', Misner-Thorne-Wheeler 'Gravitation' 1973 W.H. Freeman, Wald 'General Relativity' 1984 Chicago University Press, EHT Collaboration 2019 ApJ Letters, LIGO-Virgo Collaboration. NASA Goddard Space Flight Center, Caltech IPAC, MIT Kavli Institute for Astrophysics.

Updated July 2026