What is the Schwarzschild radius and how is it defined in physics?

The Schwarzschild radius (r_s = 2GM/c^2) is the characteristic physical radius defining the event horizon of a non-rotating, spherically symmetric black hole in General Relativity. If an object with mass M is compressed into a sphere smaller than its Schwarzschild radius, its escape velocity equals the speed of light c, preventing anything—including light—from escaping its gravitational boundary.

What is the Schwarzschild radius of Earth and the Sun?

If compressed into a black hole, Earth's mass (5.972 * 10^24 kg) would have a Schwarzschild radius of approximately 8.87 millimeters (roughly the size of a marble or peanut). The Sun (1.989 * 10^30 kg) would have a Schwarzschild radius of approximately 2.95 kilometers (roughly 1.83 miles, the size of a small city center).

What is the difference between the Event Horizon, Photon Sphere, and ISCO?

The Event Horizon (r_s = 2GM/c^2) is the point of no return. The Photon Sphere (r_ph = 1.5 r_s = 3GM/c^2) is the boundary where light rays can orbit the black hole in unstable circular orbits. The Innermost Stable Circular Orbit (ISCO = 3.0 r_s = 6GM/c^2) is the closest distance that matter (like gas in an accretion disk) can maintain a stable orbit before plunging ballistically into the singularity.

Why are Supermassive Black Holes less dense than water or air?

A black hole's Schwarzschild radius scales linearly with mass (r_s proportional to M), but its volume scales with the cube of the radius (V proportional to r_s^3 proportional to M^3). Therefore, average horizon density scales inversely with the square of mass: rho proportional to 1/M^2. Supermassive black holes with billions of solar masses (like M87* or TON 618) have average densities far lower than water (1,000 kg/m^3) and even air (1.2 kg/m^3), whereas stellar-mass black holes have densities exceeding atomic nuclei (>10^17 kg/m^3).

What is Hawking Radiation and what is the temperature of a black hole?

In 1974, Stephen Hawking showed that quantum vacuum fluctuations near the event horizon cause black holes to emit thermal blackbody radiation. The Hawking temperature is inversely proportional to mass: T_H = (hbar * c^3) / (8 * pi * G * M * k_B) ≈ 6.17 * 10^-8 K * (M_sun / M). Solar-mass black holes are virtually freezing (a few nanokelvin), while microscopic black holes radiate intensely at trillions of degrees.

How long does it take for a black hole to evaporate completely?

A black hole's evaporation lifetime scales with the cube of its mass: tau_evap = (5120 * pi * G^2 * M^3) / (hbar * c^4) ≈ 2.1 * 10^67 years * (M/M_sun)^3. A stellar-mass black hole will take over 10^67 years to evaporate (far longer than the current 1.38 * 10^10 year age of the universe), while a supermassive black hole can persist for 10^100 years.

What is Spaghettification and how do tidal forces work near a black hole?

Spaghettification is the extreme vertical stretching and horizontal compression of an object caused by steep gravitational gradients: Delta g ≈ (2GM/r^3) * Delta r. Paradoxically, tidal forces at the event horizon of a supermassive black hole are gentle enough for a human to cross unharmed, whereas approaching the horizon of a stellar-mass black hole produces lethal tidal forces of millions of g-forces that tear atoms apart.

Can any object become a black hole if compressed enough?

Yes. In classical General Relativity, any mass M—whether a human, a mountain, a planet, or a star—becomes a black hole if compressed within its Schwarzschild radius (2GM/c^2). However, only massive stars (>20 M_sun) have sufficient gravitational self-attraction at the end of their lives to naturally collapse past electron and neutron degeneracy pressures into a stellar black hole.

Who discovered the Schwarzschild radius and the Schwarzschild metric?

German physicist and astronomer Karl Schwarzschild derived the exact solution to Albert Einstein's field equations of General Relativity in 1915 while serving on the Russian front during World War I, just months after Einstein published his theory. It was the first exact non-trivial solution in General Relativity.

How does this Schwarzschild Radius Calculator compute black hole metrics?

The calculator uses exact fundamental physical constants (G = 6.67430 * 10^-11 m^3/kg*s^2, c = 299,792,458 m/s, hbar, k_B) to solve r_s = 2GM/c^2 bidirectionally. It simultaneously calculates event horizon area (4*pi*r_s^2), volume, average density, photon sphere (1.5 r_s), ISCO (3.0 r_s), Hawking temperature, surface gravity (kappa), tidal force gradients, and evaporation lifetimes.

100% Free • General Relativity & Event Horizon Simulator

Schwarzschild Radius Calculator

Calculate the Schwarzschild radius ($r_s = 2GM/c^2$) and event horizon of any mass. Compute photon sphere radius, ISCO plunge orbit, average black hole density, Hawking temperature, surface gravity, spaghettification tidal forces, and evaporation lifetimes.

Celestial Mass Presets: Tap to load

Mass & Event Horizon Solver

Enter an object's mass to calculate its Schwarzschild event horizon and relativistic properties.

General Relativity Metric

Calculation Mode

Object Mass ($M$) ~1.988 × 10³⁰ kg

Relativistic Metric Horizons Spectrum

Event Horizon (r_s) 2.95 km

Photon Sphere (1.5 r_s) 4.43 km

ISCO Orbit (3.0 r_s) 8.86 km

Schwarzschild Event Horizon Radius

2.953 km

~1.83 miles • Diameter: ~5.91 km

Black Hole Thermodynamic Telemetry

🕳️ Event Horizon Radius ($r_s$)	2.953 km
💡 Photon Sphere Radius ($1.5\,r_s$)	4.430 km
🪐 ISCO Plunge Radius ($3.0\,r_s$)	8.859 km
🧊 Average Horizon Density ($\rho$)	1.84 × 10¹⁹ kg/m³
📐 Horizon Surface Area	1.096 × 10⁸ m²
🌡️ Hawking Temperature ($T_H$)	6.17 × 10⁻⁸ K (61.7 nK)
⏳ Evaporation Lifetime ($\tau$)	2.09 × 10⁶⁷ Years
⚓ Surface Gravity ($\kappa$)	1.52 × 10¹³ m/s²

The Physics and Derivation of the Schwarzschild Metric

In 1915, Karl Schwarzschild derived the first exact non-trivial solution to Albert Einstein's field equations of General Relativity ($G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$) for a static, spherically symmetric, uncharged vacuum spacetime. The metric line element is given by:

Schwarzschild Metric Line Element:

$$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2)$$

When the radial coordinate $r$ approaches the Schwarzschild radius $r_s = \frac{2GM}{c^2}$, the temporal component $g_{00} \to 0$ and the radial component $g_{rr} \to \infty$. This coordinate singularity marks the Event Horizon—a causal boundary from which no future-directed timelike or null path can escape.

The 4 Canonical Black Hole Metrics in General Relativity (The No-Hair Theorem)

1. Schwarzschild Black Hole (1915)

Static, uncharged ($M \ne 0, J = 0, Q = 0$). Spherically symmetric vacuum metric with a single event horizon and a central point-like singularity.

2. Kerr Black Hole (1963)

Rotating, uncharged ($M \ne 0, J \ne 0, Q = 0$). Axially symmetric metric featuring an outer Ergosphere, frame-dragging (Lense-Thirring effect), and a ring singularity.

3. Reissner-Nordström Black Hole (1916–1918)

Static, electrically charged ($M \ne 0, J = 0, Q \ne 0$). Spherically symmetric with an outer event horizon and an inner Cauchy horizon.

4. Kerr-Newman Black Hole (1965)

Rotating, electrically charged ($M \ne 0, J \ne 0, Q \ne 0$). The most general stationary electro-vacuum solution completely determined by mass, charge, and spin.

Master Black Hole Horizon Benchmark Matrix

Comprehensive comparison of mass scales, event horizons, photon spheres, densities, and Hawking temperatures.

Relativistic Data

Celestial Object	Mass	Radius ($r_s$)	Photon Sphere	Average Density	Hawking Temp	Physical Comparison
Human Body	$70\,\text{kg}$	$1.04 \times 10^{-25}\,\text{m}$	$1.56 \times 10^{-25}\,\text{m}$	$1.49 \times 10^{76}\,\text{kg/m}^3$	$1.75 \times 10^{21}\,\text{K}$	Sub-subatomic particle scale
Planet Earth	$5.97 \times 10^{24}\,\text{kg}$	$8.87\,\text{mm}$	$13.31\,\text{mm}$	$2.04 \times 10^{30}\,\text{kg/m}^3$	$0.0205\,\text{K}$	Peanut / Small Marble
The Sun	$1.99 \times 10^{30}\,\text{kg}$	$2.95\,\text{km}$	$4.43\,\text{km}$	$1.84 \times 10^{19}\,\text{kg/m}^3$	$61.7\,\text{nK}$	Small Mountain / City Center
Sagittarius A*	$4.15 \times 10^6\,M_\odot$	$12.3\,\text{Million km}$	$18.4\,\text{Million km}$	$1.07 \times 10^6\,\text{kg/m}^3$	$1.49 \times 10^{-14}\,\text{K}$	$0.082\,\text{AU}$ (Inside Mercury Orbit)
M87* (EHT Black Hole)	$6.5 \times 10^9\,M_\odot$	$19.2\,\text{Billion km}$	$28.8\,\text{Billion km}$	$0.436\,\text{kg/m}^3$	$9.50 \times 10^{-18}\,\text{K}$	$128\,\text{AU}$ (Less dense than air!)
TON 618 (Ultramassive)	$6.6 \times 10^{10}\,M_\odot$	$195.0\,\text{Billion km}$	$292.5\,\text{Billion km}$	$0.0042\,\text{kg/m}^3$	$9.35 \times 10^{-19}\,\text{K}$	$1,300\,\text{AU}$ (Near vacuum density)

The Black Hole Density Paradox: Why Supermassive Black Holes are Less Dense Than Air

1. The Scaling Law of Event Horizon Volume:

A black hole's Schwarzschild radius scales linearly with mass ($r_s \propto M$). However, its event horizon volume scales with the cube of the radius ($V \propto r_s^3 \propto M^3$). As a result, the average density inside the event horizon scales inversely with the square of mass: $$\rho = \frac{M}{V} = \frac{3 c^6}{32 \pi G^3 M^2} \propto \frac{1}{M^2}$$

2. Stellar-Mass vs. Supermassive Black Holes:

A $10\,M_\odot$ stellar black hole has an average density of $\sim 10^{17}\,\text{kg/m}^3$ (denser than an atomic nucleus). In contrast, supermassive black hole M87* ($6.5 \times 10^9\,M_\odot$) has an average density of only $0.44\,\text{kg/m}^3$—less dense than Earth's sea-level air ($1.2\,\text{kg/m}^3$)!

Gravitational Time Dilation & Spaghettification Tidal Physics

1. Gravitational Time Dilation at the Horizon:

As an infalling astronaut approaches $r \to r_s$, an observer at infinity sees their clock tick slower and slower: $$\Delta t_{\text{obs}} = \frac{\Delta t_{\text{proper}}}{\sqrt{1 - \frac{r_s}{r}}}$$ At the horizon, light undergoes infinite gravitational redshift ($z \to \infty$), rendering the astronaut invisible. However, in the astronaut's own proper reference frame, they cross the horizon in a finite number of seconds.

2. Tidal Forces & The Spaghettification Gradient:

Tidal acceleration across a body of height $\Delta r$ scales as $\Delta g \approx \frac{2GM}{r^3} \Delta r$. At the horizon ($r = r_s$), this tidal force scales inversely with mass squared ($\propto 1/M^2$). Consequently, falling into a supermassive black hole is completely painless at the horizon, whereas approaching a stellar-mass black hole produces millions of g-forces tearing molecules apart miles before reaching the event horizon.

Frequently Asked Questions (FAQ)

Authoritative answers to common questions about Schwarzschild radius, event horizons, photon spheres, black hole densities, and Hawking radiation.

Celestial Object	Mass	Radius (\(r_s\))	Photon Sphere	Average Density	Hawking Temp	Physical Comparison
Human Body	\(70\,\text{kg}\)	\(1.04 \times 10^{-25}\,\text{m}\)	\(1.56 \times 10^{-25}\,\text{m}\)	\(1.49 \times 10^{76}\,\text{kg/m}^3\)	\(1.75 \times 10^{21}\,\text{K}\)	Sub-subatomic particle scale
Planet Earth	\(5.97 \times 10^{24}\,\text{kg}\)	\(8.87\,\text{mm}\)	\(13.31\,\text{mm}\)	\(2.04 \times 10^{30}\,\text{kg/m}^3\)	\(0.0205\,\text{K}\)	Peanut / Small Marble
The Sun	\(1.99 \times 10^{30}\,\text{kg}\)	\(2.95\,\text{km}\)	\(4.43\,\text{km}\)	\(1.84 \times 10^{19}\,\text{kg/m}^3\)	\(61.7\,\text{nK}\)	Small Mountain / City Center
Sagittarius A*	\(4.15 \times 10^6\,M_\odot\)	\(12.3\,\text{Million km}\)	\(18.4\,\text{Million km}\)	\(1.07 \times 10^6\,\text{kg/m}^3\)	\(1.49 \times 10^{-14}\,\text{K}\)	\(0.082\,\text{AU}\) (Inside Mercury Orbit)
M87* (EHT Black Hole)	\(6.5 \times 10^9\,M_\odot\)	\(19.2\,\text{Billion km}\)	\(28.8\,\text{Billion km}\)	\(0.436\,\text{kg/m}^3\)	\(9.50 \times 10^{-18}\,\text{K}\)	\(128\,\text{AU}\) (Less dense than air!)
TON 618 (Ultramassive)	\(6.6 \times 10^{10}\,M_\odot\)	\(195.0\,\text{Billion km}\)	\(292.5\,\text{Billion km}\)	\(0.0042\,\text{kg/m}^3\)	\(9.35 \times 10^{-19}\,\text{K}\)	\(1,300\,\text{AU}\) (Near vacuum density)

🕳️ Event Horizon Radius (\(r_s\))	2.953 km
💡 Photon Sphere Radius (\(1.5\,r_s\))	4.430 km
🪐 ISCO Plunge Radius (\(3.0\,r_s\))	8.859 km
🧊 Average Horizon Density (\(\rho\))	1.84 × 10¹⁹ kg/m³
📐 Horizon Surface Area	1.096 × 10⁸ m²
🌡️ Hawking Temperature (\(T_H\))	6.17 × 10⁻⁸ K (61.7 nK)
⏳ Evaporation Lifetime (\(\tau\))	2.09 × 10⁶⁷ Years
⚓ Surface Gravity (\(\kappa\))	1.52 × 10¹³ m/s²