Black Hole Calculator

A black hole is defined entirely by three quantities: mass, spin, and charge. From mass alone (for non-rotating, uncharged Schwarzschild black holes), all fundamental properties — the size of the event horizon, the temperature at which it radiates, the degree to which time is distorted near it — are calculable from exact formulas derived from general relativity. The black hole calculator computes all these properties simultaneously for any mass input.

The Schwarzschild Radius: Event Horizon Size

The Schwarzschild radius is the radius at which the escape velocity equals the speed of light — the boundary of the event horizon:

r_s = 2GM / c²

where G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant), c = 2.998 × 10⁸ m/s (speed of light), M = mass in kg.

Simplified for solar masses (M_☉ = 1.989 × 10³⁰ kg): r_s ≈ 2.95 × M/M_☉ km

Examples:

• Sun (if it became a black hole): r_s = 2.95 km (radius 2.95 km; the Sun's actual radius is 696,000 km)
• 10 solar mass black hole: r_s = 29.5 km — typical stellar-mass black hole
• Sagittarius A* (4.15 million M_☉, Milky Way center): r_s ≈ 12.2 million km — about 17× the Sun's radius

Use this online calculator for any mass in solar masses, Earth masses, or kilograms. The black hole temperature calculator focuses specifically on Hawking radiation.

Hawking Radiation Temperature

Stephen Hawking's 1974 discovery that black holes emit thermal radiation due to quantum effects near the event horizon:

T_H = ℏc³ / (8πGMk_B)

where ℏ = 1.055 × 10⁻³⁴ J·s (reduced Planck constant), k_B = 1.381 × 10⁻²³ J/K (Boltzmann constant).

Simplified: T_H ≈ 6.17 × 10⁻⁸ K × (M_☉/M)

Practical values: a 10 M_☉ black hole has T_H ≈ 6 × 10⁻⁹ K — immeasurably colder than the cosmic microwave background (2.73 K); Hawking radiation is completely undetectable for any astrophysically realistic black hole. Only hypothetical primordial micro-black holes with masses below about 10¹² kg would have high enough temperatures to radiate detectably.

Event Horizon Surface Area and the Bekenstein-Hawking Entropy

The event horizon area: A = 4πr_s² = 16πG²M²/c⁴

Bekenstein-Hawking entropy: S = k_B × A / (4l_P²) where l_P = √(ℏG/c³) is the Planck length.

For a 10 solar mass black hole: A = 4π × (29,500 m)² ≈ 1.1 × 10¹⁰ m² (11 billion m² ≈ the area of Morocco); entropy ≈ 10⁷⁹ k_B — the immense entropy of black holes underpins the holographic principle and the black hole information paradox. The astronomical calculators and astrophysics calculators provide complementary space science tools.

Gravitational Time Dilation at the Event Horizon

General relativity predicts that time runs slower in stronger gravitational fields. At a distance r from a Schwarzschild black hole: time dilation factor = √(1 − r_s/r). As r approaches r_s, this factor approaches 0 — time appears to freeze at the event horizon from a distant observer's perspective. An infalling observer experiences no frozen time and crosses the event horizon in finite proper time; but a distant observer sees the infalling object asymptotically approach the horizon, redshifting to invisibility, never quite crossing — an illustration of the observer-dependent nature of event horizons in general relativity.