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Black Hole Temperature Calculator

Calculate black hole temperature using Hawking's formula

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Formula

T = ℏc³/(8πGMkB)

Black hole temperature is inversely proportional to mass. A solar mass black hole has T ≈ 6×10⁻⁸ K.

What Is Hawking Temperature?

In 1974, Stephen Hawking made one of the most profound discoveries in theoretical physics: black holes are not completely black. Combining general relativity with quantum field theory in curved spacetime, he showed that black holes emit thermal radiation — now called Hawking radiation — with a temperature inversely proportional to their mass. A black hole's temperature T = ħc³/(8πGMk_B) is astonishingly low: a solar-mass black hole has T ≈ 6×10⁻⁸ K, far colder than the cosmic microwave background at 2.7 K.

This temperature has profound implications: black holes have entropy (proportional to their surface area), they can evaporate (losing mass as they radiate), and they connect three fundamental constants of nature — Planck's constant ħ (quantum mechanics), the speed of light c (relativity), and Newton's gravitational constant G (gravity). The Hawking temperature formula is one of the few equations that unifies all three pillars of modern physics.

Hawking Temperature Formula

The temperature depends only on the black hole's mass, and the relationship is inverse — smaller black holes are hotter:

Hawking Radiation Equations

T = ħc³/(8πGMk_B) R_s = 2GM/c² Surface gravity: κ = c⁴/(4GM) Evaporation time: t_evap ~ M³

Where:

  • T= Hawking temperature (Kelvin)
  • M= Black hole mass (kg)
  • ħ= Reduced Planck constant = 1.0546×10⁻³⁴ J·s
  • c= Speed of light = 2.9979×10⁸ m/s
  • G= Gravitational constant = 6.6743×10⁻¹¹ N·m²/kg²
  • k_B= Boltzmann constant = 1.3806×10⁻²³ J/K
  • R_s= Schwarzschild radius — the event horizon radius (m)

Black Hole Temperatures at Different Scales

Black Hole MassTemperatureLifetimeImplication
1 solar mass6.17×10⁻⁸ K~10⁶⁷ yearsColder than CMB (2.7K) — net absorber of radiation
Earth mass0.02 K~10⁵⁰ yearsStill extremely cold
10¹² kg (asteroid)~10¹¹ K~age of universeExtremely hot — would be explosively evaporating now

How to Use This Calculator

  1. Select Mode: Convert mass to temperature or temperature to mass.
  2. Mass to Temperature: Choose the mass unit (solar masses, Earth masses, Moon masses, or kg), enter the value, and get the Hawking temperature plus Schwarzschild radius and surface gravity.
  3. Temperature to Mass: Enter the temperature in Kelvin to find what mass black hole would have that temperature. Useful for understanding what primordial black holes would be hot enough to detect.
  4. The calculator also displays the Schwarzschild radius (event horizon size) and surface gravity — for a solar-mass black hole, the event horizon is about 3 km across with surface gravity ~1.5×10¹³ m/s².

Significance and Applications

Hawking radiation has profound implications for black hole thermodynamics. The four laws of black hole mechanics — analogous to the laws of thermodynamics — were formulated by Bardeen, Carter, and Hawking. The first law: dM = (κ/8π) dA + Ω dJ + Φ dQ, where surface gravity κ plays the role of temperature and horizon area A plays the role of entropy. Hawking's calculation confirmed that the analogy is exact: T = κ/(2π) and S = A/(4G). Black holes truly are thermodynamic objects.

For primordial black holes — hypothetical black holes formed in the early universe with masses less than ~10¹⁵ g — Hawking radiation would be detectable. A 10¹² kg primordial black hole would have T ≈ 10¹¹ K and be explosively evaporating today. Gamma-ray telescopes search for the characteristic final-stage burst of Hawking evaporation. Non-detection of such bursts places constraints on the density of primordial black holes and on early-universe physics.

Worked Examples

Solar-Mass Black Hole Temperature

Problem:

What is the Hawking temperature of a black hole with one solar mass (M = 1.989×10³⁰ kg)?

Solution Steps:

  1. 1T = ħc³/(8πGMk_B)
  2. 2Numerator: 1.0546×10⁻³⁴ × (2.9979×10⁸)³ = 1.0546×10⁻³⁴ × 2.6944×10²⁵ = 2.841×10⁻⁹
  3. 3Denominator: 8π × 6.6743×10⁻¹¹ × 1.989×10³⁰ × 1.3806×10⁻²³ = 4.61×10⁻²
  4. 4T = 2.841×10⁻⁹ / 4.61×10⁻² = 6.17×10⁻⁸ K
  5. 5Schwarzschild radius: R_s = 2GM/c² = 2.95 km

Result:

T = 6.17×10⁻⁸ K — one of the coldest possible temperatures in the current universe. The event horizon is only 2.95 km across. This black hole absorbs more CMB radiation than it emits.

Primordial Black Hole Evaporation

Problem:

A primordial black hole with mass 5×10¹¹ kg has what temperature? At what mass would it have temperature 300 K (room temperature)?

Solution Steps:

  1. 1For M = 5×10¹¹ kg: T = 6.17×10⁻⁸ × (1.989×10³⁰/5×10¹¹) = 2.45×10¹¹ K
  2. 2For T = 300 K: M = M_solar × (T_solar/T) = 1.989×10³⁰ × (6.17×10⁻⁸/300) = 4.09×10²⁰ kg
  3. 3That's about 2×10⁻¹⁰ solar masses — roughly the mass of asteroid 433 Eros

Result:

A 5×10¹¹ kg black hole has T ≈ 2.45×10¹¹ K — hot enough to emit gamma rays. A room-temperature (300 K) black hole would have mass ~4×10²⁰ kg, comparable to a large asteroid.

Micro Black Hole at LHC

Problem:

If the Large Hadron Collider produced a black hole with mass 1×10⁻²³ kg (about 5 TeV/c²), what would its temperature and lifetime be?

Solution Steps:

  1. 1T = 6.17×10⁻⁸ × (1.989×10³⁰/10⁻²³) = 1.23×10⁴⁶ K
  2. 2This is far hotter than the Planck temperature (~10³² K) — physics breaks down before reaching this scale
  3. 3Evaporation would be essentially instantaneous on any measurable timescale
  4. 4No such black holes have been observed at the LHC — their non-production confirms extra-dimension theories require higher energies

Result:

Theoretical temperature is ~10⁴⁶ K, but at this scale semi-classical gravity (Hawking's calculation) breaks down. A full theory of quantum gravity is needed. LHC data excludes certain TeV-scale gravity models.

Tips & Best Practices

  • ✓Black hole temperature is inversely proportional to mass — smaller black holes are hotter, not colder
  • ✓A solar-mass black hole has T ≈ 6×10⁻⁸ K — about 50 million times colder than the cosmic microwave background
  • ✓The Schwarzschild radius scales linearly with mass: R_s = 2GM/c² ≈ 3 km for one solar mass
  • ✓Hawking radiation has never been observed from real black holes, but analog systems in condensed matter physics have confirmed the phenomenon
  • ✓Primordial black holes with masses below ~10¹² kg would have evaporated completely by the present epoch

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Frequently Asked Questions

The evaporation timescale is proportional to M³ — for a solar-mass black hole, it's ~10⁶⁷ years, vastly longer than the age of the universe (~1.38×10¹⁰ years). Only primordial black holes with masses below ~10¹² kg would have had time to evaporate within the current age of the universe. Additionally, any black hole above ~10²³ kg (roughly the Moon's mass) is colder than the CMB at 2.7 K, so it currently absorbs more radiation than it emits, growing rather than shrinking.
Hawking applied quantum field theory to the curved spacetime near a black hole's event horizon. Quantum vacuum fluctuations near the horizon produce virtual particle-antiparticle pairs. Occasionally, one particle falls in while the other escapes to infinity. To a distant observer, the black hole appears to emit a thermal spectrum of particles. The mathematics shows that the radiation has exactly a blackbody spectrum with temperature T = ħc³/(8πGMk_B), equivalent to the Unruh temperature an accelerating observer would experience at the horizon's surface gravity.
If a black hole evaporates completely into thermal radiation, information about what fell in appears to be destroyed — this violates the unitarity principle of quantum mechanics, which says information must be conserved. The Hawking radiation spectrum is perfectly thermal, carrying no information about the black hole's formation history. This 'black hole information paradox' has been one of the deepest problems in theoretical physics for 50 years. Recent work suggests information is encoded in subtle correlations in the radiation, but a complete resolution remains elusive.
Not from astrophysical black holes — they're far too cold (nanokelvins). However, analog systems using sound waves in Bose-Einstein condensates ('dumb holes') and optical fibers have demonstrated Hawking-like radiation in the lab. Jeff Steinhauer's group at Technion observed thermal phonon emission from an analog sonic black hole in 2016. While not true gravitational Hawking radiation, these experiments confirm the underlying quantum physics in curved spacetime geometries.
Whether a black hole net-gains or loses mass depends on the balance between Hawking radiation (outgoing) and absorbed radiation from the cosmic microwave background and other sources (incoming). The CMB temperature is 2.725 K, corresponding to a black hole mass of about 4.5×10²² kg (~0.7 Moon masses). Black holes above this mass are colder than the CMB and gain mass; below this mass, they're hotter than the CMB and evaporate. As the universe expands and the CMB cools further, this crossover mass will increase.

Sources & References

Last updated: 2026-06-06

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: University Physics

by Young & Freedman

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.