Black Hole Temperature Calculator

Compute Hawking temperature, Schwarzschild radius, and evaporation time. Explore the thermodynamics of black holes.

Hawking temperature: T = ħc³ / (8πG M kB)

Numerical approximation: T (K) ≈ 1.227×10²³ / M (kg)

Schwarzschild radius: Rs = 2GM/c²

1 M☉ (stellar)
10 M☉
10⁶ M☉ (SMBH)
1e12 kg (micro)
4 M☉ (minimal stellar)
1 M⊕
Computing quantum effects...

Hawking Radiation Explained

In 1974, Stephen Hawking combined quantum field theory with general relativity to predict that black holes are not completely black. They emit a perfect blackbody spectrum of particles (now called Hawking radiation) due to quantum effects near the event horizon. This was a revolutionary idea that linked gravity, quantum mechanics, and thermodynamics.

Physical origin: Virtual particles

In quantum field theory, vacuum fluctuations constantly create pairs of virtual particles (e.g., an electron and a positron, or two photons) that ordinarily annihilate within the Heisenberg time. Near a black hole's event horizon, the immense tidal gravity can separate the pair: one particle falls into the black hole while the other escapes to infinity. The escaping particle becomes real, carrying positive energy away. To conserve total energy, the black hole must lose mass — it effectively "evaporates". This process gives the black hole a temperature.

Key formula: T = ħc³ / (8πG M kB)

For a solar mass black hole (M☉ ≈ 2×10³⁰ kg), T ≈ 6.17×10⁻⁸ K — far below the cosmic microwave background temperature (2.7 K). That's why stellar black holes are currently absorbing more radiation than they emit.

Why does temperature increase as mass decreases? The surface gravity (acceleration due to gravity at the horizon) is κ ∝ 1/M. In quantum field theory in curved spacetime, the temperature is proportional to this surface gravity: T ∝ κ. Thus, smaller black holes are much hotter. A black hole of mass 10¹² kg (about the mass of a mountain) would have T ≈ 10¹¹ K — hot enough to emit gamma rays.

Consequences of Hawking radiation

  • Black hole evaporation: As the black hole radiates, it loses mass. Since T ∝ 1/M, the temperature rises, leading to runaway evaporation. The final stages would be an explosive burst of high‑energy particles.
  • Primordial black holes: Tiny black holes formed in the early universe (mass ~10¹² kg) could be completing their evaporation today, producing detectable gamma‑ray flashes — though none have been confirmed yet.
  • Information paradox: Hawking radiation appears thermal and seemingly carries no information about the matter that formed the black hole. This conflicts with quantum unitarity and remains one of the deepest puzzles in theoretical physics.

Approximation formulas used

Quantity Formula Numerical value (M in kg)
Temperature (K) T = ħc³/(8πGMkB) 1.227×10²³ / M
Radius (m) Rs = 2GM/c² 1.485×10⁻²⁷ × M
Lifetime (s) τ = (5120πG²M³)/(ħc⁴) 8.41×10⁻¹⁷ × M³
The evaporation time formula assumes a non‑rotating, uncharged Schwarzschild black hole and no accretion. For example, a 1 M☉ black hole would take about 2×10⁶⁷ years to evaporate — immensely longer than the current age of the universe.

Current status & observations

Hawking radiation has never been directly observed, because astrophysical black holes are too cold. However, it is widely accepted by theorists as a robust prediction of semiclassical gravity. Analogous effects have been observed in laboratory systems (e.g., fluid analogues, optical lattices), lending indirect support. Future gamma‑ray telescopes might detect the final explosions of primordial black holes.

Frequently Asked Questions

The Hawking temperature is the effective blackbody temperature at which a black hole radiates due to quantum effects. It is inversely proportional to the black hole's mass. For a Schwarzschild black hole, T = ħc³/(8πGMkB).

Temperature is proportional to surface gravity, which is stronger for smaller black holes. The horizon curvature is larger, so the virtual particles are more energetic. A micro black hole (mass ~10¹² kg) would have T ~ 10¹¹ K, while a supermassive black hole (10⁹ M☉) has T ~ 10⁻¹⁴ K — far below the CMB.

Not yet. Stellar black holes are too cold (nK). Primordial black holes of ~10¹² kg might explode today with high‑energy gamma rays, but no definitive detection has been made. Analogue gravity experiments (e.g., in Bose‑Einstein condensates) have observed stimulated Hawking radiation, providing indirect evidence for the mechanism.

Hawking radiation is purely thermal and seems to carry no information about the matter that formed the black hole. If the black hole evaporates completely, the information would be lost — violating the principle of unitarity in quantum mechanics. Proposed resolutions include holography, firewalls, or that information is encoded in subtle correlations in the radiation.