Established Physics (1974)

Hawking Temperature

The temperature at which black holes emit thermal radiation, unifying quantum mechanics, general relativity, and thermodynamics.

TH = ℏc³/(8πGMkB) = ℏκ/(2πckB)

Discovered by Stephen Hawking in 1974 | Foundation of Black Hole Thermodynamics

What Does This Formula Mean?

"Black holes are not completely black—they emit thermal radiation at a temperature inversely proportional to their mass."

TH: Hawking Temperature

The effective temperature of thermal radiation emitted by a black hole. Smaller black holes are hotter!

κ: Surface Gravity

The gravitational acceleration at the event horizon. Related to the black hole's mass: κ = GM/rs².

Quantum + Gravity

Contains ℏ (quantum), G (gravity), and c (relativity)—a window into quantum gravity.

TH = ℏc³/(8πGMkB)
Established
TH
Hawking Temperature
The characteristic temperature of black hole radiation.
Units: Kelvin (K)
For a solar mass black hole: TH ≈ 60 nanokelvin (much colder than the CMB!)
Thermodynamic Parameter
ℏ = h/(2π)
Reduced Planck Constant
ℏ = 1.054571817... × 10-34 J·s
The fundamental quantum of action. Its presence indicates quantum mechanical origin.
Wikipedia: Planck Constant →
c
Speed of Light
c = 299,792,458 m/s (exact by definition)
Connects space and time in special and general relativity.
Fundamental Constant
G
Newton's Gravitational Constant
G = 6.67430(15) × 10-11 m³/(kg·s²)
The strength of gravity. Appears in Einstein's field equations.
Wikipedia: Gravitational Constant →
M
Black Hole Mass
The total mass-energy of the black hole.
TH ∝ 1/M: Larger black holes are colder.
Schwarzschild radius: rs = 2GM/c²
System Parameter
kB
Boltzmann Constant
kB = 1.380649 × 10-23 J/K (exact as of 2019)
Relates temperature to energy: E = kBT
Connects microscopic to macroscopic thermodynamics.
See Boltzmann Entropy →
κ
Surface Gravity
For Schwarzschild black hole: κ = c⁴/(4GM)
Alternative form: TH = ℏκ/(2πckB)
Measures acceleration at the horizon (in the limit approaching from infinity).
Wikipedia: Surface Gravity →
Bekenstein-Hawking Entropy
Black Hole Entropy
SBH = kBA/(4ℓP²) = kBc³A/(4ℏG)
Where A = 16πG²M²/c⁴ is the horizon area.
First Law: dM = (κ/8πG)dA connects temperature to entropy.
Wikipedia: Black Hole Thermodynamics →
Foundation Chain
→ General Relativity (Einstein, 1915) Gravity
→ Schwarzschild Solution (1916) Black Holes
→ Quantum Field Theory in Curved Spacetime (1960s) QFT + GR
→ Bekenstein Entropy (1972-1973) BH Thermodynamics
→ Hawking Radiation (1974) Landmark Discovery

Visual Understanding: Black Hole Evaporation

Hawking radiation arises from quantum vacuum fluctuations near the event horizon:

rs = 2GM/c² Black Hole Event Horizon Hawking Radiation Negative Energy Temperature Formula TH = ℏc³/(8πGMkB) Inverse relationship: TH ∝ 1/M Smaller holes → Higher temperature Larger holes → Lower temperature Example Temperatures M☉ (Sun): TH ≈ 60 nK 10 M☉: TH ≈ 6 nK 10⁶ M☉ (galactic): TH ≈ 60 fK 10¹⁵ g (primordial): TH ≈ 10¹¹ K (hot!) CMB: T = 2.725 K (All stellar-mass BHs are colder than CMB)

Virtual particle pairs near the event horizon: one particle escapes as Hawking radiation (green), while its partner falls into the black hole (red), carrying negative energy and reducing the black hole's mass.

Key Concepts to Understand

1. The Physical Mechanism

Hawking radiation arises from quantum field theory in curved spacetime:

  • Vacuum fluctuations: Quantum fields continuously create virtual particle-antiparticle pairs
  • Horizon separation: Near the event horizon, pairs can be separated before annihilating
  • Energy extraction: One particle falls in (negative energy), one escapes (positive energy)
  • Mass loss: The black hole loses mass equal to the energy of escaping radiation

2. Black Hole Evaporation

Since TH ∝ 1/M, smaller black holes are hotter and evaporate faster:

tevap = (5120πG²M³)/(ℏc⁴) ≈ 1067 years × (M/M☉)³ Evaporation time for Schwarzschild black hole

A solar-mass black hole would take ~1067 years to evaporate—far longer than the current age of the universe (1010 years). But a primordial black hole of mass 1015 g would evaporate in ~1010 years and could be evaporating now!

3. The First Law of Black Hole Thermodynamics

Black holes obey thermodynamic laws analogous to ordinary thermodynamics:

dM = (κ/8πG)dA + ΩHdJ + ΦHdQ First Law: Energy = Temperature × Entropy + Work terms

Comparing with dE = TdS + work gives the identification: TH = ℏκ/(2πckB) and SBH = kBA/(4ℓP²).

4. Connection to the Unruh Effect

The Unruh effect is closely related to Hawking radiation:

Unruh Temperature

An accelerating observer with proper acceleration a sees the Minkowski vacuum as thermal radiation at TU = ℏa/(2πckB).

Hawking vs Unruh

Hawking radiation is related to the equivalence principle: observers hovering at the horizon experience constant acceleration a = κ.

5. The Information Paradox

Hawking radiation is thermal (maximum entropy), but black holes can form from pure quantum states (zero entropy):

The Black Hole Information Paradox

If a black hole evaporates completely into thermal radiation, what happens to the quantum information that fell in? This appears to violate quantum mechanics (unitarity). Potential resolutions include:

  • Information encoded in correlations: Subtle correlations in Hawking radiation carry information
  • Black hole remnants: Evaporation stops at Planck mass, leaving a remnant
  • Holography: Information stored on the horizon (AdS/CFT correspondence)
  • Quantum gravity effects: Breakdown of semiclassical approximation near endpoint

Learning Resources

YouTube Video Explanations

Hawking Radiation - PBS Space Time

Excellent visual explanation of how black holes emit radiation and evaporate.

Watch on YouTube → 13 min

Black Hole Thermodynamics - Leonard Susskind

Detailed lecture on black hole entropy, temperature, and the information paradox.

Search Lectures → Advanced

The Information Paradox - PBS Space Time

Deep dive into the black hole information paradox and proposed resolutions.

Search Videos → Expert

Quantum Field Theory in Curved Spacetime

Technical introduction to the framework used to derive Hawking radiation.

Search Course → Graduate

Articles & Textbooks

  • Wikipedia: Hawking Radiation | Black Hole Thermodynamics | Unruh Effect
  • Original Papers: Hawking, S.W. (1974) "Black hole explosions?" Nature 248: 30–31 [DOI]
  • Original Papers: Hawking, S.W. (1975) "Particle Creation by Black Holes" Commun. Math. Phys. 43: 199–220 [DOI]
  • Textbook (Intermediate): "Black Holes: An Introduction" by Derek Raine & Edwin Thomas [WorldCat]
  • Textbook (Advanced): "Quantum Fields in Curved Space" by N.D. Birrell & P.C.W. Davies [Cambridge]
  • Textbook (Graduate): "General Relativity" by Robert Wald (Chapter 14: Black Holes) [Chicago Press]
  • Review Article: Jacobson, T. (2003) "Introduction to Black Hole Thermodynamics" [arXiv]

Interactive Resources

Key Terms & Concepts

Event Horizon

The boundary beyond which nothing, not even light, can escape. For a Schwarzschild black hole: rs = 2GM/c².

Learn more →

Surface Gravity

The gravitational acceleration at the event horizon: κ = c⁴/(4GM). Analogous to temperature in black hole thermodynamics.

Learn more →

Bekenstein Bound

Maximum entropy contained in a region: S ≤ 2πkR E/(ℏc). Black holes saturate this bound.

Learn more →

Unruh Effect

An accelerating observer sees thermal radiation at temperature TU = ℏa/(2πckB), even in Minkowski vacuum.

Learn more →

KMS Condition

Mathematical characterization of thermal states in quantum field theory. Hawking radiation satisfies the KMS condition.

Learn more →

Holographic Principle

The idea that all information in a volume can be encoded on its boundary. Motivated by black hole entropy S ∝ Area.

Learn more →

Connection to Principia Metaphysica

Hawking temperature plays a crucial role in PM's treatment of black holes and dimensional reduction:

Thermal Time Hypothesis

In PM, Hawking radiation provides a concrete example of the thermal time hypothesis:

  • KMS states at horizons: The vacuum state satisfies KMS condition with βH = 1/(kBTH)
  • Modular flow as time: The Tomita-Takesaki modular flow generates evolution at the horizon
  • Temperature from geometry: Surface gravity κ (geometric) determines temperature (thermodynamic)
  • Observer dependence: Hawking temperature depends on the observer's reference frame (horizon)

Black Holes in Higher Dimensions

PM extends black hole thermodynamics to the D bulk:

SBH = kB Ad-2 / (4ℓP,dd-2) Bekenstein-Hawking entropy in d spacetime dimensions
  • Higher-dimensional horizons: Event horizons in D have different topology and thermodynamics
  • Kaluza-Klein black holes: Compactified dimensions affect black hole properties
  • Holographic entropy bounds: Connection to AdS/CFT in higher dimensions
  • Microscopic origin: String theory provides microscopic counting of black hole microstates

Quantum Gravity Regime

Hawking temperature indicates where quantum gravity becomes important:

MPlanck = √(ℏc/G) ≈ 2.18 × 10-8 kg Planck mass: quantum gravity scale
  • Planck temperature: TPlanck ≈ 1.42 × 1032 K (TH when M = MPlanck)
  • Breakdown of semiclassical approximation: Near M ~ MPlanck, full quantum gravity needed
  • PM quantum geometry: Higher-dimensional structure becomes relevant at Planck scale
  • Evaporation endpoint: What happens when M → MPlanck? Remnants or complete evaporation?

See also: KMS Condition, Tomita-Takesaki Theory, and Boltzmann Entropy for related thermodynamic foundations.

Advanced Topics

1. Derivation Outline

The Hawking temperature can be derived by analyzing quantum field theory in Schwarzschild spacetime:

  1. Bogoliubov transformation: Relate "in" and "out" vacuum states near horizon
  2. Kruskal coordinates: Use maximally extended Schwarzschild spacetime
  3. Particle creation: Calculate expectation value ⟨N⟩ for particle number operator
  4. Thermal spectrum: Show ⟨N(ω)⟩ = 1/(eℏω/kBTH - 1) (Planck distribution)
  5. Surface gravity: Extract TH = ℏκ/(2πckB)

2. Rotating Black Holes (Kerr)

For rotating black holes with angular momentum J, the temperature is modified:

TH = ℏκ+/(2πckB) = ℏc³(r+ - r-) / (4πGkB(r+² + a²)) Kerr black hole temperature, where a = J/(Mc) is spin parameter

As a → M (extremal limit), TH → 0. Extremal black holes do not radiate.

3. Charged Black Holes (Reissner-Nordström)

For charged black holes with charge Q:

κ = (r+ - r-) / (2(r+² + Q²/c⁴)) Surface gravity for Reissner-Nordström, r± = GM/c² ± √((GM/c²)² - GQ²/c⁴)

When Q² = GM²/c², the black hole is extremal (TH = 0) with degenerate horizons r+ = r-.

4. Page Curve and Entanglement

Recent work on the information paradox involves the Page curve—the evolution of entanglement entropy during evaporation:

  • Early time: Srad grows linearly as radiation is emitted
  • Page time: tPage ~ tevap/2, when Srad = SBH
  • Late time: Srad decreases as black hole shrinks (unitarity preserved)
  • Island formula: Recent developments using quantum extremal surfaces in AdS/CFT

Practice Problems

Test your understanding with these exercises:

Problem 1: Solar Mass Black Hole Temperature

Calculate the Hawking temperature for a black hole with M = M☉ = 2.0 × 1030 kg. Compare this to the cosmic microwave background temperature TCMB = 2.725 K.

Solution

TH = ℏc³/(8πGMkB)
= (1.055 × 10-34 J·s)(3 × 108 m/s)³ / [8π(6.67 × 10-11 m³/kg·s²)(2 × 1030 kg)(1.38 × 10-23 J/K)]
≈ 6.2 × 10-8 K = 62 nanokelvin

This is ~1010 times colder than the CMB! Such a black hole cannot currently evaporate.

Problem 2: Evaporation Time

How long would it take for a solar-mass black hole to completely evaporate? Use tevap ≈ (5120πG²M³)/(ℏc⁴).

Solution

tevap ≈ 2.1 × 1067 years

This is ~1057 times the current age of the universe! Only primordial or microscopic black holes can evaporate on cosmologically relevant timescales.

Problem 3: Primordial Black Holes

A primordial black hole with initial mass M0 = 1015 g formed in the early universe. Would it still exist today (tuniverse ≈ 13.8 × 109 years)?

Hint

Calculate tevap for M = 1015 g = 1012 kg.
tevap ≈ (5120πG²M³)/(ℏc⁴) ≈ 2.1 × 1010 years × (M/M☉)³

Solution

tevap ≈ 1010 years, comparable to the age of the universe!
Such primordial black holes would be evaporating now, potentially detectable as gamma-ray bursts.

Problem 4: Black Hole Entropy

Show that the Bekenstein-Hawking entropy SBH = kBA/(4ℓP²) is consistent with the first law dM = THdS for a Schwarzschild black hole.

Hint

Use A = 4πrs² = 16πG²M²/c⁴ and κ = c⁴/(4GM).
Calculate dS/dM and check if dM/dS gives the correct temperature.

Where Hawking Temperature Is Used in PM

This foundational physics appears in the following sections of Principia Metaphysica:

Thermal Time

KMS condition and modular flow at horizons

Read More →

Black Hole Physics

Higher-dimensional black holes and holography

Read More →

Quantum Gravity

Planck scale physics and UV completion

Read More →
Browse All Theory Sections →
← All Foundations KMS Condition →