Hawking Radiation

Prerequisites: Black Hole, Canonical Quantum Field Theory. Full path (31 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

see also black_hole

Intuitive

In classical general relativity a black hole is perfectly black: nothing that crosses the horizon comes out again. In 1974 Stephen Hawking showed that this changes once quantum field theory is taken into account. A black hole emits radiation with a thermal spectrum, as if it were a hot body with a definite temperature.

The reason lies in what "empty space" means in quantum field theory. The vacuum is a state of the quantum fields, and whether a state contains particles depends on which modes one uses to count them. Near the horizon, waves that end up reaching a distant observer have been stretched by an enormous gravitational redshift. When we follow them back in time they come from extremely short wavelengths very close to the horizon. A state that looks empty to an observer falling freely through the horizon then looks to an observer far away like a steady flux of thermal particles.

A popular picture says that pairs of virtual particles appear near the horizon, one partner falls in and the other escapes. Hawking used this picture himself as a heuristic, but the actual calculation does not track individual pairs.

The temperature is inversely proportional to the mass, so heavy black holes are cold and light ones are hot. A black hole with the mass of the sun has a temperature far below that of the cosmic microwave background, so today it absorbs more radiation than it emits. Since the radiation carries away energy, an isolated black hole loses mass, gets hotter and radiates faster. This suggests that black holes eventually evaporate, and it raises the question of what happens to the information about what fell in, the black hole information problem.

Concrete

$$T= \frac{\hbar c^3}{8 \pi G M k_B} ,$$

where $k_B$ is the Boltzmann constant, $c$ the speed of light, $G$ the gravitational constant, $\hbar$ the reduced Planck constant and $M$ the mass of the black hole.

The temperature of a black hole is tiny. Putting in the numbers yields

$$T= 6.169 \cdot 10^{-8} \text{ K } \ \frac{M_\odot }{M},$$ where $M_\odot$ is the mass of the sun. In words this means that black hole with a mass equal to the mass of our sun would have a temperature of only about $6 \cdot 10^{-8}$ K. If the black hole is heavier, the temperature gets even tinier.

• For a nice explicit discussion of the question "Where does Hawking radiation originate?", see Hawking radiation, the Stefan-Boltzmann law, and unitarization by Steven B. Giddings

Abstract

Hawking radiation is a result of quantum field theory on a fixed curved background spacetime. In this section $\hbar = c = k_B = 1$ while $G$ is kept explicit.

A free field on a globally hyperbolic spacetime has no preferred vacuum unless there is a timelike Killing vector. For a black hole formed by collapse we expand the field in two complete sets of modes: modes $f_\omega$ that are positive frequency with respect to the Killing time at past null infinity $\mathscr I^-$, and modes $p_\omega$ that are positive frequency at future null infinity $\mathscr I^+$. They are related by a Bogoliubov transformation

$$p_\omega = \int d\omega' \left( \alpha_{\omega\omega'} f_{\omega'} + \beta_{\omega\omega'} f^*_{\omega'} \right) .$$

The field starts in the vacuum defined by the $f_\omega$. An outgoing late-time mode, traced backwards, passes just outside the horizon, where its phase is exponentially redshifted: the advanced time $v$ at $\mathscr I^-$ at which the mode started is related to its retarded time $u$ at $\mathscr I^+$ by $v_0 - v \propto e^{-\kappa u}$, where $v_0$ is the last advanced time from which light still escapes and $\kappa$ is the surface gravity. This exponential relation gives $|\alpha_{\omega\omega'}| = e^{\pi\omega/\kappa}\, |\beta_{\omega\omega'}|$, and the normalization of the modes then fixes the expected number of particles reaching $\mathscr I^+$ in the mode $\omega$ to

$$\langle N_\omega \rangle = \frac{\Gamma_\omega}{e^{2\pi\omega/\kappa} - 1} .$$

This is a Planck spectrum at the Hawking temperature

$$T_H = \frac{\kappa}{2\pi} ,$$

multiplied by the greybody factor $\Gamma_\omega$, the probability that a wave of that frequency (and angular momentum) passes the potential barrier outside the horizon. For Schwarzschild $\kappa = 1/(4GM)$, which reproduces $T = 1/(8\pi G M)$ from the Concrete section. The late-time result is independent of the details of the collapse.

The same temperature appears in other ways:

• Unruh effect. By the Bisognano and Wichmann theorem, the Minkowski vacuum restricted to a Rindler wedge is a KMS state at inverse temperature $2\pi$ with respect to the boost generator. An observer with proper acceleration $a$ therefore sees a thermal bath at $T = a/2\pi$. Near a non-extremal horizon the geometry is approximately Rindler, and redshifting the local temperature to infinity gives $\kappa/2\pi$.
• Euclidean geometry. Continuing $t \to -i\tau$ in the Schwarzschild metric, the Euclidean section is smooth at the horizon only if $\tau$ is periodic with period $\beta = 2\pi/\kappa = 8\pi G M$. Field theory on this geometry gives thermal correlation functions at that temperature, the Hartle and Hawking state.

Together with the first law of black hole mechanics, $dM = \frac{\kappa}{8\pi G}\, dA + \ldots$, the identification $T = \kappa/2\pi$ fixes the Bekenstein and Hawking entropy $S = A/4G$. For Schwarzschild $S = 4\pi G M^2$ and indeed $dS/dM = 8\pi G M = 1/T$.

The derivation uses modes of arbitrarily high frequency near the horizon (the trans-Planckian problem) and neglects backreaction. Including backreaction semiclassically, the mass decreases with $dM/dt \propto -1/(G^2 M^2)$, so the evaporation time scales as $G^2 M^3$. Whether the final state of evaporation is pure is the information problem, which semiclassical gravity alone does not settle.

Why is it interesting?

This formula for the Hawking radiation shows why black holes are so important and interesting. In this little formula everything comes together:

• Quantum mechanics, in the form of $\hbar$
• Gravity, in the form of $G$

It tells us that black holes are laboratories for quantum gravity.

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Hilbert Space
7. Legendre Transformation
8. Manifold
9. Newton's Second Law
10. Vector Calculus
11. Curvature
12. Fiber Bundles
13. Newton's law of Gravity
14. Newtonian Mechanics
15. Parallel Transport
16. Variational Calculus
17. Configuration Space
18. Connections
19. Euler-Lagrange Equations
20. Special Relativity
21. Lagrangian Formalism
22. Levi-Civita Connection
23. Phase Space
24. Classical Field Theory
25. Einstein Equation
26. Hamiltonian Formalism
27. General Relativity
28. Schrödinger Equation
29. Black Hole
30. Canonical Quantum Mechanics
31. Canonical Quantum Field Theory