Black Hole Thermodynamics
Surface gravity, Hawking radiation, and the four laws of black hole mechanics
The discovery that black holes obey laws formally identical to the laws of thermodynamics is one of the most profound results in theoretical physics. It connects gravity, quantum mechanics, and statistical mechanics, and has driven research in quantum gravity for over five decades.
Surface Gravity
The surface gravity κ of a black hole measures the acceleration of a static observer at the horizon as measured from infinity. For the Schwarzschild black hole, it can be computed from the Killing vector ξμ = (∂/∂t)μ that is null on the horizon:
$$\nabla_\mu(\xi_\nu \xi^\nu) = -2\kappa \, \xi_\mu \quad \text{evaluated on the horizon}$$
For the Schwarzschild metric, computing the norm of the time Killing vector:
$$\xi_\nu\xi^\nu = g_{tt} = -\left(1 - \frac{2M}{r}\right)$$
$$\nabla_r(\xi_\nu\xi^\nu) = \frac{2M}{r^2} \quad \Longrightarrow \quad \kappa = \lim_{r\to 2M} \frac{1}{2}\frac{|g_{tt,r}|}{\sqrt{-g_{tt}\,g^{rr}}}$$
Evaluating this limit gives:
Surface Gravity of Schwarzschild Black Hole
$$\kappa = \frac{1}{4M} = \frac{c^4}{4GM}$$
Inversely proportional to mass: smaller black holes have larger surface gravity
For a solar-mass black hole, κ ≈ 1.5 × 10¹² m/s² — an enormous acceleration. Yet for supermassive black holes like Sagittarius A* (4 million solar masses), κ is relatively mild.
Hawking Temperature
In 1974, Stephen Hawking showed that when quantum field theory is applied in curved spacetime, black holes are not truly black — they emit thermal radiation at a temperature determined by the surface gravity. The Hawking temperature is:
Hawking Temperature
$$T_H = \frac{\hbar\kappa}{2\pi c k_B} = \frac{\hbar c^3}{8\pi G M k_B}$$
In geometrized units (G = c = kB = 1):
$$T_H = \frac{\hbar}{8\pi M}$$
The physical intuition comes from the Unruh effect: an accelerating observer in flat spacetime perceives the vacuum as a thermal bath at temperature T = ℏa/(2πc). At the horizon, the surface gravity plays the role of acceleration, yielding the Hawking temperature.
Numerical values:
- Solar mass BH: TH ≈ 6 × 10⁻⁸ K (far below CMB temperature of 2.7 K)
- Earth mass BH: TH ≈ 0.02 K
- Mountain mass BH (10¹² kg): TH ≈ 10¹¹ K (astrophysically relevant for primordial BHs)
A key feature: since T ∝ 1/M, as the black hole radiates it loses mass, gets hotter, radiates faster, gets even hotter — a runaway process. The evaporation time scales as:
$$t_{\text{evap}} \sim \frac{M^3}{\hbar} \sim 10^{67}\left(\frac{M}{M_\odot}\right)^3 \text{ years}$$
Bekenstein-Hawking Entropy
Bekenstein (1973) proposed that black holes carry entropy proportional to their horizon area, based on the analogy with thermodynamics and the area theorem. Hawking's temperature calculation then fixed the proportionality constant exactly:
Bekenstein-Hawking Entropy
$$S_{\text{BH}} = \frac{k_B c^3 A}{4\hbar G} = \frac{A}{4\ell_P^2}$$
where A is the horizon area and ℓP = √(ℏG/c³) ≈ 1.6 × 10⁻³⁵ m is the Planck length
For a Schwarzschild black hole, the horizon area is A = 4π(2M)² = 16πM², giving:
$$S_{\text{BH}} = \frac{4\pi M^2}{\hbar} = 4\pi G M^2 / (\hbar c) \quad \text{(in SI)}$$
For a solar-mass black hole, SBH ≈ 10⁷⁷ kB — enormously larger than the entropy of the Sun (≈ 10⁵⁸ kB). Black holes are the highest-entropy objects in the universe at any given mass scale. This suggests the number of microstates is eS ≈ e10⁷⁷, a number beyond comprehension.
The Four Laws of Black Hole Mechanics
Bardeen, Carter, and Hawking (1973) established four laws governing black hole mechanics that are exact analogs of the four laws of thermodynamics:
Zeroth Law
The surface gravity κ is constant over the event horizon of a stationary black hole.
$$\kappa = \text{const on } \mathcal{H}^+ \qquad \longleftrightarrow \qquad T = \text{const in thermal equilibrium}$$
First Law
For perturbations between stationary black hole states:
$$dM = \frac{\kappa}{8\pi}\,dA + \Omega_H\,dJ + \Phi_H\,dQ$$
$$\longleftrightarrow \qquad dE = T\,dS + \text{work terms}$$
Second Law (Area Theorem)
In any classical process, the total horizon area never decreases (Hawking's area theorem, 1971):
$$\delta A \geq 0 \qquad \longleftrightarrow \qquad \delta S \geq 0$$
The generalized second law (including Hawking radiation): the total entropy SBH + Smatternever decreases.
Third Law
It is impossible to reduce the surface gravity κ to zero in a finite number of physical operations.
$$\kappa \to 0 \text{ is unattainable} \qquad \longleftrightarrow \qquad T \to 0 \text{ is unattainable (Nernst)}$$
This corresponds to the impossibility of creating an extremal (κ = 0) black hole from a non-extremal one.
The Information Paradox
Hawking radiation creates a profound tension between general relativity and quantum mechanics. The paradox arises from two seemingly contradictory facts:
1. Hawking radiation is thermal: The emitted radiation has a precisely thermal spectrum characterized only by temperature (and hence only by mass). It carries no information about what fell into the black hole.
2. Unitarity of quantum mechanics: In quantum mechanics, time evolution is unitary — information is never truly lost. A pure quantum state must evolve to a pure state.
3. Complete evaporation: If the black hole evaporates entirely (as Hawking's calculation suggests), the original pure state that formed the black hole has evolved into a mixed thermal state — violating unitarity.
The resolution of this paradox is believed to require a full theory of quantum gravity. Leading proposals include:
Page Curve
The entropy of Hawking radiation should follow the Page curve: initially increasing (as thermal radiation is emitted), then decreasing after the Page time (≈ half the evaporation time) as correlations build up and information is recovered.
Island Formula
Recent work using the "island formula" for entanglement entropy has shown that gravitational path integrals naturally reproduce the Page curve, suggesting information is recovered via quantum gravitational effects (quantum extremal surfaces).
Key Concepts Summary — Schwarzschild Solution
Page 1: Derivation
- Spherically symmetric vacuum ansatz
- Solving Rμν = 0 gives unique solution
- Birkhoff's theorem: static is automatic
- Coordinate singularity (r = 2M) vs true singularity (r = 0)
- Kretschner scalar K = 48M²/r⁶
Page 2: Geodesics
- Killing vectors → conserved E and L
- Effective potential with -ML²/r³ correction
- ISCO at r = 6M (5.72% efficiency)
- Photon sphere at r = 3M
- Mercury precession: 43"/century
Page 3: Classical Tests
- Light deflection: Δφ = 4GM/(c²b) = 1.75"
- Shapiro time delay: confirmed to 0.1%
- Gravitational redshift: z = GM/(c²r)
- Eddington-Finkelstein coordinates
- Kruskal-Szekeres maximal extension (4 regions)
Page 4: Thermodynamics
- Surface gravity: κ = 1/(4M)
- Hawking temperature: T = ℏ/(8πM)
- Bekenstein-Hawking entropy: S = A/(4ℏG)
- Four laws of BH mechanics
- Information paradox and Page curve