Table of Contents
- Introduction
- Classical Black Holes and the Laws of Mechanics
- The Zeroth Law
- The First Law
- The Second Law
- The Third Law
- Bekenstein’s Entropy
- Hawking Radiation
- Temperature of a Black Hole
- Black Hole Entropy Formula
- Area Law for Entropy
- Generalized Second Law
- Statistical Interpretation of Entropy
- Entropy from Quantum Gravity
- Loop Quantum Gravity and Entropy
- String Theory Microstates
- Black Hole Evaporation
- Information Paradox
- Firewall Hypothesis
- Black Hole Complementarity
- Page Curve and Entanglement Entropy
- Holographic Principle
- AdS/CFT and Black Hole Thermodynamics
- Observational Implications
- Conclusion
1. Introduction
Black hole thermodynamics is the study of black holes using the laws of thermodynamics. Surprisingly, black holes exhibit temperature, entropy, and obey laws that closely mirror classical thermodynamic laws. This profound connection provides a bridge between general relativity, quantum mechanics, and statistical physics.
2. Classical Black Holes and the Laws of Mechanics
The laws of black hole mechanics, discovered in the 1970s, mirror the laws of thermodynamics:
- Surface gravity \( \kappa \): analogous to temperature
- Horizon area \( A \): analogous to entropy
- Mass \( M \): analogous to energy
3. The Zeroth Law
Zeroth Law of Black Hole Mechanics:
The surface gravity \( \kappa \) is constant over the event horizon of a stationary black hole.
Analogous to the thermodynamic zeroth law: temperature is uniform at equilibrium.
4. The First Law
\[
dM = \frac{\kappa}{8\pi} dA + \Omega dJ + \Phi dQ
\]
- \( M \): mass
- \( A \): horizon area
- \( J \): angular momentum
- \( Q \): charge
- \( \Omega \), \( \Phi \): horizon angular velocity, electric potential
Matches the first law of thermodynamics: \( dE = TdS + \dots \)
5. The Second Law
Hawking’s Area Theorem:
The area of a classical black hole horizon never decreases:
\[
\frac{dA}{dt} \geq 0
\]
Corresponds to the second law: entropy increases.
6. The Third Law
It is impossible to reduce the surface gravity \( \kappa \) to zero in a finite number of steps.
Analogous to: \( T \to 0 \) cannot be reached by finite processes.
7. Bekenstein’s Entropy
Jacob Bekenstein proposed that black holes carry entropy proportional to their area:
\[
S \propto A
\]
This made the analogy between thermodynamics and black holes more than formal.
8. Hawking Radiation
Stephen Hawking showed quantum effects lead black holes to emit thermal radiation:
\[
T_H = \frac{\hbar \kappa}{2\pi c k_B}
\]
This gives black holes a real temperature and confirms Bekenstein’s idea.
9. Temperature of a Black Hole
For Schwarzschild black holes:
\[
T_H = \frac{\hbar c^3}{8\pi G M k_B}
\]
As mass decreases, temperature increases — black holes get hotter as they evaporate.
10. Black Hole Entropy Formula
\[
S = \frac{k_B A}{4 \ell_P^2}
\]
- \( A \): event horizon area
- \( \ell_P = \sqrt{\frac{\hbar G}{c^3}} \): Planck length
This is known as the Bekenstein–Hawking entropy.
11. Area Law for Entropy
Entropy is proportional to area, not volume — a deep insight suggesting that the information content of a region resides on its boundary.
12. Generalized Second Law
The total entropy of matter + black hole does not decrease:
\[
\frac{d}{dt} (S_{\text{matter}} + S_{\text{BH}}) \geq 0
\]
This generalizes the second law to include gravitational entropy.
13. Statistical Interpretation of Entropy
The formula \( S = k_B \log \Omega \) begs the question: what are the microstates \( \Omega \) of a black hole?
Various approaches attempt to answer this using quantum gravity.
14. Entropy from Quantum Gravity
A quantum theory of gravity should explain black hole entropy microscopically. Two major frameworks:
- Loop Quantum Gravity
- String Theory
15. Loop Quantum Gravity and Entropy
In LQG, black holes are modeled using isolated horizons. Counting spin network punctures on the horizon yields:
\[
S = \frac{k_B A}{4 \ell_P^2}
\]
after fixing the Immirzi parameter.
16. String Theory Microstates
Strominger and Vafa showed that counting D-brane configurations in string theory matches the Bekenstein–Hawking entropy for extremal black holes.
17. Black Hole Evaporation
Due to Hawking radiation, black holes lose mass and eventually evaporate completely, unless new physics intervenes.
18. Information Paradox
Evaporation produces thermal radiation — seemingly uncorrelated with the initial state. Does information disappear?
This violates unitarity in quantum mechanics, leading to the black hole information paradox.
19. Firewall Hypothesis
One radical proposal is the firewall: a high-energy surface at the event horizon that destroys infalling information. Controversial and debated.
20. Black Hole Complementarity
Suggests:
- No single observer sees a paradox
- Information is both reflected and absorbed
- No contradiction due to causal limitations
An attempt to resolve the information loss paradox.
21. Page Curve and Entanglement Entropy
Don Page proposed the Page curve: entropy of Hawking radiation rises and then falls, consistent with unitary evolution. Recent work using replica wormholes supports this view.
22. Holographic Principle
The area law inspires the holographic principle: all information in a volume can be encoded on its boundary. Central to modern theories like AdS/CFT.
23. AdS/CFT and Black Hole Thermodynamics
Black holes in AdS space are dual to thermal states in CFT. This duality provides a unitary description of black hole dynamics, including entropy and evaporation.
24. Observational Implications
While Hawking radiation is too weak to observe for astrophysical black holes, analog systems (e.g., sonic black holes) may provide indirect evidence.
25. Conclusion
Black hole thermodynamics reveals deep connections between gravity, thermodynamics, and quantum mechanics. The laws governing black holes echo those of entropy and temperature, while quantum effects reveal a rich structure behind the classical event horizon. As a testing ground for quantum gravity, black holes remain central to our quest for a unified theory of the fundamental forces.
