Laws of Black Hole Mechanics

1. The Four Laws of Black Hole Mechanics

The striking parallels between the properties of stationary black holes and the laws of thermodynamics were first systematically laid out in a foundational 1973 paper by James Bardeen, Brandon Carter, and Stephen Hawking. Initially, this was understood as a powerful analogy, as a classical black hole, being a perfect absorber, should have a physical temperature of absolute zero.

Let's begin by studying the section of this historic paper where the laws are first articulated.

Now, let's break down these laws one by one, supplementing the original statements with modern interpretations and context.

The Second Law: The Area Theorem

The Second Law: In any physical process, the total surface area $A$ of the event horizons of all black holes in the universe can never decrease. $\delta A\ge 0$.

This is Hawking's Area Theorem, which you have encountered before. It provides the analogy between horizon area $A$ and entropy $S$. Just as the total entropy of an isolated system can never decrease, neither can the total horizon area.

A key difference from classical thermodynamics is that area, unlike entropy, cannot be transferred from one black hole to another. Black holes can merge, but they cannot split apart. The theorem guarantees that if two black holes of areas $A_1$ and $A_2$ merge, the final black hole will have an area $A_3\ge A_1+A_2$.

This law elegantly resolves a puzzle related to the Penrose process. While the process extracts energy by decreasing the black hole's mass $M$, it does so by increasing its horizon area (or keeping it constant in the maximally efficient, reversible limit).

The First Law: Conservation of Energy

The First Law: A small change in a stationary black hole's parameters is related by:
$dM=\frac{\kappa }{8\pi G}dA+{\Omega }_{H}dJ+{\Phi }_{H}dQ$

This law connects the change in a black hole's mass $M$ (its total energy) to changes in its area $A$, angular momentum $J$, and electric charge $Q$.

Let's dissect the analogy with the first law of thermodynamics, $dE=TdS-pdV+\mu dN$:

• $dM$ is analogous to the change in energy, $dE$.
• The term $\frac{\kappa }{8\pi G}dA$ is analogous to the heat flow, $TdS$. This reinforces the link between surface gravity $\kappa$ and temperature $T$, and area $A$ and entropy $S$.
• ${\Omega }_{H}dJ$ is a work term, analogous to $-pdV$. Here, ${\Omega }_H$ is the horizon's angular velocity, and the term represents the work done on the black hole by changing its rotation.
• ${\Phi }_{H}dQ$ is another work term, analogous to the chemical potential term $\mu dN$. Here, ${\Phi }_H$ is the electrostatic potential at the horizon.

This law is a precise mathematical statement relating neighboring stationary black hole solutions. It's a statement about equilibrium states, just like its thermodynamic counterpart.

The Third Law: The Unattainability of $\kappa =0$

The Third Law: It is impossible, by any finite sequence of physical operations, to reduce the surface gravity $\kappa$ of a black hole to zero.

This is analogous to the Nernst or "unattainability" formulation of the third law of thermodynamics, which states that it's impossible to cool a system to absolute zero temperature in a finite number of steps.

The physical significance of this law is profound. A black hole with $\kappa =0$ is an extremal black hole. For a Kerr black hole, this occurs when $a=M$ (or $J=M^2$). If you could add just a tiny bit more angular momentum, you would have $J>M^2$, which corresponds to a naked singularity. The third law therefore acts as a form of cosmic censorship, preventing the creation of these causally pathological objects from black holes.

As Jacobson's notes hint, any attempt to "over-spin" or "over-charge" a black hole to reach the extremal state fails in a subtle way. For example, as a particle with the required angular momentum approaches the horizon, it either just misses being captured or is repelled by a spin-spin interaction, preventing the final step to $\kappa =0$.

2. From Analogy to Physics: Bekenstein and Hawking

For years, the "laws of black hole mechanics" were viewed as a tantalizing but formal analogy. The turning point came from a thought experiment by Jacob Bekenstein.

Bekenstein argued that if the second law of thermodynamics is to be universally valid, a black hole must possess an entropy. Otherwise, one could simply throw a hot object (with high entropy) into a black hole, seemingly causing the entropy of the universe to decrease. He proposed the Generalized Second Law (GSL), stating that the total entropy, defined as the sum of the ordinary entropy outside the black hole ($S_{\text{out}}$) and the black hole's own entropy ($S_{\text{BH}}$), must never decrease:
$\delta S_{\text{total}}=\delta (S_{\text{out}}+S_{\text{BH}})\ge 0$
Based on the area theorem, he identified the black hole's entropy with its horizon area: $S_{\text{BH}}\propto A$.

The final, crucial piece of the puzzle was provided by Stephen Hawking. Initially skeptical of Bekenstein's idea, Hawking set out to prove him wrong. Instead, by applying quantum field theory in the curved spacetime of a black hole, he made a revolutionary discovery.

Hawking discovered that black holes radiate particles as if they were a perfect blackbody with a physical temperature:
$T_H=\frac{\hslash c^3}{8\pi G{k}_{B}M}=\frac{\hslash }{2\pi k_{B}c}\kappa$
A black hole has a real, non-zero temperature! This discovery, known as Hawking radiation, elevated the analogy to a physical identity. With a real temperature, one could now use the First Law to definitively calculate the entropy.

Let's see how this is done.

By equating the two forms of the First Law,
$dM=T_{H}d{S}_{\text{BH}}\text{and}dM=\frac{\kappa }{8\pi G}dA$
and substituting $T_H=\frac{\hslash \kappa }{2\pi k_B}$ (in units where c=1), we get:
$\frac{\hslash \kappa }{2\pi k_B}d{S}_{\text{BH}}=\frac{\kappa }{8\pi G}dA$
Solving for $d{S}_{\text{BH}}$ gives:
$d{S}_{\text{BH}}=\frac{k_B}{4G\hslash }dA$
Integrating this yields the celebrated Bekenstein-Hawking entropy formula:
$S_{\text{BH}}=\frac{k_B{c}^3}{4G\hslash }A$
This result is one of the most profound in theoretical physics, linking gravity ($G$), quantum mechanics ($\hslash$), thermodynamics ($k_B$), and relativity ($c$) in a single equation describing the entropy of a black hole. It implies that the information about the matter that formed the black hole is somehow encoded on its two-dimensional surface area, a key insight that leads to the holographic principle.

Conclusion

In this lesson, we have completed the remarkable story of the laws of black hole thermodynamics.

Key Takeaways:

• The properties of stationary black holes are governed by four laws that are in perfect analogy with the laws of thermodynamics.
• The First Law relates changes in mass to changes in area, angular momentum, and charge: $dM=\frac{\kappa }{8\pi G}dA+{\Omega }_{H}dJ+{\Phi }_{H}dQ$.
• The Second Law is Hawking's Area Theorem, stating that the total horizon area can never decrease ($\delta A\ge 0$), making area the analogue of entropy.
• The Third Law states that surface gravity cannot be reduced to zero in a finite number of steps, which is analogous to the unattainability of absolute zero temperature.
• Bekenstein proposed that the analogy was a physical reality and that black holes must carry entropy proportional to their area to save the second law of thermodynamics.
• Hawking's discovery of black hole radiation confirmed this, showing that black holes have a physical temperature $T_H\propto \kappa$.
• Combining these results yields the Bekenstein-Hawking entropy formula, $S_{\text{BH}}=A/(4G\hslash )$, a cornerstone of modern physics that suggests information is stored holographically on the event horizon.

Preview of Next Lesson:

Having established these deep connections between geometry and thermodynamics, we will pivot back to the pure geometry of spacetime. We will derive the Raychaudhuri equation, a fundamental equation describing the evolution of a congruence of geodesics. This powerful tool will then allow us to prove the famous Penrose-Hawking singularity theorems, which show that under very general physical conditions, the formation of singularities in General Relativity is inevitable.