Black hole thermodynamics

Laws of classical black hole mechanics

These four laws of black hole mechanics are relationships between physical properties of black holes assuming general relativity but no quantum effects. This form is analogous to the laws of thermodynamics but without quantum effects the black hole temperature must be zero and simply absorbs all radiation. This is the form proposed by Bardeen, Brandon Carter, and Hawking in 1973 (expressed in geometrized units).

Zeroth law

For a stationary black hole (one in mechanical equilibrium) the surface gravity, ${\displaystyle \kappa }$, is constant on the event horizon. This is the analog of the zeroth law of thermodynamics on the equivalence of temperature across a system at equilibrium with surface gravity being analogous to temperature.

First law

For perturbations of stationary black holes, the change of energy is related to change of area, angular momentum, and electric charge by

$${\displaystyle dE=\frac{\kappa }{8\pi }dA+\mathrm{\Omega }dJ+\mathrm{\Phi }dQ,}$$

where ${\displaystyle E}$ is the energy, ${\displaystyle \kappa }$ is the surface gravity, ${\displaystyle A}$ is the horizon area, ${\displaystyle \mathrm{\Omega }}$ is the angular velocity, ${\displaystyle J}$ is the angular momentum, ${\displaystyle \mathrm{\Phi }}$ is the electrostatic potential and ${\displaystyle Q}$ is the electric charge. Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right side a term equal to temperature times change in entropy, ${\displaystyle TdS}$.

Second law

The horizon area is, assuming the weak energy condition, a non-decreasing function of time:

$${\displaystyle \frac{dA}{dt}\ge 0.}$$ Here black hole surface area is analogous to thermodynamic entropy and the laws say these properties can never spontaneously decrease.

Third law

It is not possible to form a black hole with vanishing surface gravity. That is, ${\displaystyle \kappa =0}$ cannot be achieved. This is analogous to some but not all forms of the third law of thermodynamics.

Quantum effects

The generalized second law of thermodynamics (GSL) is needed to present the second law of thermodynamics as valid. This is because the second law of thermodynamics, as a result of the disappearance of entropy near the exterior of black holes, is not useful. The GSL allows for the application of the law because it allows for the measurement of interior, common entropy. The validity of the GSL can be established by studying an example, such as looking at a system having entropy that falls into a bigger, non-moving black hole, and establishing upper and lower entropy bounds for the increase in the black hole entropy and entropy of the system, respectively. The GSL also holds for theories of gravity such as general relativity, Lovelock gravity, or braneworld gravity, because the conditions to use it for these theories can be met.

However, on the topic of black hole formation, the question becomes whether or not the generalized second law of thermodynamics will be valid, and if it is, it will have been proved valid for all situations. Because black hole formation is not a stationary event, proving that the GSL holds is difficult. Proving it is generally valid would require quantum-statistical mechanics, because the GSL is both a quantum and statistical law. This discipline does not exist so the GSL can be assumed to be useful in general, as well as for prediction. For example, one can use the GSL to predict that, for a cold, non-rotating assembly of ${\displaystyle N}$ nucleons, ${\displaystyle S_{\text{BH}}-S>0}$, where ${\displaystyle S_{\text{BH}}}$ is the entropy of a black hole and ${\displaystyle S}$ is the sum of the ordinary entropy.

Third law

The third law of black hole thermodynamics is controversial. Specific counterexamples called extremal black holes fail to obey the rule. The classical third law of thermodynamics, or Nernst's theorem, which says the entropy of a system must go to zero as the temperature goes to absolute zero, is also not a universal law. However, systems that fail the classical third law have not been realized in practice, leading to the suggestion that the extremal black holes may not represent the physics of black holes generally.

A weaker form of the classical third law known as the unattainability principle states that an infinite number of steps are required to put a system into its ground state. This form of the third law does have an analog in black hole physics.

Interpretation of the laws

The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If black holes are only considered classically, then they have zero temperature, and by the no-hair theorem, zero entropy, and the laws of black hole mechanics remain an analogy. However, when quantum-mechanical effects are taken into account, it is found that black holes emit thermal radiation (Hawking radiation) at the Hawking temperature

$${\displaystyle T_{\text{H}}=\frac{\kappa }{2\pi }.}$$

From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein–Hawking entropy, which is (in geometrized units)

$${\displaystyle S_{\text{BH}}=\frac{A}{4}.}$$

which is the entropy of the black hole in Einstein's general relativity. Quantum field theory in curved spacetime can be utilized to calculate the entropy for a black hole in any covariant theory for gravity, known as the Wald entropy.

Critique

While black hole thermodynamics (BHT) has been regarded as one of the deepest clues to a quantum theory of gravity, there remains criticism that "the analogy is not nearly as good as is commonly supposed", that it "is often based on a kind of caricature of thermodynamics" and "it's unclear what the systems in BHT are supposed to be".

Others have come to the opposite conclusion believing, "stationary black holes are not analogous to thermodynamic systems: they are thermodynamic systems, in the fullest sense."