Black hole thermodynamics

In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.^[1]

An artist depiction of two black holes merging, a process in which the laws of thermodynamics are upheld.

Black hole

The only way to satisfy the second law of thermodynamics is to admit that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.

Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. Bekenstein suggested $(1/2\cdot\ln{2})/4\pi$ as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year, Hawking showed that black holes emit thermal Hawking radiation^[2]^[3] corresponding to a certain temperature (Hawking temperature).^[4]^[5] Using the thermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at $1/4$ :^[6]

S_{\text{BH}} = \frac{kA}{4\ell_{\mathrm{P}}^2}

where $A$ is the area of the event horizon, calculated at $4\pi R^2$ , $k$ is Boltzmann's constant, and $\ell_{\mathrm{P}}=\sqrt{G\hbar / c^3}$ is the Planck length. This is often referred to as the Bekenstein–Hawking formula. The subscript BH either stands for "black hole" or "Bekenstein-Hawking". The black hole entropy is proportional to the area of its event horizon $A$ . The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound (wherein the Bekenstein bound becomes an equality) was the main observation that led to the holographic principle.^[1]

Although Hawking's calculations gave further thermodynamic evidence for black hole entropy, until 1995 no one was able to make a controlled calculation of black hole entropy based on statistical mechanics, which associates entropy with a large number of microstates. In fact, so called "no hair"^[7] theorems appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated ^[8] the right Bekenstein-Hawking entropy of a supersymmetric black hole in string theory, using methods based on D-branes and string duality. Their calculation was followed by many similar computations of entropy of large classes of other extremal and near-extremal black holes, and the result always agreed with the Bekenstein-Hawking formula. However, for the Schwarzschild black hole, viewed as the most far-from-extremal black hole, the relationship between micro and macro-states is expected to be clarified from the string theoretical viewpoint. Various studies are in progress, but this has not yet been elucidated.

In Loop quantum gravity (LQG)^[9] it is possible to associate a geometrical interpretation to the microstates: these are the quantum geometries of the horizon. LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon.^[10]^[11] It is possible to derive, from the covariant formulation of full quantum theory (Spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy.^[12] The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes. There seems to be also discussed the calculation of Bekenstein-Hawking entropy from the point of view of LQG.

The laws of black hole mechanics

The four laws of black hole mechanics are physical properties that black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawking, and James Bardeen.

Statement of the laws

The laws of black hole mechanics are expressed in geometrized units.

The Zeroth Law

The horizon has constant surface gravity for a stationary black hole.

The First Law

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

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

where $E$ is the energy, $\displaystyle \kappa$ is the surface gravity, $A$ is the horizon area, $\Omega$ is the angular velocity, $J$ is the angular momentum, $\Phi$ is the electrostatic potential and $Q$ is the electric charge.

The Second Law

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

\frac{dA}{dt} \geq 0.

This "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time.

The Third Law

It is not possible to form a black hole with vanishing surface gravity. $\displaystyle \kappa$ = 0 is not possible to achieve.