Laws of black hole mechanics

The four laws of black hole mechanics are analogous to the laws of thermodynamics.

Statement of the Laws

The Zeroth Law

The horizon has constant surface gravity $\kappa$.

The First Law

We have

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

where $M$ is the mass, $A$ 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,

$dA\geq 0$.

The Third Law

It is not possible to form a black hole with vanishing surface gravity

Discussion of the Laws

The Zeroth Law

The zeroth law is analagous to the zeroth law of thermodynamics which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the surface gravity is analogous to temperature.

The First Law

The left hand side, $dM$, is the change in mass/energy. Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right hand side represent changes in energy due to rotation and electromagnetism. Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right hand side the term $TdS$.

The Second Law

The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the entropy of a closed system is a non-decreasing function of time, suggesting a link between entropy and the area of a black hole horizon.

The Third Law

Extremal black holes have vanishing surface gravity. The third law is analogous to the third law of thermodynamics which, in its weak formulation, states that it is impossible to reach absolute zero temperature in a physical process. The strong version of the third law of thermodynamics, which states that as the temperature approaches zero, the entropy also approaches zero, does not have an analogue for black holes.

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 one only considers black holes classically, then they have zero temperature and, by the no hair theorem, infinite entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation) at temperature

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

From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein-Hawking entropy which is

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

References

• J. M. Bardeen, B. Carter and S. W. Hawking, "The four laws of black hole mechanics", Commun. Math. Phys. 31, 161 (1973).
• J. D. Bekenstein, "Black holes and entropy", Phys. Rev. D 7, 2333 (1973).
• S. W. Hawking, "Black hole explosions?", Nature 248, 30 (1974).
• S. W. Hawking, "Particle creation by black holes", Commun. Math. Phys. 43, 199 (1975).
• S. W. Hawking and G. F. R. Ellis, "The large-scale structure of space-time", Cambridge University Press (1973).