Black hole thermodynamics

In 1972, Jacob Bekenstein conjectured that black holes should have an entropy proportional to the area of the event horizon,[3] where by the same year, he proposed no-hair theorems.

[6][7] Using the thermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at

The black hole entropy is proportional to the area of its event horizon

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.

[2] This area relationship was generalized to arbitrary regions via the Ryu–Takayanagi formula, which relates the entanglement entropy of a boundary conformal field theory to a specific surface in its dual gravitational theory.

In fact, so called "no-hair" theorems[11] appeared to suggest that black holes could have only a single microstate.

The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated[12] 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.

Efforts to develop an adequate answer within the framework of string theory continue.

LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon.

[13][14] 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.

[15] 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 loop quantum gravity.

The partition function for black holes results in a negative heat capacity.

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

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

The horizon area is, assuming the weak energy condition, a non-decreasing function of time: This "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time.

Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right side the term

Analogously, the second law of thermodynamics states that the change in entropy in an isolated system will be greater than or equal to 0 for a spontaneous process, suggesting a link between entropy and the area of a black hole horizon.

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.

Because a black hole formation is not stationary, but instead moving, proving that the GSL holds is difficult.

[20] Specific counterexamples called extremal black holes fail to obey the rule.

[19]: 10 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.

However, when quantum-mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation) at a temperature From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein–Hawking entropy, which is (in geometrized units) which is the entropy of the black hole in Einstein's general relativity.

[24] While black hole thermodynamics (BHT) has been regarded as one of the deepest clues to a quantum theory of gravity, there remain a philosophical 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".

More fundamentally, Gerard 't Hooft and Leonard Susskind used the laws of black hole thermodynamics to argue for a general holographic principle of nature, which asserts that consistent theories of gravity and quantum mechanics must be lower-dimensional.

Though not yet fully understood in general, the holographic principle is central to theories like the AdS/CFT correspondence.