Black hole thermodynamics

In physics, black hole thermodynamics is a set of physical relationships between the properties of black holes that stands in direct relationship to classical laws of thermodynamics. The equivalence is developed by replacing entropy with black hole horizon area and replacing temperature with black hole horizon surface gravity. Having temperature implies that a black hole must emit radiation, that is, Hawking radiation. though Hawking's area law has already been tested by analyzing gravitational waves.

History

In 1972, Jacob Bekenstein conjectured that black holes should have an entropy proportional to the area of the event horizon, where by the same year, he proposed the no-hair theorem. In 1973 Bekenstein heuristically suggested <math>\frac{\ln{2{8\ \!\pi}\approx 0.0276</math> as the constant of proportionality. The next year, in 1974, Stephen Hawking showed that black holes emit thermal Hawking radiation corresponding to a certain temperature (Hawking temperature). Using the thermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at <math>1/4</math>:

where <math>A</math> is the area of the event horizon, <math>k_\text{B}</math> is the Boltzmann constant, and <math>\ell_\text{P} = \sqrt{G\hbar / c^3}</math> 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 <math>A</math>. 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.

In the early 1990's Gerard 't Hooft and Leonard Susskind generalized the relationship between a black hole's surface area and its entropy, applying to all of spacetime. This was an early form of the holographic principle, a universal relationship between geometry and information. The idea is that the area surrounding any volume in spacetime limits the information content of the volume. Thus the number of degrees of freedom in any volume is bounded and not infinite.

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. Some work, called no-hair theorems, suggested that black holes could have only a single microstate. The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated the 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 furthest-from-extremal black hole, the relationship between micro- and macrostates has not been characterized. Efforts to develop an adequate answer within the framework of string theory continue.

In loop quantum gravity (LQG) it is possible to associate a geometrical interpretation with 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. It is possible to derive, from the covariant formulation of full quantum theory (spin foam), the correct relation between energy and area (first law), the Unruh temperature and the distribution that yields Hawking entropy. 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 current accepted microstate ensemble for black holes is the microcanonical ensemble. The partition function for black holes results in a negative heat capacity. In canonical ensembles, there is limitation for a positive heat capacity, whereas microcanonical ensembles can exist at a negative heat capacity.

Analyses of gravitational waves emitted by GW250114 confirmed Hawking's assertion that the surface area of a black hole is a non-decreasing function of time.

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 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 <math>N</math> nucleons, <math>S_\text{BH} - S > 0</math>, where <math>S_\text{BH}</math> is the entropy of a black hole and <math>S</math> 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. 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.

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."

Notes

References

External links

Bekenstein-Hawking entropy on Scholarpedia
Black Hole Thermodynamics
Black hole entropy on arxiv.org