In physics, **black hole thermodynamics**[1] 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 development 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.[2]

The second law of thermodynamics requires 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.

In 1972, Jacob Bekenstein conjectured that black holes should have an entropy,[3] where by the same year, he proposed no hair theorems.

In 1973 Bekenstein suggested ln 2 8 π ≈ 0.276 {\displaystyle {\frac {\ln {2}}{8\pi }}\approx 0.276} as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year, in 1974, Stephen Hawking showed that black holes emit thermal Hawking radiation[4][5] corresponding to a certain temperature (Hawking temperature).[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 1 / 4 {\displaystyle 1/4} :[8][9]

where
A
{\displaystyle A}
is the area of the event horizon,
k
B
{\displaystyle k_{\text{B}}}
is the Boltzmann constant, and
ℓ
P
=
G
ℏ
/
c
3
{\displaystyle \ell _{\text{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
{\displaystyle 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.[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.[10]