# Black hole entropy Black holes have entropy. The entropy of a black hole is given by the Bekenstein-Hawking entropy, $S=\frac{A c^3}{4 G \hbar},$which, very interestingly, is proportional to the *area* of the horizon, $A$. This is very surprising because entropy is an extensive quantity and is proportional to the *volume* for a box a gas. This fact turns out to be a manifestation of the holographic principle, i.e. the information content of the black hole is somehow encoded in some lower-dimensional space. In [[0554 Einstein gravity|General Relativity]], the area of the black hole can only [[0005 Black hole second law|increase]] in time, and a change in the area is related to other quantities such as mass, angular momentum and charge. These can all be summarised as [[0127 Black hole thermodynamics|laws of black hole thermodynamics]], and they look just the same as ordinary thermodynamic laws. It turned out, interestingly, that black holes are in fact thermodynamical objects just like everything else. This realisation was a result of [[0304 Hawking radiation|Hawking's calculation]], which established that black holes radiate with a spectrum dependent on the temperature just like any ordinary object with a temperature. ## Refs - classical BH entropy - [[0005 Black hole second law]] - with matter - [[0082 Generalised second law]] - [[0145 Generalised area|holographic entanglement entropy functional]] - arbitrary region - [[0044 Extended phase space|Extended phase space]] - approach of Casini et al - [[0127 Black hole thermodynamics]] ## Arbitrary region - [[2012#Bianchi, Myers]] - holographic holes - [[BalasubramanianCzechChowdhuryDeBoer2013]][](https://arxiv.org/abs/1305.0856) - entropy of a spherical hole is $A/4$ - [[BalasubramanianChowdhuryCzechDeBoerHeller2013]][](https://arxiv.org/abs/1310.4204) - embedding in AAdS - in higher dimensions - [[MyersRaoSugishita2014]][](https://arxiv.org/abs/1403.3416) ## Methods - Euclidean methods - has $U(1)$ symmetry - solutions changes while keeping interior smooth - uses a background metric - Euclidean method origin: [[GibbonsHawking1977]] - 4 method listed in [[1995#Iyer, Wald]] - Wald method - independent of boundary condition - shown recently by R. B. Mann - have [[0018 JKM ambiguity]] - field redefinition - [[1993#Jacobson, Kang, Myers]] - Wall method for time-dependent black holes - [[2015#Wall (Essay)]] for f(Riem) - extended phase space method - for Einstein [[2016#Donnelly, Freidel]] - higher derivative [[2017#Speranza]] - Kerr/CFT - [[2009#Azeyanagi, Compere, Ogawa, Tachikawa, Terashima]] obtains Wald entropy by choosing boundary term - Dirichlet flux - not done for general higher derivative - [[2020#Chandrasekaran, Speranza]] - boundary Noether current approach - [[MajhiPadmanabhan2012]] - [[Majhi2012]] - t'Hooft S-matrix approach - [[THooft2000]][](https://arxiv.org/abs/hep-th/0003004) - Bianchi's local approach - [[Bianchi2012]][](https://arxiv.org/abs/1211.0522) - [[0357 Imaginary action]] - [[2013#Neiman]] ## BC dependence - should not depend on BC (see e.g. [[2020#Khodabakhshi, Shirzad, Shojai, Mann]]) ## Early development - [[FloydPenrose1971]]: Area always increase in Penrose process [[1969#Penrose]], so maybe this is more general. - [[Hawking1971]]: BH entropy does not decrease in any process. - [[Bekenstein1973]]: introduces Area = entropy. - [[CallanWilczek1994]]: develop a straightforward Hamiltonian approach to geometric entropy and a less direct Euclidean (aka. heat kernel) method - show that geometric entropy is the first quantum correction to a thermodynamic entropy which reduces to the Bekenstein-Hawking entropy in the black hole context - seems to introduce the replica trick ## From string theory - [[2023#Dabholkar, Moitra (Jun)]] and refs therein