# Black hole second law Black holes have [[0004 Black hole entropy|entropy]]. The second law of thermodynamics says that entropy should not decrease in time. In [[0554 Einstein gravity|General Relativity]], the entropy of a black hole is proportional to its area, and the area of a black hole cannot decrease (classically). Even when two black holes merge, the total area must increase. This means that black holes in classical General Relativity satisfy the second law of thermodynamics. This is the famous black hole second law, or area-increasing theorem. Semiclassically, black holes can radiate, so the area-increasing theorem can be violated. Nevertheless, we have an improved version of the second law called the [[0082 Generalised second law|generalised second law]], which says that the combined entropy of the black hole and the quantum fields outside the black hole cannot decrease. This statement depends on the type of the matter field, but it has been established for certain types. Another way the classical second law can break down is when the theory is not just GR. For example, in [[0425 Gauss-Bonnet gravity|GB gravity]], black hole entropy can decrease in the event of a merger. The best we can do in this case is to study perturbations to a stationary black hole. Then at linearised or even second order, the black hole still satisfy a second law for a very general class of theories, including arbitrary [[0006 Higher-derivative gravity|higher-curvature gravity]]. ## Refs - concept introduced in Bekenstein1974 - higher-derivative gravity, linear order in perturbation to a static black hole - [[JacobsonKangMyers1995]]: $R^2$ - [[KolekarPadmanabhanSarkar2012]]: JM satisfy 2nd law for linear perturbations - [[SarkarWall2010]]: for non-linear perturbations, even JM does not satisfy second law - [[2013#Sarkar, Wall]]: f(Lovelock) - [[2015#Bhattacharjee, Sarkar, Wall]]: quadratic gravity - [[2015#Wall (Essay)]]: general argument - [[2016#Bhattacharyya, Haehl, Kundu, Loganayagam, Rangamani]]: Lovelock but higher order in EFT length scale - [[2019#Bhattacharya, Bhattacharyya, Dinda, Kundu]]: local entropy current - [[WangJiang2020]][](https://arxiv.org/pdf/2008.09774.pdf): non-linearly coupled scalar fields ([[0346 Horndeski gravity|Horndeski]]) - [[WangJiang2021]][](https://arxiv.org/pdf/2108.04402.pdf): with quadric gauge fields - [[2022#Biswas, Dhivakar, Kundu]]: with general non-minimally coupled gauge fields - [[WangJiang2022]][](https://arxiv.org/pdf/2211.06882.pdf): $\mathcal{L}=\mathcal{L}\left(g_{a b}, R_{a b c d}, F_{a b}, \phi, \nabla_a \phi, \nabla_a \nabla_b \phi\right)$ - [[2023#Dhivakar, Jalan]]: with non-minimally coupled matter - [[2024#Wall, Yan]]: with vector fields - [[2023#Deo, Dhivakar, Kundu]]: Chern-Simons gravity - higher-derivative gravity, second order in perturbation - [[2022#Hollands, Kovacs, Reall]] - [[2022#Davies, Reall]] - higher-derivative gravity, non-perturbative (within the regime of EFT) - [[2023#Davies, Reall]] - higher-spin - [[2024#Yan]] - rephrasing in the [[0415 Von Neumann algebra|vN algebra]] framework - [[2023#Ali, Suneeta]] - an integrated version - [[2024#Hollands, Wald, Zhang]] - relation of the integrated version to [[0226 Apparent horizon|apparent horizons]] and [[0298 MOTS|MOTS]] - [[2024#Visser, Yan]] - [[2025#Furugori, Nishii, Yoshida, Yoshimura]] - properties - [[2022#Bhattacharyya, Jethwani, Patra, Roy]]: reparametrisation symmetry - [[2024#Kar, Dhivakar, Roy, Panda, Shaikh]]: gauge covariance - other generalisations - [[2024#Bernamonti, Galli, Myers, Reyes]]: to [[0293 Renyi entropy|Renyi entropy]] ## Higher-derivative gravity It is curious how everything fits nicely together in General Relativity. One may then ask how general this is: is General Relativity special? After all, String Theory says the low energy effective field theory of graviton is not just General Relativity, but General Relativity with an infinite number of [[0006 Higher-derivative gravity|higher-derivative corrections]]. Therefore, it is of value to understand whether the existence of a second law is unique to special theories like General Relativity, or perhaps more universal. Some references on this are given below. ## Effects of quantum - Hawking radiation - Loop corrections renormalise the gravitational action ![[A0010 Firey hole.jpeg]]