# Aretakis instability Aretakis proved the existence of instability of massless scalar field at the horizon of extreme Kerr or RN black hole. Then [[2012#Lucietti, Reall]] proved the instability for extremal Kerr due to gravitational perturbations. Aretakis charges are some constants along the horizon. When they are non-zero, the spacetime is non-stationary and certain quantities diverge. In the case of a scalar, the Aretakis is given by $\partial_r \phi$ and the higher derivatives $\partial^n_r \phi$ go like $v^{n-1}$ along the horizon ($v$ being a EF coordinate) and are therefore divergent. Along the radial directions, they all decay, but are related to certain quantities (near infinity), e.g. $t^2 \phi$ at infinity is related to the Aretakis charge. ## Refs - originals: - Aretakis201110a[](http://arxiv.org/abs/1110.2007), Aretakis201110b[](http://arxiv.org/abs/1110.2009), Aretakis201206, Aretakis201212 - general understanding - using a symmetry (weight) argument: [[2018#Gralla, Zimmerman]][](https://arxiv.org/pdf/1804.04753.pdf) - near-extremal - [[2016#Gralla, Zimmerman, Zimmerman]]: near-extremal Kerr - [[2016#Zimmerman]]: near-extremal and the extremal limit of RN, against charged scalar perturbations - extensions (linear) - [[2012#Lucietti, Reall]]: extremal Kerr against gravitational, EM and massless scalar perturbations - [[2012#Lucietti, Murata, Reall, Tanahashi]]: extremal RN, against massive scalar, EM and gravitational perturbations - [[Murata2012]]: higher dimensions - extremal [[0086 Banados-Teitelboim-Zanelli black hole|BTZ]]: [[2019#Gralla, Ravishankar, Zimmerman]][](http://arxiv.org/abs/1911.11164) - 2d AdS - [[2012#Lucietti, Murata, Reall, Tanahashi]] - [[HadarReall2017]][](http://arxiv.org/abs/1709.09668) - [[KatagiriKimura2021]][](http://arxiv.org/abs/2101.03479) - non-linear - [[2013#Murata, Reall, Tanahashi]]: (spherical, scalar) generic instabilities decay non-linearly to non-extremal BHs; but fine-tuned instability keep growing on the horizon, even though the exterior decays - [[Hadar2018]][](http://arxiv.org/abs/1811.01022) - [[AngelopoulosAretakisGajic2019]][](http://arxiv.org/abs/1908.00115) - holography - Kerr - [[2017#Gralla, Zimmerman]] - [[2018#Casals, Zimmerman]] - planar - [[2018#Gralla, Ravishankar, Zimmerman]] - BTZ - [[2019#Gralla, Ravishankar, Zimmerman]] - observation from far distance - [[2020#Burko, Khanna, Sabharwal]] - [[2023#Aretakis, Khanna, Sabharwal]] ## Related - parent [[0339 Stability of GR solutions]] - [[0455 Black hole uniqueness theorems]] - [[0460 Aretakis-NP charge connection]]