# Stability and instability of GR solutions ## Refs - more generally see [[0402 Instability of bulk states]] - related: [[0455 Black hole uniqueness theorems]] ## Minkowski - non-linear stability of Minkowski without symmetry - [[ChristodoulouKlainerman1990]] - [[LindbladRodnianski2004]] - etc ## Flat space BHs - linear stability of Schwarzschild in double null gauge - [[DafermosHolzegelRodnianski2019]] - generalised to very slowly rotating Kerr and charged by various later papers - non-linear stability of Schwarzschild for Einstein-scalar field under spherical symmetry - [[Christodoulou1987]] - [[DafermosRodnianski2003]] - etc - non-linear stability of 5d Schwarzschild for Einstein vacuum under Bianchi IX symmetry - [[Holzegel2010]] - non-linear stability of 4d Schwarzschild for Einstein equations under polarised axisymmetry - [[KlainermanSzeffel2018]] - first example beyond effective 1+1 dimensional problem - non-linear stability of Schwarzschild without symmetry - [[DafermosHolzegelRodnianskiTaylor2021]][](https://arxiv.org/pdf/2104.08222.pdf) ## dS - non-linear stability of dS - [[Friedrich1986]] - non-linear stability for very slowly rotating Kerr-dS - [[HintzVasy2016]] ## AdS - decays are slower (bad for stability) - no decay at all for pure AdS but logarithmic decay for BH (still very slow) - pure AdS is non-linearly unstable as a solution to Einstein-Vlasov system and Einstein-scalar system - [[Moschidis2018]][](https://arxiv.org/abs/1812.04268) ## AdS BH - Schwarzschild-AdS - with spherical symmetry: is non-linearly stable - without symmetry: conjectured to be unstable (like Kerr below) - Kerr-AdS is conjectured to be non-linearly *unstable* with reflective boundary conditions at infinity - Holzegel and Smulevici ## Kerr - non-extremal Kerr is expected to be stable - extremal Kerr (and Kerr-Newman) are subject to [[0340 Aretakis instability|Aretakis instabilities]]