# Linearization Instability for Generic Gravity in AdS

## TL;DR

This paper investigates linearization instability in modified gravity theories with AdS backgrounds, revealing that some such theories exhibit instability even with non-compact Cauchy surfaces, unlike in general relativity.

## Contribution

It demonstrates that certain modified gravity theories have linearization instability in AdS backgrounds, expanding understanding beyond Einstein's theory.

## Key findings

- Some modified gravity theories show linearization instability in AdS backgrounds.
- In contrast to general relativity, instability can occur even with non-compact Cauchy surfaces.
- This explains the paradoxical behavior of conserved charges in these theories.

## Abstract

In general relativity, perturbation theory about a background solution fails if the background spacetime has a Killing symmetry and a compact spacelike Cauchy surface. This failure, dubbed as {\it linearization instability}, shows itself as non-integrability of the perturbative infinitesimal deformation to a finite deformation of the background. Namely, the linearized field equations have spurious solutions which cannot be obtained from the linearization of exact solutions. In practice, one can show the failure of the linear perturbation theory by showing that a certain quadratic (integral) constraint on the linearized solutions is not satisfied. For non-compact Cauchy surfaces, the situation is different and for example, Minkowski space having a non-compact Cauchy surface, is linearization stable. Here we study, the linearization instability in generic metric theories of gravity where Einstein's theory is modified with additional curvature terms. We show that, unlike the case of general relativity, for modified theories even in the non-compact Cauchy surface cases, there are some theories which show linearization instability about their anti-de Sitter backgrounds. Recent $D$ dimensional critical and three dimensional chiral gravity theories are two such examples. This observation sheds light on the paradoxical behavior of vanishing conserved charges (mass, angular momenta) for non-vacuum solutions, such as black holes, in these theories.