Linearization Instability in Gravity Theories

TL;DR

This paper investigates the phenomenon of linearization instability in gravity theories, highlighting cases where linear perturbation solutions do not correspond to actual solutions, with detailed examples including three-dimensional chiral gravity and higher-dimensional critical gravity.

Contribution

It provides a comprehensive analysis of linearization instability in generic gravity theories, introducing methods to identify spurious solutions and illustrating with explicit examples.

Findings

Identification of spurious solutions not arising from exact solutions

Analysis of Taub and ADT charges in the context of linearization instability

Explicit examples in chiral gravity and critical gravity theories

Abstract

In a nonlinear theory, such as gravity, physically relevant solutions are usually hard to find. Therefore, starting from a background exact solution with symmetries, one uses the perturbation theory, which albeit approximately, provides a lot of information regarding a physical solution. But even this approximate information comes with a price: the basic premise of a perturbative solution is that it should be improvable. Namely, by going to higher order perturbation theory, one should be able to improve and better approximate the physical problem or the solution. While this is often the case in many theories and many background solutions, there are important cases where the linear perturbation theory simply fails for various reasons. This issue is well known in the context of general relativity through the works that started in the early 1970s, but it has only been recently studied in modified gravity theories. This thesis is devoted to the study of linearization instability in generic gravity theories where there are spurious solutions to the linearized equations which do not come from the linearization of possible exact solutions. For this purpose we discuss the Taub charges, the ADT charges and the quadratic constraints on the linearized solutions. We give the three dimensional chiral gravity and the $D$ dimensional critical gravity as explicit examples and give a detailed ADM analysis of the topologically massive gravity with a cosmological constant.