# Gravitational Perturbations in a cavity: Nonlinearities

## TL;DR

This paper investigates nonlinear gravitational perturbations in a Minkowski spacetime with a spherical boundary, deriving simplified equations and boundary conditions, and discusses implications for nonlinear stability and potential extensions to anti-de Sitter space.

## Contribution

It provides a systematic approach to analyze nonlinear gravitational perturbations with a spherical boundary, including simplified equations and boundary conditions at arbitrary order.

## Key findings

- Derived simplified nonlinear perturbation equations for tensor, vector, scalar sectors.
- Highlighted the importance of variable choice for boundary condition simplification.
- Discussed potential for extending methods to anti-de Sitter spacetime.

## Abstract

Motivated by recent studies of nonlinear perturbations of asymptotically anti-de Sitter spacetimes, we study gravitational perturbations of $(n+2)$ dimensional Minkowski spacetime with a spherical Dirichlet wall. By considering the tensor, vector and scalar perturbations on the $n$ sphere, we present simplified nonhomogeneous equations at arbitrary order in weakly nonlinear perturbation theory for each sector. A suitable choice of perturbative variables is required at higher orders to simplify the expression for the boundary conditions and to expand the variables in terms of linear order eigenfunctions. Finally we comment on the nonlinear stability of the system. Some of the tools used can easily be generalized to study nonlinear perturbations of anti-de Sitter spacetime.