# Towards a theory of nonlinear gravitational waves: a systematic approach to nonlinear gravitational perturbations in vacuum

## TL;DR

This paper develops a systematic method for analyzing nonlinear gravitational waves in vacuum spacetimes, showing that perturbations reduce to scalar wave equations and enabling explicit metric perturbation calculations.

## Contribution

It introduces a robust approach to nonlinear gravitational perturbations, establishing a foundation for a theory of nonlinear gravitational waves in vacuum.

## Key findings

- Perturbative Einstein equations reduce to scalar wave equations at each order.
- Explicit metric perturbations can be obtained from solutions to these scalar equations.
- The concept of gravitational wave polarization extends beyond linear approximation.

## Abstract

We present a systematic and robust approach to nonlinear gravitational perturbations of vacuum spacetimes. This approach provides a basis for a theory of nonlinear gravitational waves. In particular, we show that the system of perturbative Einstein equations reduces at each perturbation order to two (for each gravitational mode in 3 + 1 dimensions on which our study is focused) scalar wave equations, and then we show how the metric perturbations can be explicitly obtained, once the solutions to these scalar wave equations are known. These results show that the concept of polarization of a gravitational wave does make sense also beyond the linear approximation.