Marconi”, Università di Roma ”La Sapienza” and Sezione INFN ROMA 1, piazzale Aldo Moro 2, I–00185 Roma, Italy'
author:
- 'Marco Bruni Leonardo Gualtieri§ and Carlos F. Sopuerta'
title: 'Two–parameter non–linear spacetime perturbations: gauge transformations and gauge invariance'
---

Introduction
============

An implicit fundamental assumption in relativistic perturbation theory is that there exists a parametric family of spacetimes such that the perturbative formalism is built as a Taylor expansion of this family around a background. The perturbations are then defined as the derivative terms of this series, evaluated on this background [@wald]. In most cases of interest one deals with an expansion in a single parameter, which can either be a formal one, as in cosmology [@BMMS; @MMB; @BMT] or in the study of quasi–normal modes of stars and black holes [@chandrabook; @kokkomodes], or can have a specific physical meaning, as in the study of binary black hole mergers via the close limit approximation [@price; @gleisclose], or in the study of quasi–normal mode excitation by a physical source (see [@kokkomodes; @my123] and references therein). Typically the perturbative expansion stops at the first order, but recent interesting developments deal with second order perturbations [@BMMS; @MMB; @BMT; @cl; @gleiser].

In some physical 