Non-Linear N-Parameter Spacetime Perturbations: Gauge Transformations

TL;DR

This paper develops a new N-parameter perturbation theory for non-linear relativistic phenomena, utilizing the Baker-Campbell-Hausdorff formula to ensure consistency and gauge invariance at any order.

Contribution

It introduces a novel formulation of perturbation theory using exponential formulas, providing a systematic way to handle gauge transformations in non-linear spacetime perturbations.

Findings

Proves the consistency of the perturbation scheme at all orders

Derives gauge transformation rules using exponential generating functions

Establishes connections with previous perturbation approaches

Abstract

We introduce N-parameter perturbation theory as a new tool for the study of non-linear relativistic phenomena. The main ingredient in this formulation is the use of the Baker-Campbell-Hausdorff formula. The associated machinery allows us to prove the main results concerning the consistency of the scheme to any perturbative order. Gauge transformations and conditions for gauge invariance at any required order can then be derived from a generating exponential formula via a simple Taylor expansion. We outline the relation between our novel formulation and previous developments.