Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations
Kouji Nakamura
TL;DR
This work presents a systematic method to construct gauge-invariant variables for two-parameter nonlinear perturbations in general relativity, given a linear-order gauge-invariant framework. By introducing a canonical Taylor expansion of two-parameter diffeomorphisms with generators $\xi_{(p,q)}^{a}$ and operators $\mathcal{L}_{(p,q)}$, the authors show how to define order-by-order gauge-invariant metric perturbations ${}^{(p,q)}\mathcal{H}_{ab}$ and corresponding gauge-variant pieces ${}^{(p,q)}X_a$, enabling a linear-like gauge-invariant extraction up to third order. They extend the procedure to matter perturbations, defining gauge-invariant perturbations $\delta^{(p,q)}\mathcal Q$ for arbitrary fields. The results are applicable to general covariant theories and provide a robust framework for analyzing complex systems, such as rotating relativistic stars, where gauge ambiguities at higher orders previously hindered precise physical interpretation. The paper also clarifies the equivalence with Bruni et al.'s representation, ensuring consistency across formalisms.
Abstract
The procedure to find gauge invariant variables for two-parameter nonlinear perturbations in general relativity is considered. For each order metric perturbation, we define the variable which is defined by the appropriate combination with lower order metric perturbations. Under the gauge transformation, this variable is transformed in the manner similar to the gauge transformation of the linear order metric perturbation. We confirm this up to third order. This implies that gauge invariant variables for higher order metric perturbations can be found by using a procedure similar to that for linear order metric perturbations. We also derive gauge invariant combinations for the perturbation of an arbitrary physical variable, other than the spacetime metric, up to third order.