Second Order Gauge Invariant Perturbation Theory -- Perturbative curvatures in the two-parameter case --

TL;DR

This paper develops explicit formulas for second order gauge invariant perturbative curvatures in general relativity, ensuring Einstein equations are expressed in a gauge invariant manner, facilitating physical applications.

Contribution

It derives new formulae for second order gauge invariant curvatures in arbitrary backgrounds, extending previous gauge invariant perturbation frameworks.

Findings

Perturbative curvatures have the same form as gauge invariant variables.

Einstein equations can be expressed in gauge invariant form.

Framework applicable to various physical scenarios.

Abstract

Based on the gauge invariant variables proposed in our previous paper [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], some formulae of the perturbative curvatures of each order are derived. We follow the general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime to derive these formulae. These perturbative curvatures do have the same form as the definitions of gauge invariant variables for arbitrary perturbative fields which are previously proposed. As a result, we explicitly see that any perturbative Einstein equations are given in terms of gauge invarinat form. We briefly discuss physical situations to which this framework should be applied.