Second-order gauge invariant perturbation theory - Perturbative curvatures in the two-parameter case

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