Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations

The procedure to find gauge invariant variables for two-parameter nonlinear perturbations in general relativity is considered. Up to third order perturbations, it is shown that appropriate combinations of nonlinear metric perturbations and lower order one are transformed as the gauge transformation of linear order metric perturbations. This implies that gauge invariant variables for the higher order metric perturbations can be found by using the similar procedure of linear order metric perturbations. We also show the gauge invariant combinations for the perturbations of physical variables other than the spacetime metric up to third order.