Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations

The procedure to find gauge invariant variables for two-parameter nonlinearperturbations in general relativity is considered. For each order metricperturbation, we define the variable which is defined by the appropriatecombination with lower order metric perturbations. Under the gaugetransformation, this variable is transformed in the manner similar to the gaugetransformation of the linear order metric perturbation. We confirm this up tothird order. This implies that gauge invariant variables for higher ordermetric perturbations can be found by using a procedure similar to that forlinear order metric perturbations. We also derive gauge invariant combinationsfor the perturbation of an arbitrary physical variable, other than thespacetime metric, up to third order.Comment: 33 pages, 1 figure, PTPTeX ver.0.8 (LateX2e), Accepted for Publication to Progress of Theoretical Physics. Typos and trivial mistakes in equations are correcte