The proper-time method for constructing perturbative dynamical gravitational fields is presented. Using the proper-time method, a perturbative analytical model of gravitational waves against the backdrop of an exact wave solution of Einstein’s equations in a Bianchi IV universe is constructed. To construct the perturbative analytical wave model a privileged wave coordinate system and a synchronous time function associated with the proper time of an observer freely moving in a gravitational wave are used. Reduction of the field equations, taking into account compatibility conditions, reduces the mathematical model of gravitational waves to a system of coupled ordinary differential equations for functions of the wave variable. Analytical solutions for the components of the gravitational wave metric have been found. The stability of the resulting perturbative solutions for the continuum domain of parameters is proven. The linear stability of the exact solution for a gravitational wave in the anisotropic Bianchi IV universe for the continuum domain of parameters is demonstrated.
Konstantin Osetrin (Thu,) studied this question.