Linear perturbations of an exact gravitational wave in the Bianchi IV universe
Konstantin Osetrin
TL;DR
This work addresses how strong gravitational waves in an anisotropic cosmology, specifically a $Bianchi\ IV$ universe, can be studied perturbatively by constructing an analytical secondary GW on top of an exact background. It introduces the proper-time method, deriving a synchronous time function $\tau$ from the geodesic Hamilton–Jacobi equation and formulating a perturbative expansion in $\epsilon$ that depends on the wave variable $x^0$ and $\tau$. The authors obtain explicit solutions for the perturbation components, prove the stability of both the perturbations and the underlying exact solution, and show that the perturbations introduce additional curvature degrees of freedom (seven independent Riemann components). This framework advances analytic modeling of gravitational-wave effects in anisotropic early-universe contexts and provides a rigorous benchmark for numerical simulations in complex GW spacetimes.
Abstract
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 were 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 is proven. The stability of the exact solution for a gravitational wave in the anisotropic Bianchi IV universe is demonstrated.