Secondary gravitational waves against a strong gravitational wave in the Bianchi VI universe

A proper-time method for constructing models of dynamic gravitational-wave fields is presented. Using the proper-time method, analytical (not numerical) models of secondary gravitational waves are constructed as perturbative solutions of linearized field equations against the background of the exact wave solution of Einstein’s equations for the vacuum in the Bianchi VI universe in a privileged wave coordinate system. Relations for the proper time of test particles against the background of a strong gravitational wave are used. The analytical form of the metric components for secondary gravitational waves is found from compatibility conditions for the field equations. From the field equations, an explicit form of ordinary differential equations and their solutions is obtained for functions included in small corrections to the metric for secondary gravitational waves. It is shown that there exists a continuum of gravitational wave parameters for which the perturbative solutions are stable.

💡 Research Summary

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The paper presents a novel “proper‑time” method for constructing analytical models of secondary (weak) gravitational waves propagating on top of an exact strong plane‑wave solution of Einstein’s vacuum equations in a Bianchi VI universe. The authors first recall the exact strong‑wave background, which is expressed in a privileged wave coordinate system (x⁰, x¹) where the metric components g₀₀ and g₁₁ vanish, so that the spacetime interval along the wave direction is null. The background metric depends on three parameters µ, ν and ϕ, but the vacuum field equations impose a single algebraic constraint, reducing the independent degrees of freedom to two angular parameters (θ, ϕ). The Riemann, Ricci and Weyl tensors are computed explicitly, showing that the space becomes conformally flat when µ = ν.

A crucial ingredient of the method is the proper time τ of a test particle freely falling in the strong wave. Using the Hamilton‑Jacobi integration performed in earlier work, τ is expressed as a complicated function of the wave variable x⁰ and the spatial coordinates (x¹, x², x³). Three distinct forms of τ arise depending on whether µ or ν equals ½, whether µ + ν = 1, or the generic case. The authors adopt the most general expression (τ given by Eq. 19) for the subsequent analysis.

The secondary wave is introduced as a linear perturbation of the background metric: \