Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes

TL;DR

The paper presents a unified explicit family of Kundt-class non-diverging spacetimes that contains Petrov types O, D, N, and II and can represent exact gravitational waves propagating in various conformally flat and type D backgrounds. It identifies new solutions, including type D pure radiation in the (anti-)Nariai universe and type II electrovacuum waves in the (anti-)Nariai background, and proves that there are no conformally flat Einstein–Maxwell fields with non-vanishing cosmological constant.

Key findings

• A single metric family (1)–(2), with constants α, β, ε and functions C(u), D(u), H(ζ, ζ̄, u), yields Kundt spacetimes of Petrov type O, D, N, or II depending on whether D = D₀ and/or H = H₀.
• The backgrounds include Minkowski, conformally flat (anti-)de Sitter, Edgar–Ludwig, Bertotti–Robinson, and type D (anti-)Nariai or Plebański–Hacyan spaces; adding H_w = H − H₀ introduces exact gravitational waves of arbitrary profile.
• New solutions reported include a type D pure radiation solution in the (anti-)Nariai universe, with Ψ₂ = −Λ/3 and R = 4Λ, and an explicit type II electrovacuum solution for electromagnetic plus gravitational waves in the (anti-)Nariai background.
• The appendix proves that there are no conformally flat Einstein–Maxwell solutions with Λ ≠ 0; for Λ = 0 the only possibilities are the Bertotti–Robinson universe for a non-null Maxwell field and the special pp-wave for a null field.

Methods

• Theoretical derivation within the Kundt class: the authors adopt an explicit metric ansatz, compute the Newman–Penrose Weyl and Ricci scalars, and classify Petrov types and matter content by conditions on D and H.
• They analyze special subclasses corresponding to conformally flat and type D backgrounds, impose Einstein–Maxwell or pure radiation conditions, and construct radiative type N/II solutions by adding a generalised Kerr–Schild term −(Q/P)H_w du².
• Physical interpretation uses the geodesic deviation equation in an orthonormal frame, decomposing effects of R, Φ₁₁, Φ₂₂, Ψ₂, and Ψ₄ into isotropic, matter, Coulomb, and gravitational-wave contributions.

Limitations

• Stated: the general solution describing different Petrov types and matter fields explicitly is still impossible; even with symmetries, solutions can usually only be given implicitly.
• Stated: the Einstein–Maxwell restrictions rule out a purely electromagnetic field in general spacetimes of the form (1)–(2); only special choices of α, β, ε, C, D, and H satisfy the field equations.
• Evident from the text: the work is a theoretical exact-solution study with no observational data or numerical analysis, and its results are restricted to the non-diverging Kundt class and the specific metric ansatz considered.