Gravitational waves in a Bianchi type-I universe

Tensor perturbation equations in a diagonal Bianchi type-I universe are derived in a preliminary study of the propagation of gravitational waves in anistropic universes. Exact solutions to these equations are obtained for an axisymmetric Kasner background, in which wave propagation is along the symmetry axis. Near the singularity the universe behaves like a generalized Kasner solution and at large time like it possesses a directed stream of radiation with energy density equal to pressure. It is shown to be equivalent to the linear limit of the Einstein-Rosen or Gowdy $T^3$ solution. The back reaction of gravitational waves on a type-I universe with matter is also studied in the long- and short-wavelength limits. High-frequency waves with sufficient energy could reverse the contracting axis of the universe into expansion, while waves propagating in all directions could bring about isotropization of the background. Long-wavelength perturbations act like an effective anisotropy potential which induces "small oscillations" on the background. The universe with such long waves can be shown to be equivalent to the weak-field limit of a "corner run" solution in the type-IX and type ${\mathrm{VII}}_0$ universes.