4D spacetimes embedded in 5D light-like Kasner universes

We consider spatially homogeneous, anisotropic cosmological models in 5D whose line element can be written as $dS^2 = {\cal{A}}(u, v)du dv - {\cal{B}}_{i j}(u, v)dx^{i}dx^{j}$, $(i, j = 1, 2, 3)$, where $u$ and $v$ are light-like coordinates. In the case where ${\cal{B}}_{i j}$ is diagonal, we construct three families of analytic solutions to the 5D vacuum field equations $R_{AB} = 0$ $(A, B = 0, 1, 2, 3, 4)$. Among them, there is a family of self-similar homothetic solutions that contains, as a particular case, the so-called light-like Kasner universes. In this work we provide a detailed study of the different types of 4D scenarios that can be embedded in such universes. For the sake of generality of the discussion, and applicability of the results, in our analysis we consider the two versions of non-compactified 5D relativity in vogue, viz., braneworld theory and induced matter theory. We find a great variety of cosmological models in 4D which are anisotropic versions of the FRW ones. We obtain models on the brane with a non-vanishing cosmological term $\Lambda_{(4)}$, which inflate {\it \`a la} de Sitter without satisfying the classical false-vacuum equation of state. Using the symmetry of the solutions, we construct a class of non-static vacuum solutions on the brane. We also develop {\it static} pancake-like distributions where the matter is concentrated in a thin surface (near $z = 0$), similar to those proposed by Zel'dovich for the shape of the first collapsed objects in an expanding anisotropic universe. The solutions discussed here can be applied in a variety of physical situations.