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We study the mode solution to the Cauchy problem of the scalar wave equation$\Box \varphi = 0$ in Kasner spacetimes. As a first result, we give theexplicit mode solution in axisymmetric Kasner spacetimes, of which flat Kasnerspacetimes are special cases. Furthermore, we give the small and large timeasymptotics of the modes in general Kasner spacetimes. Generically, the modesin non-flat Kasner spacetimes grow logarithmically for small times, while themodes in flat Kasner spacetimes stay bounded for small times. For large times,however, the modes in general Kasner spacetimes oscillate with a polynomiallydecreasing amplitude. This gives a notion of large time frequency of the modes,which we use to model the wavelength of light rays in Kasner spacetimes. Weshow that the redshift one obtains in this way actually coincides with theusual cosmological redshift.

The mode solution of the wave equation in Kasner spacetimes and redshift