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The Mode Solution of the Wave Equation in Kasner Spacetimes and Redshift

Abstract

We study the mode solution to the Cauchy problem of the scalar wave equation □φ = 0 in Kasner spacetimes. As a first result, we give the explicit mode solution in axisymmetric Kasner spacetimes, of which flat Kasner spacetimes are special cases. Furthermore, we give the small and large time asymptotics of the modes in general Kasner spacetimes. Generically, the modes in non-flat Kasner spacetimes grow logarithmically for small times, while the modes in flat Kasner spacetimes stay bounded for small times. For large times, however, the modes in general Kasner spacetimes oscillate with a polynomially decreasing 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. We show that the redshift one obtains in this way actually coincides with the usual cosmological redshift.

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Correspondence to Oliver Lindblad Petersen.

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Petersen, O.L. The Mode Solution of the Wave Equation in Kasner Spacetimes and Redshift. Math Phys Anal Geom 19, 26 (2016). https://doi.org/10.1007/s11040-016-9233-x

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  • DOI: https://doi.org/10.1007/s11040-016-9233-x

Keywords

  • General relativity
  • Wave equation
  • Kasner spacetime
  • Redshift

Mathematics Subject Classfication (2010)

  • Primary 83C15
  • Secondary 34C11