Time Variation of the Fine Structure Constant in the Spacetime of a Domain Wall

The gravitational field produced by a domain wall acts as a medium with spacetime-dependent permittivity \epsilon. Therefore, the fine structure constant \alpha = e^2/4 \pi \epsilon will be a time-dependent function at fixed position. The most stringent constraint on the time-variation of \alpha comes from the natural reactor Oklo and gives |\dot{\alpha}/\alpha| < few 10^{-17} yr^{-1}. This limit constrains the tension of a cosmic domain wall to be less than \sigma \lesssim 10^{-2} MeV^3, and then represents the most severe limit on the energy density of a cosmic wall stretching our Universe.