Variation of the fine structure constant caused by expansion of the Universe

In present paper we evaluate the fine structure constant variation, that should take place as the Universe expands and its curvature is changed adiabatically. Such variation of the fine structure constant is attributed to an energy losses by an extended physical system (consist of baryonic component and electromagnetic field) due to expansion of our Universe. Obtained ratio (d alpha)/alpha = 1. 10{-18} (per second) is only five times smaller than actually reported experimental limit on this value. For this reason obtained variation can probably be measured within a couple of years. To argue the correctness of our approach we calculate the Planck constant as adiabatic invariant of the electromagnetic field propagated on a manifold characterized by slowly varied geometry, in the framework of the pseudo- Riemannian geometry. Finally we discuss the double clock experiment based on Al+ and Hg+ clocks carried out by T. Rosenband et al. (Science 2008). We show that in this case (when the fine structure constant is changed adiabatically) the method based on double clock experiment can not be applied to measure the fine structure constant variation.