The gravitational potential phi = GM/Rc^2 at the surface of the white dwarf G191-B2B is 10,000 times stronger than that at the Earths surface. Numerous photospheric absorption features are detected, making this a suitable environment to test theories in which the fundamental constants depend on gravity. We have measured the fine structure constant, alpha, at the white dwarf surface, used a newly calibrated Hubble Space Telescope STIS spectrum of G191-B2B, two new independent sets of laboratory Fe V wavelengths, and new atomic calculations of the sensitivity parameters that quantify Fe V wavelength dependency on alpha. The two results obtained are: dalpha/alpha = 6.36 +/- [0.33(stat) + 1.94(sys)] X 10^{-5} and dalpha/alpha = 4.21 +/- [0.47(stat) + 2.35(sys)] X 10^{-5}. The measurements hint that the fine structure constant increases slightly in the presence of strong gravitational fields. A comprehensive search for systematic errors is summarised, including possible effects from line misidentifications, line blending, stratification of the white dwarf atmosphere, the quadratic Zeeman effect and electric field effects, photospheric velocity flows, long-range wavelength distortions in the HST spectrum, and variations in the relative Fe isotopic abundances. None fully account for the observed deviation but the systematic uncertainties are heavily dominated by laboratory wavelength measurement precision.
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