Limits on a Gravitational Field Dependence of the Proton--Electron Mass Ratio from H$_2$ in White Dwarf Stars

Spectra of molecular hydrogen (H$_2$) are employed to search for a possible proton-to-electron mass ratio ($mu$) dependence on gravity. The Lyman transitions of H$_2$, observed with the Hubble Space Telescope towards white dwarf stars that underwent a gravitational collapse, are compared to accurate laboratory spectra taking into account the high temperature conditions ($T sim 13,000$ K) of their photospheres. We derive sensitivity coefficients $K_i$ which define how the individual H$_2$ transitions shift due to $mu$-dependence. The spectrum of white dwarf star GD133 yields a $Deltamu/mu$ constraint of $(-2.7pm4.7_{rm stat}pm 0.2_{rm sys})times10^{-5}$ for a local environment of a gravitational potential $phisim10^4 phi_textrm{Earth}$, while that of G29$-$38 yields $Deltamu/mu=(-5.8pm3.8_{rm stat}pm 0.3_{rm sys})times10^{-5}$ for a potential of $2 times 10^4$ $phi_textrm{Earth}$.