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)oftheirphotospheres\mathrm{.}Wederivesensitivitycoefficients$K_i$whichdefinehowtheindividualH$2$transitionsshiftdueto$\mu$-dependence\mathrm{.}ThespectrumofwhitedwarfstarGD133yieldsa$\Delta\mu/\mu$constraintof$(-2.7\pm4.7{\rm stat}\pm 0.2_{\rm sys})\times10^{-5}$foralocalenvironmentofagravitationalpotential$\phi\sim10^4\ \phi_$\text{Earth}$$,whilethatofG29$-$38yields$\Delta\mu/\mu=(-5.8\pm3.8_{\rm stat}\pm 0.3_{\rm sys})\times10^{-5}$ for a potential of $2\times {10}^4$ ${\varphi }_{\text{Earth}}$.