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
$\Delta\mu/\mu$ constraint of $(-2.7\pm4.7_{\rm stat}\pm 0.2_{\rm
sys})\times10^{-5}$ for a local environment of a gravitational potential
$\phi\sim10^4\ \phi_\textrm{Earth}$, while that of G29$-$38 yields
$\Delta\mu/\mu=(-5.8\pm3.8_{\rm stat}\pm 0.3_{\rm sys})\times10^{-5}$ for a
potential of $2 \times 10^4$ $\phi_\textrm{Earth}$.
