  When neutrinos travel through a normal matter medium, the electron neutrinos
couple differently to gravity compared to the other neutrinos, due to the
presence of electrons in the medium and the absence of the other charged
leptons. We calculate the momentum-dependent part of the matter-induced
gravitational couplings of the neutrinos under such conditions, which arise at
order $g^2/M^4_W$, and determine their contribution to the neutrino dispersion
relation in the presence of a gravitational potential $\phi^{\mathrm{ext}}$.
These new contributions vanish for the muon and tau neutrinos. For electron
neutrinos with momentum $K$, they are of the order of the usual Wolfenstein
term times the factor $(K^2/M^2_W)\phi^{\mathrm{ext}}$, for high energy
neutrinos. In environments where the gravitational potential is substantial,
such as those in the vicinity of Active Galactic Nuclei, they could be the
dominant term in the neutrino dispersion relation. They must also be taken into
account in the analysis of possible violations of the Equivalence Principle in
the neutrino sector, in experimental settings involving high energy neutrinos
traveling through a matter background.
