  We study the proposed solution of the solar neutrino problem which requires a
flavor nondiagonal coupling of neutrinos to gravity. We adopt a
phenomenological point of view and investigate the consequences of the
hypothesis that the neutrino weak interaction eigenstates are linear
combinations of the gravitational eigenstates which have slightly different
couplings to gravity, $f_1G$ and $f_2G$, $|f_1-f_2| << 1$, corresponding to a
difference in red-shift between electron and muon neutrinos, $\Delta z/(1+z)
\sim |f_1 - f_2|$. We perform a $\chi^2$ analysis of the latest available solar
neutrino data and obtain the allowed regions in the space of the relevant
parameters. The existing data rule out most of the parameter space which can be
probed in solar neutrino experiments, allowing only $|f_1 - f_2| \sim 3 \times
10^{-14}$ for small values of the mixing angle ($2 \times 10^{-3} \le
\sin^2(2\theta_G) \le 10^{-2}$) and $10^{-16} \stackrel{<}{\sim} |f_1 - f_2|
\stackrel{<}{\sim}10^{-15}$ for large mixing ($0.6 \le \sin^2(2\theta_G) \le
0.9$). Measurements of the $^8{\rm B}$-neutrino energy spectrum in the SNO and
Super-Kamiokande experiments will provide stronger constraints independent of
all considerations related to solar models. We show that these measurements
will be able to exclude part of the allowed region as well as to distinguish
between conventional oscillations and oscillations due to the violation of the
equivalence principle.
