  Nucleon effective masses in neutron-rich matter are studied with the
relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space. The
neutron and proton effective masses for symmetric nuclear matter are 0.80 times
rest mass, which agrees well with the empirical values. In neutron-rich matter,
the effective mass of the neutron is found larger than that of the proton, and
the neutron-proton effective mass splittings at the empirical saturation
density are predicted as $0.187\alpha$ with $\alpha$ being the isospin
asymmetry parameter. The result is compared to other ab initio calculations and
is consistent with the constraints from the nuclear reaction and structure
measurements, such as the nucleon-nucleus scattering, the giant resonances of
$^{208}$Pb, and the Hugenholtz-Van Hove theorem with systematics of nuclear
symmetry energy and its slope. The predictions of the neutron-proton effective
mass splitting from the RBHF theory in the full Dirac space might be helpful to
constrain the isovector parameters in phenomenological density functionals.
