  Gravitationally bound neutrons have become an important tool in the
experimental searches for new physics, such as modifications to Newton's force
or candidates for dark matter particles. Here we include the relativistic
effects of mass-energy equivalence into the model of gravitationally bound
neutrons. Specifically, we investigate a correction in a gravitationally bound
neutron's Hamiltonian due to the presence of an external magnetic field. We
show that the neutron's additional weight due to mass-energy equivalence will
cause a small shift in the neutron's eigenenergies and eigenstates, and examine
how this relativistic correction would affect experiments with trapped
neutrons. We further consider the ultimate precision in estimating the
relativistic correction to the precession frequency and find that, at short
times, a joint measurement of both the spin and motional degrees of freedom
provides a metrological enhancement as compared to a measurement of the spin
alone.
