  Common-envelope evolution is important in the formation of neutron star
binaries within the isolated binary formation channel. As a neutron star
inspirals within the envelope of a primary massive star, it accretes and spins
up. Because neutron stars are in the strong-gravity regime, they have a
substantial relativistic mass deficit, i.e., their gravitational mass is less
than their baryonic mass. This effect causes some fraction of the accreted
baryonic mass to convert into neutron star binding energy. The relativistic
mass deficit also depends on the nuclear equation of state, since more compact
neutron stars will have larger binding energies. We model the mass growth and
spin-up of neutron stars inspiraling within common-envelope environments and
quantify how different initial binary conditions and hadronic equations of
state affect the post-common-envelope neutron star's mass and spin. From these
models, we find that neutron star mass growth is suppressed by $\approx
15-30\%$. We also find that for a given amount of accreted baryonic mass, more
compact neutron stars will spin-up faster while gaining less gravitational
mass, and vice versa. This work demonstrates that a neutron star's strong
gravity and nuclear microphysics plays a role in neutron-star-common-envelope
evolution, in addition to the macroscopic astrophysics of the envelope. Strong
gravity and the nuclear equation of state may thus affect both the population
properties of neutron star binaries and the cosmic double neutron star merger
rate.
