  Dark matter can capture in neutron stars and heat them to observable
luminosities. We study relativistic scattering of dark matter on highly
degenerate electrons. We develop a Lorentz invariant formalism to calculate the
capture probability of dark matter that accounts for the relativistic motion of
the target particles and Pauli exclusion principle. We find that the actual
capture probability can be five orders of magnitude larger than the one
estimated using a nonrelativistic approach. For dark matter masses $10~{\rm
eV}\textup{--}10~{\rm PeV}$, neutron star heating complements and can be more
sensitive than terrestrial direct detection searches. The projected sensitivity
regions exhibit characteristic features that demonstrate a rich interplay
between kinematics and Pauli blocking of the DM--electron system. Our results
show that old neutron stars could be the most promising target for discovering
leptophilic dark matter.
