  We analyze the coupling between the internal degrees of freedom of neutron
stars in a close binary, and the stars' orbital motion. Our analysis is based
on the method of matched asymptotic expansions and is valid to all orders in
the strength of internal gravity in each star, but is perturbative in the
``tidal expansion parameter'' (stellar radius)/(orbital separation). At first
order in the tidal expansion parameter, we show that the internal structure of
each star is unaffected by its companion, in agreement with post-1-Newtonian
results of Wiseman (gr-qc/9704018). We also show that relativistic interactions
that scale as higher powers of the tidal expansion parameter produce
qualitatively similar effects to their Newtonian counterparts: there are
corrections to the Newtonian tidal distortion of each star, both of which occur
at third order in the tidal expansion parameter, and there are corrections to
the Newtonian decrease in central density of each star (Newtonian ``tidal
stabilization''), both of which are sixth order in the tidal expansion
parameter. There are additional interactions with no Newtonian analogs, but
these do not change the central density of each star up to sixth order in the
tidal expansion parameter. These results, in combination with previous analyses
of Newtonian tidal interactions, indicate that (i) there are no large
general-relativistic crushing forces that could cause the stars to collapse to
black holes prior to the dynamical orbital instability, and (ii) the
conventional wisdom with respect to coalescing binary neutron stars as sources
of gravitational-wave bursts is correct: namely, the finite-stellar-size
corrections to the gravitational waveform will be unimportant for the purpose
of detecting the coalescences.
