  Recent observations of PSR J0045-7319, a radio pulsar in a close eccentric
orbit with a massive B-star companion, indicate that the system's orbital
period is decreasing on a timescale of $\approx 5 \times10^{5}$ years, which is
much shorter than the timescale of $\approx$ 10^9 years given by the standard
theory of tidal dissipation in radiative stars. Observations also provide
strong evidence that the B-star is rotating rapidly, perhaps at nearly its
break up speed. We show that the dissipation of the dynamical tide in a star
rotating in the same direction as the orbital motion of its companion (prograde
rotation) with a speed greater than the orbital angular speed of the star at
periastron results in an increase in the orbital period of the binary system
with time. Thus, since the observed time derivative of the orbital period is
large and negative, the B-star in the PSR J0045-7319 binary must have
retrograde rotation if tidal effects are to account for the orbital decay. We
also show that the time scale for the synchronization of the B-star's spin with
the orbital angular speed of the star at periastron is comparable to the
orbital evolution time. From the work of Goldreich and Nicholson (1989) we
therefore expect that the B-star should be rotating differentially, with the
outer layers rotating more slowly than the interior. We show that the
dissipation of the dynamical tide in such a differentially rotating B-star is
enhanced by almost three orders of magnitude leading to an orbital evolution
time for the PSR J0045-7319 Binary that is consistent with the observations.
