Tully-Fisher relation, key to dark matter companion of baryonic matter
  Rotation curves of spiral galaxies \emph{i}) fall off much less steeply than
the Keplerian curves do, and \emph{ii}) have asymptotic speeds almost
proportional to the fourth root of the mass of the galaxy, the Tully-Fisher
relation. These features alone are sufficient for assigning a dark companion to
the galaxy in an unambiguous way. In regions outside a spherical system, we
design a spherically symmetric spacetime to accommodate the peculiarities just
mentioned. Gravitation emerges in excess of what the observable matter can
produce. We attribute the excess gravitation to a hypothetical, dark, perfect
fluid companion to the galaxy and resort to the Tully-Fisher relation to deduce
its density and pressure. The dark density turns out to be proportional to the
square root of the mass of the galaxy and to fall off as $r^{-(2+\alpha)},
  \alpha\ll 1$. The dark equation of state is barrotropic. For the interior of
the configuration, we require the continuity of the total force field at the
boundary of the system. This enables us to determine the size and the
distribution of the interior dark density and pressure in terms of the
structure of the observable matter. The formalism is nonlocal and nonlinear,
and the density and pressure of the dark matter at any spacetime point turn out
to depend on certain integrals of the baryonic matter over all or parts of the
system in a nonlinear manner.
