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Rotation curves of spiral galaxies \emph{i}) fall off much less steeply thanthe Keplerian curves do, and \emph{ii}) have asymptotic speeds almostproportional to the fourth root of the mass of the galaxy, the Tully-Fisherrelation. These features alone are sufficient for assigning a dark companion tothe galaxy in an unambiguous way. In regions outside a spherical system, wedesign a spherically symmetric spacetime to accommodate the peculiarities justmentioned. Gravitation emerges in excess of what the observable matter canproduce. We attribute the excess gravitation to a hypothetical, dark, perfectfluid companion to the galaxy and resort to the Tully-Fisher relation to deduceits density and pressure. The dark density turns out to be proportional to thesquare 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 ofthe configuration, we require the continuity of the total force field at theboundary of the system. This enables us to determine the size and thedistribution of the interior dark density and pressure in terms of thestructure of the observable matter. The formalism is nonlocal and nonlinear,and the density and pressure of the dark matter at any spacetime point turn outto depend on certain integrals of the baryonic matter over all or parts of thesystem in a nonlinear manner.

Tully-Fisher relation, key to dark matter companion of baryonic matter