Dark companion of Baryonic matter, III

Wherever one talks of dark matter, one does so where there is an observable matter and an associated unsolved dynamical issue to be settled. We promote this observation to the status of an axiom and conjecture that there is a dark companion to every baryonic matter, subject to certain rules as regards its size, distribution. To pursue the proposition in a systematic way we resort to the rotation curves of spiral galaxies. They have non classical features. First, we design a spacetime metric around the galaxy to accommodate these features. Next we calculate the density and pressure of a hypothetical dark matter that could generate such a spacetime. In the weak field regime and for a spherical distribution of mass $M$, we are able to assign a dark perfect gas companion, whose density is almost proportional to $M^{1/2}$ and fades away almost as $r^{-2}$. However, in view of this orderly relation between the observable mass and its dark companion, one may choose to interpret the whole scenario as an alternative theory of gravitation.