Conservative Transformation Group: Dark Matter Halos

Pandres has proposed a theory which extends the geometrical structure of a real four-dimensional space-time via a field of orthonormal tetrads with an enlarged transformation group. This new transformation group, called the conservation group, contains the group of diffeomorphisms as a proper subgroup and we hypothesize that it is the foundational group for quantum geometry. The fundamental geometric object of the new geometry is the curvature vector, $\mathrm{C}_{\mathrm{\mu }}$. Using the scalar Lagrangian density, $\mathrm{C}^{\mathrm{\mu }}C_{\mu }\sqrt {-g\thinspace } $ field equations for the free field have been obtained which are invariant under the conservation group. We present spherically symmetric solutions for the corresponding free field and a solution similar to the external Schwarzschild solution is obtained. The theory implies that the external stress-energy tensor has non-compact support and hence may give the geometrical foundation for dark matter. Flat velocity curves are obtained under suitable thermodynamic conditions.