Static Thin Disks with Haloes as Sources of Conformastatic Spacetimes

Abstract: Two new families of exact solutions to the Einstein equations for a conformastatic spacetime with axial symmetry are presented which describe thin disks of dust immersed in a spheroidal halo. The solutions are obtained by expressing the metric function in terms of an auxiliary function which satisfies the Laplace equation, a characteristic property of the conformastatic spacetimes. The first family of solutions is obtained from the displacement, cut and reflexion method, which introduces a discontinuity in the first $z$-derivate of the metric tensor across the plane of the disk. The second family of solutions is obtained by using the oblate spheroidal coordinates because they adapt to the shape of the source and introduce naturally a cutting radius for the disk. The energy densities of the disk and the halo are everywhere positive and well behaved and their energy-momentum tensor agrees with all the energy conditions. Some particular solutions for the energy density of the disk and the halo are presented and the rotational curves are obtained by solving the geodesic equation for a particle that moves in circular orbits in the plane of the disk.