Dark matter effects in vacuum spacetime

We analyze a toy model describing an empty spacetime in which the motion of a test mass (and the trajectories of photons) evidence the presence of a continuous and homogeneous distribution of matter; however, since the energy-momentum tensor vanishes, no real matter or energy distribution is present at all. Thus, a hypothetical observer will conclude that he is immersed in some sort of dark matter, even though he has no chance to directly detect it. This suggests yet another possibility of explaining the elusive dark matter as a purely dynamical effect due to the curvature of spacetime.

Figure 1

$\rho \mathrm{\text{−}}t$ section of the spacetime described by metric (1). The shaded, infinite cylinder represents the asymptotic region, accessible to a hypothetical “Minkowskian” observer. In the bands above the singularity, ${\partial }_t$ becomes spacelike.