  In this work we study the properties of rigidly rotating neutral dust
solutions in general relativity. This class of solutions gained relevance
recently due to applications to the dynamics of spiral galaxies. We show that
this class could be interpreted as a rigid body in general relativity and we
analyze the different properties respect to the rigidly rotating disk in
special relativity: for example, the general relativistic counterpart shows no
Doppler effect for a light signal emitted and received from any two points at
rest respect to the rigid body. This effect can be important to test the
validity of the assumed model for our galaxy. In the second part we approach
the problem from a low energy expansion perspective and we write down a
generalization of the virial theorem for stationary spacetimes. The
non-Newtonian contributions can lead to a re-weighting of dark matter in
galaxies.
