Low-energy limit of stationary and axisymmetric solutions in general relativity

We study the low-energy limit of general relativity in the presence of stationarity and axial symmetry, coupled to dust. Specifically, we demonstrate that differences between the dynamics of general relativity and those of Newtonian gravity persist even in the weak-field and slow-motion regime. Notably, these differences are driven by dragging terms that are not necessarily small, as is typically the case in the well-known gravitomagnetic limit. To highlight this distinction, we use the concept of strong gravitomagnetism that we introduced in previous works. We provide a pedagogical discussion of how these discrepancies arise and outline a systematic procedure to solve the equations of motion for such systems. Furthermore, we present analytical results for specific cases and also give the general solution for the vacuum case. A particularly notable result is our demonstration of how general relativity can naturally account for a Tully-Fisher-like relation.