On Coordinate Frames in Axisymmetric Static Vacuum Spacetimes and Implications for Observations

While a physical theory should be independent of the coordinate frame chosen by any observer, the observations themselves in fact depend on the choice of coordinates. In particular, different coordinate frames reflect different symmetries seen by a local observer. In this work, we discuss the applicability of different coordinate choices and the resulting line elements for static axisymmetric vacuum spacetimes. We find that the effective potential experienced by a local observer in the low-velocity limit is highly dependent on the form of the line element and thus on the coordinates chosen in the description. For example, this affects the form of a rotation curve expected by such an observer. We thus conclude that it is important to review the choices of local (coordinate frame of the observer) and global symmetries carefully to understand observations from a generally relativistic point of view.