Equatorial Lensing in the Balasin-Grumiller Galaxy Model
The Balasin-Grumiller model has been the first model employed as an attempt towards providing a fully general relativistic description of the dynamics of a disc galaxy. In this paper, we compute the equatorial gravitational lensing observables of the model. Indeed, our purpose is to investigate the role that gravitational lensing plays as an observable in distinguishing between the state-of-the-art galaxy models and the fully general relativistic ones, with the latter stressing the role of frame-dragging and hence conceivably pointing to a possible re-weighting of the dark matter content of disc galaxies. We obtain for the Balasin-Grumiller model the exact formula for the bending angle of light and we provide a corresponding estimate for the time delay between images in the equatorial plane. For a reasonable choice for the values of the parameters of the solution (bulge and scale radiuses, and average rotational star speeds), the values that we obtain for the bending angle are in agreement with those observed for typical disc galaxies. On the other hand, the calculated time delay, which is directly tied to the frame-dragging generated by the angular momentum of the galaxy, turns out to be some orders of magnitude larger than the ones measured for the class of galaxies that the Balasin-Grumiller model would claim to describe. We believe this abnormal discrepancy to be due to the very nature of the Balasin-Grumiller model. Namely, it being rigidly rotating, hence providing an unphysical amount of frame-dragging. Therefore, we conclude that, in spite of its simplicity and its unquestionable didactical value, the Balasin-Grumiller model is far too crude to provide an instrument for a reliable general relativistic description of a disc galaxy and that further work in the fully general relativistic modelling of galaxies is required to reach a satisfactory stage.