Extrapolating the projected potential of gravitational lens models: property-preserving degeneracies

While gravitational lens inversion holds great promise to reveal the structure of the light-deflecting mass distribution, both light and dark, the existence of various kinds of degeneracies implies that care must be taken when interpreting the resulting lens models. This article illustrates how thinking in terms of the projected potential helps to gain insight into these matters. Additionally it is shown explicitly how, when starting from a discretized version of the projected potential of one particular lens model, the technique of quadratic programming can be used to create a multitude of equivalent lens models that preserve all or a subset of lens properties. This method is applied to a number of scenarios, showing the lack of grasp on the mass outside the strong lensing region, revisiting mass redistribution in between images, and applying this to a recent model of the SDSS J1004+4112 cluster, as well as illustrating the generalized mass sheet degeneracy and source-position transformation. In the case of J1004, we show that this mass redistribution did not succeed at completely eliminating a dark mass clump recovered by GRALE near one of the quasar images.