Difficulties of quantitative tests of the Kerr-hypothesis with X-ray observations of mass accreting black holes

X-ray studies of stellar mass black holes in X-ray binaries and mass-accreting supermassive black holes in Active Galactic Nuclei have achieved a high degree of maturity and have delivered detailed information about the astrophysical sources and the physics of black hole accretion. In this article, I review recent progress made towards using the X-ray observations for testing the "Kerr hypothesis" that the background spacetimes of all astrophysical quasi-stationary black holes are described by the Kerr metric. Although the observations have indeed revealed clear evidence for relativistic effects in strong-field gravity, quantitative tests of the Kerr hypothesis still struggle with theoretical and practical difficulties. In this article, I describe several recently introduced test metrics and review the status of constraining the background spacetimes of mass accreting stellar mass and supermassive black holes with these test metrics. The main conclusion of the discussion is that astrophysical uncertainties are large compared to the rather small observational differences between the Kerr and non-Kerr metrics precluding quantitative constraints on deviations from the Kerr metric at this point in time. I conclude with discussing future progress enabled by more detailed numerical simulations and by future X-ray spectroscopy, timing, polarimetry, and interferometry missions.