Lensing-band approach to spacetime constraints

General relativity's prediction that all black holes are described by the Kerr metric, irrespective of their size, can now be empirically tested using electromagnetic observations of supermassive black holes and gravitational waves from mergers of stellar-mass black holes. In this work, we focus on the electromagnetic side of this test and quantify the constraining power of very-long-baseline-interferometry (VLBI) observations of emission generated by hot gas surrounding supermassive black holes. We demonstrate how to use lensing bands-annular regions on the observer's screen surrounding the critical curve-to constrain the underlying spacetime geometry. Contingent upon a detection of a lensed VLBI feature, the resulting lensing-band framework allows us to exclude spacetimes for which said feature cannot arise from geodesics that traversed the equatorial plane more than once. Focusing on the first indirect image and tests of black-hole uniqueness, we employ a parametrized spacetime as a case study. We find that resolving geometric information that goes beyond the apparent size of the critical curve has the potential to lift degeneracies between different spacetime parameters. Our work thereby quantifies a conservative estimate of the constraining power of VLBI measurements and contributes to a larger effort to simultaneously constrain geometry and astrophysics.