Decoding a black hole metric from the interferometric pattern of the relativistic images of a compact source

Photons emitted by light sources in the neighbourhood of a black hole can wind several times around it before fleeing towards the observer. For spherically symmetric black holes, two infinite sequences of images are created for any given source, asymptotically approaching the shadow border with decreasing magnitude. These sequences are reflected by a characteristic staircase structure in the complex visibility function that may be used to decode the properties of the black hole metric. Recalling the formalism of gravitational lensing in the strong deflection limit, we derive analytical formulae for the height, the width and the periodicities of the steps in the visibility as functions of the black hole parameters for the case of a single compact and distant source. With respect to diffuse emission by the whole accretion flow, this ideal framework provides clean insight and model-independent information on the metric. These basic formulae can then be used to build visibilities for more complicated sources and track the changes induced by alternative metrics and ultimately test General Relativity. As simple examples, we include visibilities for Reissner-Nordstr\"om and Janis-Newman-Winicour metrics.