Prospects for future experimental tests of gravity with black hole imaging: Spherical symmetry

Astrophysical black holes (BHs) are universally expected to be described by the Kerr metric, a stationary, vacuum solution of general relativity (GR). Indeed, by imaging M87* and Sgr A* and measuring the size of their shadows, we have substantiated this hypothesis through successful null tests. Here we discuss the potential of upcoming improved imaging observations in constraining deviations of the spacetime geometry from that of a Schwarzschild BH (the nonspinning, vacuum GR solution), with a focus on the photon ring. The photon ring comprises a series of time -delayed, self -similarly nested higher -order images of the accretion flow, and is located close to the boundary of the shadow. In spherical spacetimes, these images are indexed by the number of half -loops executed around the BH by the photons that arrive in them. The delay time offers an independent shadow size estimate, enabling tests of shadow achromaticity, as predicted by GR. The image self -similarity relies on the lensing Lyapunov exponent, which is linked to photon orbit instability near the unstable circular orbit. Notably, this critical exponent, specific to the spacetime, is sensitive to the rr-component of the metric, and also offers insights into curvature, beyond the capabilities of currently available shadow size measurements. The Lyapunov time, a characteristic instability timescale, provides yet another probe of metric and curvature. The ratio of the Lyapunov and the delay times also yields the lensing Lyapunov exponent, providing alternative measurement pathways. Remarkably, the width of the first -order image can also serve as a discriminator of the spacetime. Each of these observables, potentially accessible in the near future, offers spacetime constraints that are orthogonal to those of the shadow size, enabling precision tests of GR.