Measuring the Shape of Kerr Black Holes at the Photon Orbit

Abstract The bright ringlike structures observed in the images of M87* and Sgr A* captured by the Event Horizon Telescope strongly support the validity of general relativity. Lensed images of the emission region, often referred to as photon rings in this context, are a direct consequence of the unstable dynamics of null geodesics near the spherical photon orbit in the Kerr spacetime. The order of the lensed image can be characterized by the number of half-orbits the photons complete before reaching the observer, with higher-order photon rings produced by null geodesics that circle the black hole more times. However, low-order rings are significantly influenced by the astrophysical environment. The relative radii of highly lensed photon rings are a function of the purely gravitational quantity called the Lyapunov exponent ( γ ). This instability of spherical photon orbits encodes fundamental information about the near-horizon geometry, which provides a tool for probing gravity. One way of doing so is by evaluating the value of γ using the near-horizon observations. Measuring γ requires probing the exponentially small differences between successive photon rings or between photon rings and the boundary of the black hole shadow (the critical curve). We investigate potential astrophysical sources of systematic error in the estimation of γ , including the location of the observed emission, especially at low photon ring order. We show that it is nevertheless possible to measure this purely gravitational quantity to roughly 10% and 1% systematic uncertainty by resolving the n = 2 and n = 3 photon rings, respectively, together with the radius of the shadow boundary . Therefore, forthcoming black hole imaging efforts to capture the n = 2 photon ring could result in a measurement of the Lyapunov exponent that is not limited by astrophysical uncertainties.