The symplectic geometry of the black hole photon shell

The images of supermassive black holes published by the Event Horizon Telescope (EHT) collaboration show a bright annulus of light with a dark center, the detailed structure of which is still not resolved. General relativity predicts that within this annulus of light there is a thin bright ring known as the ``photon ring", resulting from nearly-bound photons which orbit the black hole a number of times on the way from the source to the detector. This set of nearly bound null geodesics defines a submanifold of the null geodesic phase space known as the ``photon shell". In this work, we study a new property of the famous Kerr solution: the symplectic geometry and differential phase-space volume of the photon shell. We begin by considering exactly critical, bound photon orbits, which theoretically orbit the BH an infinite number of times. We give special attention to the near extremal limit, in which the near-horizon-extremal-Kerr (NHEK) geometry is known to govern a significant fraction of the photon orbits. With our calculations, we quantify the relative volume of phase space in the (near-) NHEK region. We also explore the differential phase space volume of the near-critical photon orbits, which are able to reach an observer at infinity. Going forward, our approach, combined with the stochastic nature of the astrophysical emission, could be applied to provide new insights into properties of black hole images.