The symplectic geometry of the black hole photon shell

Abstract

The unstably bound, critical null geodesics of the Kerr spacetime form a distinguished class of orbits whose properties govern observables such as the photon ring and the high-frequency component of black-hole ringdown. This set of orbits defines a codimension-two submanifold of the null-geodesic phase space known as the photon shell. In this work we investigate the photon shell's intrinsic symplectic geometry. Using the induced symplectic form, we construct the canonical volume form on the shell and compute the differential phase-space volume it encloses as a function of radius -- equivalently, the radial density of states. In the near-extremal limit the photon shell bifurcates into near-horizon and far-region components; we find that approximately $3\%$ of the shell's phase-space volume resides in the near-horizon component. We also analyze a thickening of the photon shell that includes near-critical orbits, and compute its differential phase-space volume. Beyond their intrinsic theoretical interest, these results may inform the interpretation of high-resolution observations of spinning black holes.