The symplectic geometry of the black hole photon shell

TL;DR

This paper explores the symplectic geometry of the photon shell in Kerr black holes, revealing how its phase-space volume and structure influence observable phenomena like photon rings and black hole ringdowns.

Contribution

It introduces a detailed symplectic geometric analysis of the photon shell, including volume calculations and bifurcation behavior in near-extremal black holes.

Findings

Approximately 3% of phase-space volume is near-horizon in near-extremal cases

Constructed the canonical volume form on the photon shell

Analyzed the thickening of the photon shell including near-critical orbits

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.