The symplectic geometry of the black hole photon shell

TL;DR

The paper develops the intrinsic symplectic geometry of the Kerr photon shell by deriving the induced 4D volume form from the full 6D null geodesic phase space and computing the radial density of states $dV_{PS}/dE$ as a function of the photon-shell radius. It analyzes both low-spin (Schwarzschild) and high-spin (near-extremal) limits, revealing a bifurcation of the shell into far and near-NHEK regions in the extremal limit, with a near-horizon contribution of order a few percent. It further extends the framework to near-critical (thickened) phase space slices labeled by half-orbits, deriving their volume elements and Schwarzschild limits, to better connect with photon-ring observations. The results provide coordinate-invariant geometric quantities that may inform phase-space modeling of black-hole images and offer potential links to quasinormal mode structure and holographic symmetries in Kerr spacetimes.

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.

Figures (5)

• Figure 1: The cross section of the Kerr photon shell in the $(r,\theta)$ plane in Boyer-Lindquist coordinates.
• Figure 2: The photon ring is the brightness enhancement in black hole images which follows closely the critical curve $\mathcal{C}$, indicated by the dashed green circle-like line. This snapshot is taken from cardenas2023adaptive.
• Figure 3: Differential phase-space volume of the Kerr photon shell as a function of the radius $\tilde{r}$ and the spin $a$. The differential volume is largest at the Schwarzschild limit where the photon shell is a sphere of radius $\tilde{r}=3M$. The density depends nontrivially on $\tilde{r}$. As the spin increases, the density reduces and the photon shell region becomes larger. In the extremal limit ($a\to M)$ it extends in the range $M\le \tilde{r} \le 4M$.
• Figure 4: The total (integrated over $\tilde{r}$) differential-in-energy phase-space volume of the photon shell $\frac{dV_\mathrm{PS}}{dE}$ as a function of the spin $a$, at fixed $M$. It is largest at the Schwarzschild limit (blue) and smallest at the near-extremal limit (red).
• Figure 5: Differential phase-space volume of the near-extremal Kerr photon shell as a function of the proximity to extremality and to the horizon. The horizontal axis parameterizes the deviation from extremality via $\ln \kappa$, where $\kappa=\sqrt{1-a^2/M^2}$. The vertical axis parameterizes the distance from the horizon via a "NHEK-band" radial coordinate, which we choose to define here as $\tilde{r}=M(1+\frac{2}{\sqrt{3}}\kappa^p)$ so that $p=1$ corresponds to $\tilde{r}_- + \mathcal{O}(\kappa^2)$. The colorscale quantifies the logarithm of the differential (in $p$ and $E$) phase-space volume $\frac{d\mathrm{V_{PS}}}{dE dp}$. As extremality is approached, moving left on the diagram, the phase space bifurcates into two branches: the near-NHEK region ($p=1$) and the far region ($p=0$), and a gap opens up between these branches.