Interferometric Signatures of Higher-Order Photon Subrings in Polarized Supermassive Black Hole Accretion

1. Spacetime Geometry & Null Geodesics

In the vicinity of a Kerr black hole with event horizon $r_+ = M + \sqrt{M^2 - a^2}$, null geodesics experience extreme gravitational deflection. Photons passing near the critical impact parameter $\tilde{b}_c$ execute multiple half-orbits around the photon orbit sphere before escaping to infinity.

These trajectories produce an infinite sequence of self-similar subrings indexed by integer $n \in \{0, 1, 2, \dots\}$, where $n$ denotes the count of half-orbits executed in the Kerr ergosphere.

2. Lyapunov Exponent & Universal Scaling

The thickness $w_n$ and flux density $F_n$ of successive subrings scale exponentially according to the principal Lyapunov exponent $\gamma$:

In the Fourier $(u, v)$ interferometric domain, the $n=1$ subring produces a damped sinusoidal ripple with period $(u^2 + v^2)^{1/2} = 1/d_{\text{ring}}$, which can be decoupled from the diffuse $n=0$ accretion disk.

3. Space-VLBI Feasibility (Moon / L2 Baselines)

While ground-based millimeter VLBI is limited to baselines $B \le 10,000\text{ km}$ ($~20\,\mu\text{as}$ resolution at 345 GHz), deploying a 3-meter sub-millimeter dish to the Lunar surface or Sun-Earth L2 creates baselines up to $B \approx 384,000\text{ km}$, yielding angular resolution down to $0.5\,\mu\text{as}$, allowing clean isolation of the $n=1$ subring.