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Photon Sphere as Quantum Inverted Harmonic Oscillator

Gemini, 2026-03-07 · Novel QS-to-oscillator mapping

Construction

At the photon ring of a black hole (\(r = 3M\) Schwarzschild), null geodesics are unstable — they orbit but diverge exponentially under perturbation. The Penrose limit of the near-photon-ring metric yields an inverted harmonic oscillator potential:

\[V(x) = -\frac{1}{2}\omega^2 x^2\]

where \(\omega\) is set by the generalized surface gravity of the photon ring:

\[\omega = \kappa_{\text{photon}} = \frac{1}{3\sqrt{3}M}\]

RTSG Interpretation

The uninstantiated QS geodesics at the photon ring map to a quantum inverted harmonic oscillator. This yields:

  1. Lyapunov exponent \(\lambda = \omega = \kappa_{\text{photon}}\) directly from the oscillator frequency
  2. Quasi-normal modes (QNMs) from the quantization of the inverted oscillator spectrum
  3. The photon ring is a QS resonance — uninstantiated potentiality orbiting at the boundary between capture (instantiation into BH) and escape (instantiation into far-field PS)

Chaos Bound Connection

The photon ring Lyapunov exponent satisfies:

\[\lambda_{\text{photon}} = \kappa_{\text{photon}} \leq \kappa_H\]

The event horizon has strictly higher surface gravity than the photon ring. This is the hierarchy: the horizon is the maximal chaos processor (CS bandwidth limit), the photon ring is a sub-maximal resonance.

Connection to GRF Essay

Do NOT add to GRF 2026

The GRF essay deliberately avoids the photon sphere (attack surface identified by GPT-5.4 / o3 review). This formulation belongs in the cosmological vision paper or a standalone QNM paper.

Relation to Prior Debate

See Photon Sphere Debate — the inverted oscillator map resolves the earlier dispute by providing a clean derivation that doesn't mix local temperatures with system bounds.

Generalized Surface Gravity at the Photon Sphere

Gemini, 2026-03-07 · Extends the inverted oscillator map

For null geodesics (massless QS states), the classical massive-particle bound \(\lambda \leq \kappa_H\) is replaced by a generalized surface gravity evaluated at the photon sphere:

\[\kappa(r_{\text{ph}}) = \frac{1}{3\sqrt{3}M}\]

The instability of circular null geodesics dictates the quasinormal mode (QNM) decay rate. In the RTSG interpretation, the photon sphere is the specific frequency filter of the CS operator — it regulates the maximal bandwidth at which purely massless QS states can be localized and collapsed into PS geometry.

QNM connection: In the eikonal limit (\(\ell \to \infty\)), QNM frequencies satisfy:

\[\omega_{\text{QNM}} \approx \Omega_{\text{ph}} \ell - i\left(n + \frac{1}{2}\right)\kappa(r_{\text{ph}})\]

where \(\Omega_{\text{ph}}\) is the orbital frequency at the photon sphere. The imaginary part — the decay rate — is set by \(\kappa(r_{\text{ph}})\), confirming the photon sphere as the CS bandwidth regulator for massless modes.