Schwarzschild Geodesics & Lyapunov Exponent¶
Painlevé-Gullstrand Setup¶
\[ds^2 = -\!\left(1 - \frac{2M}{r}\right)dT^2 + 2\sqrt{\frac{2M}{r}}\,dT\,dr + dr^2 + r^2\,d\Omega^2\]
Regular at r = 2M (no coordinate singularity). Time T = PG time.
Derivation: λ = κ¶
Outgoing radial null geodesic: dr/dT = 1 − √(2M/r).
Near horizon: let r = 2M + ε, ε ≪ 2M.
\[\frac{d\varepsilon}{dT} = \kappa\varepsilon + O(\varepsilon^2), \qquad \kappa = \frac{1}{4M}\]
Solution: ε(T) = ε(0)·e^{κT}
Lyapunov exponent:
\[\lambda = \kappa = \frac{1}{4M}\]
Corrected Error (v5 → v6)
Earlier drafts contained a contradiction: claiming λ = 0 at the horizon (S-N bifurcation) while also claiming λ = κ > 0 (MSS saturation). These are mutually exclusive. Resolution: λ = κ > 0 is the correct statement for radial null congruences. The bifurcation (λ = 0 transition) is a different physical regime — the interior attractor.
MSS Comparison¶
| Surface | λ | Status |
|---|---|---|
| Event horizon r = 2M | κ = 1/(4M) ≈ 0.25/M | MSS saturated ← unique |
| Photon sphere r = 3M | 1/(3√3 M) ≈ 0.192/M | Below MSS bound |
| Asymptotic r → ∞ | 0 | No divergence |
Thermodynamic Reinterpretation¶
| Classical expression | Lyapunov reading |
|---|---|
| T_H = κ/(2π) | Thermal scale of geodesic chaos |
| S_BH = ∫dM/T_H | Integral of reciprocal chaos rate ∫(2π/λ)dM |
| Negative heat capacity | Smaller M → larger λ → faster scrambling |
MSS Bound Statement¶
\[\lambda_{\text{OTOC}} \leq 2\pi T\]
For Schwarzschild: λ = κ = 2πT_H. Exactly saturated at horizon and only at horizon.