Hidden SO(4) Symmetry and the Laplace-Runge-Lenz Vector in Kepler and Kerr Spacetimes — E8 Intelligence Research

FINDING: The Laplace-Runge-Lenz (LRL) vector reveals a hidden SO(4) symmetry in the Coulomb-Kepler problem, causing accidental degeneracy and closed elliptical orbits; its perturbative extension to Kerr spacetime yields perihelion precession. | MATH: Conserved LRL vector \(\mathbf{A} = \mathbf{p} \times \mathbf{L} - m k \hat{\mathbf{r}}\); for bound orbits, eccentricity \(e = |\mathbf{A}|/(m k)\); orbit equation \(r = \frac{L^2/mk}{1 + e \cos\theta}\); degeneracy \(n^2\) (hydrogen) from SO(4) ≅ SU(2)×SU(2); precession rate in Kerr: \(\Delta \omega \propto a/r^{3/2}\) (first-order in spin \(a\)). | CONNECTION: The eccentricity \(e\) directly links to the golden ratio via the apsidal angle: for \(e = 0.618\) (inverse golden ratio conjugate), the orbit's radial oscillation period ratio yields \(\omega_r/\omega_\theta = 1/\sqrt{1-e^2} = 1/\sqrt{1-0.3819} = 1/\sqrt{0.618} = 1.272\) — a near-\(5/4\) resonance; the SO(4) symmetry group is isomorphic to the root system of \(D_2\) (dihedral), w

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com