A Parameter-Free Closed-Form Approximation for the Period of the Relativistic Harmonic Oscillator

A parameter-free closed-form approximation is derived for the period of the special-relativistic harmonic oscillator. The exact period is expressed in terms of complete elliptic integrals K and E with modulus k² = (γ−1)/(γ+1), where γ is the Lorentz factor at maximum velocity. A normalization function q(k′) = Tk′/8 renders the period smooth and bounded on k′ ∈ [0, 1], with all four boundary values and slopes expressible in π alone. A rational approximation satisfying all four anchor conditions exactly is constructed with coefficients a = 3−π and b = 16/π−5, achieving a maximum error of +0.128% at γ ≈ 6.62. The derivation follows the normalization-and-anchoring methodology of the author's prior work on classical oscillator periods. Part of a series: AGM · Ellipse perimeter · Nonlinear pendulum · Quartic anharmonic oscillator · Asymmetric cubic oscillator