Null Geodesics and Redshift in Static Randers Spacetime

We present a concise, rigorous derivation of null geodesics and the associated frequency shift in a static Randers (Finsler–Randers) spacetime on a Minkowski background. We state explicit Lorentz–Finsler admissibility conditions, derive the canonical momentum, show which components are conserved for static one-form backgrounds, and obtain an exact expression for the locally measured photon frequency. We then perform a controlled small-A expansion that yields the exponential redshift law 1+z = exp[Φ(xobs) − Φ(xemit)] only under explicit, stated assumptions; otherwise the first-order result is linear in endpoint differences. Appendices provide a robust inversion pµ ↔ yµ for admissible null directions and a symbolic verification of the first-order expansion.