A Compactness Obstruction to Linear Growth Along Null Geodesics

Let (M,g) be a compact semi-Riemannian manifold of indefinite signature whose null geodesics are complete. We prove that no C¹ one-form η can have the property that, along every nonconstant affinely parametrized null geodesic γ: ℝ → M, the function η(γ̇) is affine with nonzero slope. This gives a negative answer to Question 9.2.1 in the Burns–Matveev list of open problems about geodesics. The proof normalizes the null cone by an auxiliary Riemannian metric. Compactness then gives a uniform lower bound for the quadratic form (∇_vη)(v) on normalized null vectors, while homogeneity forces the speed of any fixed null geodesic to remain bounded. Compactness also bounds the norm of η, contradicting the asserted nonzero linear growth. Only null completeness, rather than full geodesic completeness, is used. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.