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 \(C^1\) one-form \(\eta\) can have the property that, along every nonconstant affinely parametrized null geodesic \(\gamma:\R\to M\) , the function \(\eta(\dot\gamma)\) 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 \((\nabla_v\eta)(v)\) on normalized null vectors, while homogeneity forces the speed of any fixed null geodesic to remain bounded. Compactness also bounds the norm of \(\eta\) , contradicting the asserted nonzero linear growth. Only null completeness, rather than full geodesic completeness, is used.