# Critical growth and geodesic connectedness in Lorentz manifold

What is the deep ("heuristic") reason why the quadratic growth of $$\beta$$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $$\mathcal M = \mathcal M_0 \times \mathbb{R}$$, where $$\langle \cdot, \cdot \rangle_L = \langle \cdot, \cdot \rangle_R - \beta(x) dt$$ and $$\mathcal M$$ is a complete Riemannian manifold?

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