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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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