= Reverse triangle inequality for future timelike vectors
{title2=$\sqrt{g(U+V,U+V)}\geq\sqrt{g(U,U)}+\sqrt{g(V,V)}$}
Use signature $(+---)$ and future-pointing timelike vectors $U,V$. Their time components exceed their spatial magnitudes, so the same holds for their sum by the Euclidean triangle inequality. In the rest frame of $U$, write $U=(a,0)$, $a=\sqrt{g(U,U)}>0$, and let $b=\sqrt{g(V,V)}>0$. Then $g(U,V)=aV^0\geq ab$. Expanding the squared norm of $U+V$ proves the displayed inequality. Equality requires $V$ to have zero spatial component in this frame, equivalently $V$ is a positive scalar multiple of $U$. The shared future time orientation is essential.
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