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In $n$ dimensions, the Ricci scalar curvature $R$ at a point measures how much the volume $V$ of a small $n$-dimensional ball of radius $\epsilon$ around that point differs from the Euclidean value $V_E$: $$\frac{V}{V_E}=1-\frac{R}{6(n+2)}\epsilon^2+O(\epsilon^4).$$ I am not aware of similar formulas for higher-order invariants, but that doesn’t mean they ...

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Provided all observers agree when it is zero, they will all agree on the same speed of light. As far as I can see, the universal speed of light per se imposes no other requirement. But the universal speed of light is not the only input. We also require there to be a single underlying reality behind the different observations made in the different frames. ...

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The physical implication would be that different inertial observers would find different values for the speed of light. It is the assumption that c is constant for all observers that leads to that particular equation for the interval.

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