In this German Classical Mechanics lecture by Frederic Schuller, it is given that a Newtonian spacetime with an absolute inertial frame is one in which

$$ \nabla_{v} G=0$$

Where $\nabla_v$ is the covariant derivative in the direction of vector $v$, $G$ is space-metric, and, the above true for any vector $v$.

I understand the above condition means that the space part of vector is homogenous i.e: in Newtonian space time, the lengths and angles measured of an object is independent of where you are in space for a fixed point in time.. but, why is this the condition of an inertial system? Would it be that in a non inertial system for different points in space, the rule for measuring distances are different? It doesn't make sense to me..



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