Parallel transport, as one means of quantifying the curvature of a coordinate space, enables(adsbygoogle = window.adsbygoogle || []).push({});

changes in a vector's components, when it is carried around variously oriented loops in that space, to beproperlymeasured, i.e. by comparisons made at thesamelocation. Those changes which are independent of the size of the loop are measurable manifestations of local curvature, and can be coded into components of the Riemann tensor. Have I got this right?

Now spacetime has four dimensions, three of space and one of time. Transporting anything

around a loop takes time, so a one-way leg along the time dimension must in principle be part of any loop. It is therefore neverquitepossible --- especially in cosmology! --- to compare the original vector with its parallel-transported version at the same location in spacetime, as is possible with a loop on the 2-D Earth's surface (sometimes used to explain how parallel transport measures curvature).

How could the curvature of spacetime on say, a cosmological scale then be measured, even in thought experiments? And is the separation of spacetime curvature into that of space sections and of time thus moot?

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# Measuring curvature with parallel transport

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