Measuring curvature with parallel transport

[oldman]

Parallel transport, as one means of quantifying the curvature of a coordinate space, enables
changes in a vector's components, when it is carried around variously oriented loops in that space, to be properly measured, i.e. by comparisons made at the same location. 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 never quite possible --- 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?

 

[George Jones]

How could the curvature of spacetime on say, a cosmological scale then be measured, even in thought experiments?

At any event in spacetime, spacetime curvature can be measured by looking at tidal forces. Take a collection of test particles that is slightly extended in space and measure how the shaped of the collection deforms over a small time interval.

And is the separation of spacetime curvature into that of space sections and of time thus moot?

Why?

 

[oldman]

At any event in spacetime, spacetime curvature can be measured by looking at tidal forces. Take a collection of test particles that is slightly extended in space and measure how the shaped of the collection deforms over a small time interval.

Yes. I agree. As with John Baez's clusters of coffee grounds. But I have cosmological puzzles in mind. For example, if one waited long enough, would the test particles move apart because 'the universe is expanding'? Or would they move together as Peacock described recently? I was hoping to avoid this kind of puzzle by thinking of curvature measured by parallel transport instead. But then I have difficulty with parallel transport, too, as I explained.

As to

why?

I just can't see how this method would work, even in principle, in spacetime and allow one to find components of the Riemann tensor and then talk of space foliations, etc.

But perhaps I'm just getting too muddled.