We recently had a long thread https://www.physicsforums.com/showthread.php?t=666861 about cases where raising and lowering indices isn't completely natural, i.e., where a vector "naturally" wants to be upper-index or lower-index.(adsbygoogle = window.adsbygoogle || []).push({});

If you have a metric, then it's pretty clear to me what real-world measurement processes correspond to the use of the metric. In a Riemannian space, they're basically the ones referred to in Euclid: compass and straightedge or, in some weaker sense, a marked straightedge. In a semi-Riemannian space, you need the universe to contain at least one clock. (A ruler will also work.) Geroch has a very nice popular-level book, Relativity from A to B, that systematically works out this interpretation of GR.

What is the right model of measurement for a space where you don't have a metric but you do have parallelism, e.g., the tangent space on a manifold where no metric exists? In affine geometry, you have a measure of area and you have a measure of length along a line and lines parallel to it. You don't have angular measure, a notion of perpendicularity, congruence, or comparability of vectors that are not parallel. You can take products of vectors with dual vectors, giving real numbers. The best metaphor I've been able to come up with is the ghost of a clock. What I mean by this is that you have a clock that cruises like a ghost through space, moving inertially. You can't interact with it, so you can't, e.g., force it to travel along a curved world-line and measure its proper time. It can move at c or at speeds higher than c. Unlike a physically manipulable clock, it doesn't suffice to establish a frame of reference. You can make copies of it moving along parallel world-lines, but you can't synchronize the copies (because synchronization is perpendicularity). Basically if you take the product of a dual vector with a vector, [itex]\omega\cdot v[/itex], you can think of it as comparing the rate of one ghost-clock against the rate of another ghost-clock.

Is this a correct model? Is there a simpler, more accurate, or more natural model, or one stated in terms of more realistic measuring devices?

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# Physical model of measurement for affine geometry, dual

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