Not so fast. Let me remind you that a statement by you is part of the dispute, and despite more than ample indication that we might not have the same meanings in mind when using the same words, you have not clearly committed to one yet.
The disputed part is emphasized like this here (it was...
OK, one last try. Remember this?
It's somewhat unfortunate you truncated my explanations right before the moment they had a chance to avoid a misunderstanding. The full quote is:
It should be clear by now the 1st physical measurement in question was always meant to yield the g_{ab} at e, and...
What a pity. Well: g and \Gamma are obtained by measurement; h is obtained by integration.
This is not the point; not necessarily an uninteresting or irrelevant one, mark. Let me just remind you that you've had the following info:
You've also had clear indications that interferometry is how...
Yes, vectors and other multi-dimensional quantities can be measured. You do it by specifying a reference frame, and then, you get simultaneous readings -scalars assuredly, present and the same for all to see, whatever their position- which are the components of what you want measured, a 3-dim...
Okay... Only there's a bit of the way you'll have to do by yourself. Which is to realize:
1) all of this is done on one and same manifold M, in a neighborhood of one 4-point e, so we pick one set of coordinates around e and write everything in the basis of tangent vectors.
2) not everything...
Ah, that's the main contention then.
Measurement #1: interferometry. Yields components of the fundamental metric. Allows SR approximation in an infinitesimal neighborhood of event p, i. e. in the tangent space at p.
Measurement #2: gravimetry. Yields coeffs of the actual connection, the one...
Gee... How do you understand this then?
Please quote the post where I claimed otherwise.
Continuing with the metaphor, " the altimiter reading would be based on J and the gravimeter reading would be based on K" makes sense if, when discussing altimeter readings we are required to use...
Yeah right, that's what I mean: same manifold, same point, same coordinates so same tangent vectors; for I assume this is what your J^{\mu}'s are; and YET: different lengths obtain.
A bit like in "Heck, my altimeter says we're at 4792m on the summit where IGN's gravimeters said 4810m". See...
Sorry, this time it's me not making of sense of what you wrote.
2 different metrics mean different physical measurements, by 2 different mechanisms, at the same point yield 2 different results whereas GR *posits* they must be equal.
1st physical measurement at point e: establish the metric...
A metric tensor on M is just a tensor field required to satisfy certain properties, and certainly you don't need to wait for M to be Riemannian to state what these properties are; cf. the WP definition:
http://en.wikipedia.org/wiki/Metric_tensor
They take care to mark the difference with...
Planet Math http://planetmath.org/encyclopedia/RiemannNormalCoordinates.html has one where I don't see any dot product. It is labelled "Riemann normal coordinates" however it relies solely on the existence of geodesics, and coincides with what WikiPedia labels "geodesic normal coordinates"...