I should think it was obvious that the metric tensor, M, at a point g*Mg gives the length of g.

I generally ignore these questions because you're asking questions that take some exposition which is found in any text book and many online sites.  I also recommend "Relativity Demystified" by David McMahon.

Bretn


On 9/2/2024 5:25 PM, Alan Grayson wrote:
Brent; do me a small favor. If you are unable to answer my question, just say so. OK? AG

On Saturday, August 31, 2024 at 8:48:13 PM UTC-6 Alan Grayson wrote:

    You seem to know a lot about relativity, so please explain how the
    metric tensor can be defined unambiguously at some point P on the
    underlying manifold, spacetime, if there is an uncountable set of
    pairs on a vector space on the tangent space at some point P on
    which the metric tensor is defined (as a bilinear function which
    maps to the real numbers.) TY, AG

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