Looks like I'm spinning my wheels and getting nowhere, since the Metric 
Tensor has nothing to do any particular spacetime one is assuming and its 
energy distribution, when writing and trying to solve EFE's, since the 
inner products on the basis vectors in V, are completely arbitrary, which 
implies the same situation for the associated tensor product space, which 
defines the dual map. It's hard to see why this tensor has any value. AG

On Saturday, May 14, 2022 at 11:38:00 AM UTC-6 Alan Grayson wrote:

> It appears in Einstein's Field Equations and presumably allows us to 
> calculate the Metric of spacetime, which is a bilinear map from a pair of 
> spacetime coordinates to a real number. If the foregoing is correct, what 
> is the definition of distance between any pair of spacetime points, and 
> does it depend, as I think it does, on the energy distribution of the type 
> of spacetime under consideration? That is, is it the free fall path length 
> between any two spacetime points, or if not, then what? TY, AG

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