The Metric Tensor

[Alan Grayson]

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

[Alan Grayson]

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

[Alan Grayson]

I'm getting it. Defined abstractly, without reference to any particular spacetime, the inner products from which the metric tensor derives, must be abstract. It's only by applying the EFE's that we can SOLVE for the specific metric tensor associated with a particular spacetime. And its value as a tensor field DOES depend on the energy distribution of the spacetime under consideration. AG

[Brent Meeker]

Of course it depends on the energy distribution.  That's what's on the right side of Einstein's equation and if it's given you can solve for the metric on the left side.  The proper time between two time-like events is the metric measure along a geodesic between them, which is  force-free path.  But it's possible, in a curved spacetime, for there to be more than one geodesic between the same two events.

Brent

On 5/14/2022 10:38 AM, 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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[Alan Grayson]

On Tuesday, May 17, 2022 at 11:42:37 AM UTC-6 meeke...@gmail.com wrote:

Of course it depends on the energy distribution.  That's what's on the right side of Einstein's equation and if it's given you can solve for the metric on the left side.  The proper time between two time-like events is the metric measure along a geodesic between them, which is  force-free path.  But it's possible, in a curved spacetime, for there to be more than one geodesic between the same two events.

Brent

The author of the GR YouTube videos I am following initially confused me -- by doing a general case of constructing a metric tensor from an arbitrary linear metric defined on the basis vectors of a 4-D vector space. Later, using various spacetimes defined by energy distributions and well-known in the GR literature, he does what you suggest -- but for me those lectures are still TBD's. TY, AG