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Summary:
- What are the mathematical constraints on the metric?
Main Question or Discussion Point
Aside from being symmetric, are there any other mathematical constraints on the metric?
Not the ones you state. It is perfectly possible to have a metric tensor that violates your conditions, if the coordinates are chosen appropriately.Physical interpretation requires some other features
That’s clearly true for the sign of ##g_{00}##, but for the statement about the determinant?but that in no way requires the condition you impose on the particular components.
The determinant of a 3-submatrix is not constrained. The determinant of the full metric is, but I don't think that's the determinant that the poster in post #2 meant, since he used the indexes ##ik##, which usually means just the "spatial" indexes. He's welcome to correct me if I am wrong.for the statement about the determinant?
I intended i,k=0,1,2,3. Thanks.He's welcome to correct me if I am wrong.
Ah, ok. Then your constraint on the determinant is correct, but your constraint on ##g_{00}##, as noted, is not.I intended i,k=0,1,2,3.
There are plenty of examples. Just a few off the top of my head:Now I know for an example in rotating system ##g_{00}<0## for region r > c / ##\omega## where no real body cannot stay still to represent coordinate (r,##\phi##). Thanks.