Constraints on Metric Tensor

quickAndLucky
TL;DR Summary
What are the mathematical constraints on the metric?
Aside from being symmetric, are there any other mathematical constraints on the metric?

Gold Member
Physical interpretation requires some other features like
$$g_{00}>0, g=det(g_{ik})<0$$
in (+---) 0123 convention.

Last edited:
Mentor
Physical interpretation requires some other features

Not the ones you state. It is perfectly possible to have a metric tensor that violates your conditions, if the coordinates are chosen appropriately.

Physically, the metric tensor must have a (1, 3) signature (or (3, 1) if we choose the opposite signature convention), but that in no way requires the condition you impose on the particular components.

anuttarasammyak
Mentor
but that in no way requires the condition you impose on the particular components.
That’s clearly true for the sign of ##g_{00}##, but for the statement about the determinant?

anuttarasammyak
Mentor
for the statement about the determinant?

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.

Gold Member
He's welcome to correct me if I am wrong.
I intended i,k=0,1,2,3. Thanks.

Mentor
I intended i,k=0,1,2,3.

Ah, ok. Then your constraint on the determinant is correct, but your constraint on ##g_{00}##, as noted, is not.

anuttarasammyak
Gold Member
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.

Mentor
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.

There are plenty of examples. Just a few off the top of my head:

Null coordinates in Minkowski spacetime, and the various kinds of null charts in curved spacetime (for example, Eddington-Finkelstein, Kerr-Schild).

Painleve coordinates in Schwarzschild spacetime, at and inside the event horizon.

Static coordinates in de Sitter spacetime, at and outside the cosmological horizon.

anuttarasammyak