I just wanted to clarify some notation regarding the metric tensor. In books I often read the metric has signature -+++, does this mean I can make a change of coordinates s.t. the metric g always looks like the 4x4 identity matrix with the g_11 component being -1? And if so the determinant is always -1 cause determinant is an invariant under arbitrary coordinate transformations? And in the simple case of a "spatial" metric (I guess thats a GR term) the metric has signature +++ which means its the identity? Therefore the determinant is always 1 and also the determinant of the inverse is always 1? Could someone please explain? I guess its a very basic question.. thanks in advance!
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The signs are those of eigenvalues of the metric, viewed as a matrix. The determinant of this matrix changes under general coordinate transformations (in fact, $\sqrt{|g|}d^nx$ is the invariant infinitesimal $n$-dimensional volume element). However, the eigenvalues' signs don't. For a $++$ example, compare $ds^2=dx^2+dy^2=dr^2+r^2d\theta^2$. However, replacing $ds^2$ with $-ds^2$ switches to a different convention, with all signs flipped; if $n$ is odd, $g$ also changes sign.
J.G.
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Ah alright, so I know about the signature but when I have to calculate the determinant of the metric I really need to know the specific components of the metric, is that right? For instance for your first example I would get 1 as the determinant and (1+r²) for the second? – Orange123 Feb 25 '20 at 22:35
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@Orange123 Just $r^2$, actually; perhaps you were thinking of the trace. – J.G. Feb 25 '20 at 22:43
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1oh yeah sure, thanks for pointing out! And thanks for your quick help :) – Orange123 Feb 25 '20 at 22:50