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Considering a metric tensor with the signature $(-,+,+,+)$:

$g_{\mu\nu}= \begin{pmatrix} -c^2 & g_{01} & g_{02} & g_{03}\\ g_{10} & a^2 & g_{12} & g_{13}\\ g_{20} & g_{21} & a^2 & g_{23}\\ g_{30} & g_{31} & g_{32} & a^2\\ \end{pmatrix}$

If I switch the signature to $(+,-,-,-)$, what is the form of the new tensor ? $g_{\mu\nu}= \begin{pmatrix} c^2 & ? & ? & ?\\ ? & -a^2 & ? & ?\\ ? & ? & -a^2 & ?\\ ? & ? & ? & -a^2\\ \end{pmatrix}$

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up vote 2 down vote accepted

You multiply the whole metric tensor by a factor of $-1$.

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