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We have the metric given by a matrix $g_{\mu\nu}$, however, some textbooks define it as: $$g_{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$ And others as: $$g_{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix}$$ Is there is a specific physical significance to either one of the choices or is it simply a convenience?

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There is no physical significance. Choose whichever makes your calculation easier.

Of course, if you want to compute something physical like, say, the proper time, you need to put in a negative sign, or not, depending on your choice.

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