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Given a Lagrangian $$\mathscr{L} = -\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - V(\phi),\tag{1}$$ is the metric always with signature $(-, +, +, +)$? It seems to me that This post Sign Convention in Field Theory says that the metric is with this signature.

However, when I calculate the energy-momentum tensor, the $00$ component is negative, which is in contradiction to this post Does metric signature affect the stress energy tensor?.

Note I'm using the definition $$T^{\alpha\beta} = \frac{\partial \mathscr{L}}{\partial (\partial_\alpha \phi_i)}\frac{\partial \phi_i}{\partial x_\beta} - \mathscr{L}\eta^{\alpha\beta}\tag{2}$$ but I'm thinking this could depend on the signature of the metric.

Qmechanic
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Geigercounter
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    The battle of the metric signature rages on. The best way to see if the metric signature affects things, is to insert a lot of $\eta^{tt}$ terms to keep track of the signature everywhere, and see if they cancel out. IIRC, stress energy tensor has a factor of that, so it should be affected. – naturallyInconsistent May 21 '23 at 12:14
  • Comment to the post (v3): Where did you get eq. (2) from? It is incompatible with $(−,+,+,+)$. – Qmechanic May 21 '23 at 12:31
  • @Qmechanic From textbooks that use the $(+,-,-,-)$ convention... So just multiply by $-1$? – Geigercounter May 21 '23 at 12:32
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    $\uparrow$ Yes. – Qmechanic May 21 '23 at 18:45

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