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If one were to derive the stress-energy tensor for a metric with $(+,-,-,-)$ signature would it be different from the stress-energy tensor derived from the same metric but with $(-,+,+,+)$ signature?

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The sign convention for the stress-energy-momentum (SEM) tensor $T^{\mu\nu}$ is usually chosen such that the energy density $T^{00}$ is positive.

This implies that the Hilbert SEM tensor is defined as

$$\tag{1} T^{\mu\nu}~:=~\pm \frac{2}{\sqrt{|g|}}\frac{\delta S}{\delta g_{\mu\nu}}, \qquad T_{\mu\nu}~:=~\mp \frac{2}{\sqrt{|g|}}\frac{\delta S}{\delta g^{\mu\nu}},$$ for Minkowski sign convention $(\mp,\pm,\pm,\pm)$, respectively.

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The straight answer to your question is no, the value of the components of the stress energy tensor do not change according to the metric signature. Qmechanich showed you how the formulas change, depending on the metric signature, just in order to keep the component values the same.

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