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At low speeds and assuming pressure ##P=0##,

[tex]T^{\alpha \beta} = \rho U^\alpha U^\beta [/tex]

[tex] g_{\alpha \mu} g_{\gamma \beta} T^{\alpha \beta} = \rho g_{\alpha \mu} g_{\gamma \beta} U^\alpha U^\beta [/tex]

[tex]T_{\gamma \mu} = \rho U_\mu U^\beta g_{\gamma \beta} [/tex]

Setting ##\gamma = \mu = 0##:

[tex] T_{00} = \rho U_0 U^\beta g_{0 \beta} [/tex]

Since ##g_{0 \beta} \backsimeq \eta_{0 \beta} ## and the only non-zero term is ##\eta_{00} = -1##, combined with ##U_\alpha U^\alpha = -c^2##:

[tex] T_{00} = \rho U_0 U^0 g_{00} = \rho c^2 [/tex]

I'm still learning tensor calculus, would that be considered a proper derivation?

Also, is ##g_{ij} \backsimeq \eta_{ij}## the reason why ##T_{ij} \backsimeq 0##?

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# Quick one-line on lowering indices

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