- #1
unscientific
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[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##?