EFE's question regarding Ricci scalar

[unchained1978]

Quick question about the EFE's. When writing the einstein tensor G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}, and using the definition of the Ricci scalar R=g^{\mu\nu}R_{\mu\nu}, how does this not give you problems when you expand out R so that the second term becomes -\frac{1}{2}g^{\mu\nu}R_{\mu\nu}g_{\mu\nu}=-2R_{\mu\nu} when evaluating the trace, giving you the EFE's as R_{\mu\nu}=-4πGT_{\mu\nu}?
Any help would be appreciated.

 

[George Jones]

Quick question about the EFE's. When writing the einstein tensor G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}, and using the definition of the Ricci scalar R=g^{\mu\nu}R_{\mu\nu}, how does this not give you problems when you expand out R so that the second term becomes -\frac{1}{2}g^{\mu\nu}R_{\mu\nu}g_{\mu\nu}=-2R_{\mu\nu} when evaluating the trace, giving you the EFE's as R_{\mu\nu}=-4πGT_{\mu\nu}?
Any help would be appreciated.

In anyone term, an index can only appear at most twice, so -\frac{1}{2}g^{\mu\nu}R_{\mu\nu}g_{\mu\nu} is not legal. Maybe you want to write R=g^{\alpha\beta}R_{\alpha\beta}, or maybe you want do do something like
<br /> \begin{align}<br /> \left( R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} \right) g^{\mu \nu} &amp;= 8\pi T_{\mu\nu} g^{\mu \nu}\\<br /> -R = 8\pi T^\mu_\mu.<br /> \end{align}<br />
Using -R = 8\pi T^\alpha_\alpha (after relabeling to avoid the same symbol being used as both a free index and a summed index) in R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu} = 8\pi T_{\mu\nu} gives R_{\mu\nu} = 8\pi \left( T_{\mu \nu} - \frac{1}{2}T^\alpha_\alpha g_{\mu\nu} \right), another useful form of the EFE.

 

[unchained1978]

Thanks, I always thought there was some index trickery involved in resolving this, but I never knew about multiple repeated indices being disallowed.