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I know the the energy momentum tensor $T^{uv}$ and $T^{00}$ stands for the energy density. But what about $T_{00}$? I think that for the metric tensor $g_{ij}$, $T_{00}=g_{0i}T^{ij}g_{0j}$ holds. Then, what is the meaning of $T_{00}$? It seems that $T_{00}$ is supposed to be a energy density just like $T^{00}$. But, the relation $T_{00}=g_{0i}T^{ij}g_{0j}$ mixes up everything, so I am totally stuck at figuring out what $T_{00}$ really means...Could anyone please help me?

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  • $\begingroup$ $T_{\mu\nu} = T^{\alpha\beta}g_{\alpha\mu} g_{\beta\nu}$ $\endgroup$ – John Rennie Feb 4 '18 at 8:58
  • $\begingroup$ Then how can I determine that $T_{00}$ is just an energy density? $\endgroup$ – Keith Feb 4 '18 at 9:02
  • $\begingroup$ $T^{00}$ is only the energy density in flat spacetime (or I suppose if you're using the Fermi normal coordinates) and in that case $T^{00}=T_{00}$. $\endgroup$ – John Rennie Feb 4 '18 at 9:26
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You cannot select out components of a tensor a call them energy or something else. That is the point of general covariance. For instance if one has a four moment $p_\mu$ then the energy seen by an observer with four velocity $u^\mu$ is the Lorentz scalar $p.u$.

So what you want to ask is what is the energy density seen by an observer with four velocity $u$ on a spacelike infinitesimal volume $dS_\mu$. Well it's simply the scalar

$$ \rho=u.T.dS $$

where the contraction of indices is obvious. You can read more about this in the chapter on hypersurfaces in Eric Poisson's book or see an application of this for energy and angular momentum in chapter 4 of my paper https://arxiv.org/pdf/0806.2309.pdf.

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