# How to calculate $T_{00}$ of the energy-momentum tensor?

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?

• $T_{\mu\nu} = T^{\alpha\beta}g_{\alpha\mu} g_{\beta\nu}$ – John Rennie Feb 4 '18 at 8:58
• Then how can I determine that $T_{00}$ is just an energy density? – Keith Feb 4 '18 at 9:02
• $T^{00}$ is only the energy density in flat spacetime (or I suppose if you're using the Fermi normal coordinates) and in that case $T^{00}=T_{00}$. – John Rennie Feb 4 '18 at 9:26

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