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As I understand it, the component $T_{00}$ of the stress-energy tensor contains the energy density (which equals the mass density), $T_{0i}$ are the impulse flows (intuitively speaking, the velocities), and $T_{ii}$ (the diagonals of the stress tensor) are the pressures in the respective directions.

In which of these components 'goes' the kinetic energy of the matter's heat?

On a microscopic level, heat is just motion, so I suppose it should be in $T_{0i}$.

However, if you look at the sun as one homogeneous body like in a Schwarzschild solution, I'm not sure where to put it.

  • Is it just energy, so it's part of $T_{00}$? This would mean that $T_{00}$ does not equal the mass density.
  • Or is it part of the pressure, thus going to $T_{ii}$?

marked as duplicate by Kyle Kanos, John Rennie general-relativity Jul 26 '15 at 16:24

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    $\begingroup$ Hi Bastian. Have a look at my answer to the duplicate Kyle suggested. Hopefully it should be obvious that random thermal motion contributes to the diagonal elements (excluding $T_{00}$). If you think it isn't obvious comment here and I can reopen this question. $\endgroup$ – John Rennie Jul 26 '15 at 16:27
  • $\begingroup$ Thanks John! It's obvious now. Somehow I thought $T_ii$ is a directed 3-vector that represents some flow, but if it stands for an undirected pressure, then it's clear that thermal energy is part of it. $\endgroup$ – Bass Jul 29 '15 at 17:50