# What component of the stress–energy tensor contains the kinetic energy of heat? [duplicate]

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 StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jul 26 '15 at 16:24

• 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. – John Rennie Jul 26 '15 at 16:27
• 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. – Bass Jul 29 '15 at 17:50