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What does the stress-energy tensor describe?

The tt term of the stress-energy tensor expresses energy density. So we all say.

Energy density=energy per unit volume. Why in the world is an expression governing the dynamics of four dimensional spacetime talking about something over a three dimensional xyz submanifold? This seems too peculiar to me.

Do we have a problem admitting time on equal footing?

Point 2: the xt terms are alternately expressed as momentum flux and shear stress. Are these the same thing describing the same phenomena? I'm sure there are physical situations were shear stress is not always equal to momentum flux.

I hold some secret suspicions that these terms are not properly understood.

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