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I found the following definitions:

  • The stress–energy tensor (sometimes stress–energy–momentum tensor or energy–momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime.

  • Flux is a simple and ubiquitous concept throughout physics and applied mathematics describing the flow of a physical property in space, frequently also with time variation. It is the basis of the field concept in physics and mathematics.

So I presume from above that the property which flows in GR is energy and/or momentum. Is this a correct?

Under what circumstances does it flow? and from where to where?

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The stress-energy tensor describes the flow of 4-momentum through spacetime. Without worrying too much about index placement or order, let's say $T^{\mu\nu}$ is the flux of $\mu$-momentum in the $\nu$-direction.

When $\mu$ is a spatial index, this is a component of ordinary $3$-momentum, and when $\mu$ is the time index we can call the flowing quantity energy.

Similarly, when $\nu$ is spatial we have a flux in a spatial direction (technically, across a surface whose normal is spacelike). When $\nu$ is the time index the quantity is flowing in the time direction, and we can call this a density.

Thus $T^{00}$ is energy density, $T^{0i}$ is the flux of energy in the $i$-direction, $T^{i0}$ is $i$-momentum density, and $T^{ij}$ is the $j$-flux of $i$-momentum.

The distinction between flow of $3$-momentum and flow of energy is a somewhat artificial remnant of Newtonian thinking, as is the distinction between fluxes in spatial directions and densities of quantities.

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