stress-energy tensor

The stress-energy tensor is a tensor quantity in relativity. It describes the flow of energy and momentum and is therefore sometimes referred to as energy-momentum tensor. It satisfies the continuity equation

\nabla_a T^{ab}=T^{ab}{}_{;a}=0

The quantity

\int d^3x T^{0b}

over a spacelike slice gives the energy-momentum vector. This tensor is the Noether current associated with spacetime translations. In general relativity, this quantity acts as the source of spacetime curvature, and is the current density associated with gauge transformations (in this case coordinate transformations) by Noether's theorem. In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy-momentum vector in a general curved spacetime.

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