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The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields.
The symmetry of the tensor is as for a general stress–energy tensor in general relativity. The trace of the energy–momentum tensor is a Lorentz scalar ; the electromagnetic field (and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so its energy–momentum tensor must have a vanishing trace.
For completion, one must make hypotheses on the forms of τ and p, that is, one needs a constitutive law for the stress tensor which can be obtained for specific fluid families and on the pressure. Some of these hypotheses lead to the Euler equations (fluid dynamics) , other ones lead to the Navier–Stokes equations.
That is, the change in the internal energy of the substance within a volume is the negative of the amount carried out of the volume by the flow of material across the boundary plus the work done compressing the material on the boundary minus the flow of heat out through the boundary. More generally, it is possible to incorporate source terms. [2]
In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function. The incompressible Navier–Stokes equation is a differential algebraic equation , having the inconvenient feature that there is no explicit mechanism ...
The stress–energy tensor of a perfect fluid contains only the diagonal components. In space-positive metric signature tensor notation, the stress–energy tensor of a perfect fluid can be written in the form = (+) +,
Energy flow, flow of energy in an ecosystem through food chains; Energetics (disambiguation), the scientific study of energy in general; Stress–energy tensor, the density and flux of energy and momentum in space-time; the source of the gravitational field in general relativity; Food energy, energy in food that is available; Primary energy ...
Clearly this pseudotensor for gravitational stress–energy is constructed exclusively from the metric tensor and its first derivatives. Consequently, it vanishes at any event when the coordinate system is chosen to make the first derivatives of the metric vanish because each term in the pseudotensor is quadratic in the first derivatives of the ...