Stress energy momentum tensor

In 0(3) electrodynamics, the stress energy momentum tensor is defined [11-20] as... 

This four-tensor is sometimes called stress-energy tensor or stress-energy-momentum tensor . It describes the density and fliix of energy and momentum in spacetime, generalizing the stress tensor in Newtonian mechanics. 

For the Maxwell field, the energy-momentum tensor Tfi(A) derived from Noether s theorem is unsymmetric, and not gauge invariant, in contrast to the symmetric stress tensor derived directly from Maxwell s equations [318], Consider the symmetric tensor 0 = T + AT, where... 

We base our considerations on the symmetric energy momentum tensor 7" rather than the canonical 0". Both versions of the energy momentum tensor, of course, satisfy identical continuity equations, i.e. all physical results are independent of this choice. P represents a covariant combination of the energy, momentum and stress densities of the system. 

The Schwarzschild solution in nonempty space estimates the stress or energy-momentum tensor T p in terms of an incompressible perfect fluid medium with the same symmetry as before and serves as a simple model of a star. To get the complete picture the interior solution for a sphere of perfect fluid of radius ro is joined continuously with the free-space solution that applies at r > ro > 2m. As before m = nM/( , where M is the mass of the fluid sphere. 

According to the statements made above, the J integral can only be nonzero if there is a singularity in the energy-momentum tensor. This is indeed the case if there is a crack tip in an elastic material As shown in section 5.2.1, stresses and strains become infinite if we approach the crack tip, see equation (5.1) ... 

The Ricci tensor that represents the geometry of space is next equated with the so-called energy-momentum (stress) tensor of the matter field that defines the influence of matter and field energy... 

The energy density concepF has been formulated here in terms of stress tensor in general relativity. Electron spin vorticity has been hidden in the energy-momentum... 

It is composed of the stress tensor t — 2p S — 1/3 SijTr S)) (momentum equations), the energy flux u-r+< (energy equation) and the diffusive flux Jk... 

The possible development of gradients in the components of the interfacial stress tensor due to flow of an adjacent fluid implies that the momentum flux caused by the the flow of liquid at one side of the interface does not have to be completely transported across the interface to the second fluid but may (partly or completely) be compensated in the interface. The extent to which this is possible depends on the rheological properties of the interface. For small shear stresses the interface may behave elastically or viscoelastically. For an elastic interfacial layer the structure remains coherent the layer will only deform, while for a viscoelastic one it may or may not start to flow. The latter case has been observed for elastic networks (e.g. for proteins) that remciln intact, but inside the meshes of which liquid can flow leading to energy dissipation. At large stresses the structure may yield or fracture (collapse), leading to an increased flow. 

The expression —gwxw y (SI units N/m2) is an averaged momentum flow per unit area, and so comparable to a shear stress A force in the direction of the y-axis acts at a surface perpendicular to the a -axis. Terms of the general form —gwf-wij are called Reynolds stresses or turbulent stresses. They are symmetrical tensors. In a corresponding manner, the energy equation (3.135), contains a turbulent heat flux of the form... 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

By decomposing the work term associated with the stress tensor, and then using the momentum equation, the Newtonian constitutive relation, and Fourier heat law, along with some manipulation, we obtain the following equation for the rate of change of internal energy (see, for example, Howarth 1953) ... 

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