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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]
The stress–energy tensor of a relativistic pressureless fluid can be written in the simple form T μ ν = ρ 0 U μ U ν . {\displaystyle T^{\mu \nu }=\rho _{0}U^{\mu }U^{\nu }.} Here, the world lines of the dust particles are the integral curves of the four-velocity U μ {\displaystyle U^{\mu }} and the matter density in dust's rest frame is ...
The material derivative is defined for any tensor field y that is macroscopic, with the sense that it depends only on position and time coordinates, y = y(x, t): +, where ∇y is the covariant derivative of the tensor, and u(x, t) is the flow velocity.
Just as a small increment of energy in a mechanical system is the product of a force times a small displacement, so an increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" which, when unbalanced, cause certain generalized "displacements" to occur, with their product being the energy transferred as a result.
Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context: Total rate of energy transfer (not per unit area); [1] SI units: W = J⋅s −1. Specific rate of energy transfer (total normalized per unit area); [2] SI units: W⋅m −2 = J⋅m −2 ⋅s −1:
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 ...
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.
In the latter case, the equation of state of the perfect fluid may be used in Friedmann–Lemaître–Robertson–Walker equations to describe the evolution of the universe. In general relativity, the expression for the stress–energy tensor of a perfect fluid is written as