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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
For a perfect fluid in thermodynamic equilibrium, the stress–energy tensor takes on a particularly simple form = (+) + where is the mass–energy density (kilograms per cubic meter), is the hydrostatic pressure , is the fluid's four-velocity, and is the matrix inverse of the metric tensor.
The stress–energy tensor of a relativistic pressureless fluid can be written in the simple form =. Here, the world lines of the dust particles are the integral curves of the four-velocity and the matter density in dust's rest frame is given by the scalar function .
In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid. In astrophysics , fluid solutions are often employed as stellar models , since a perfect gas can be thought of as a special case of a perfect fluid.
One can fix the form of the stress–energy tensor (from some physical reasons, say) and study the solutions of the Einstein equations with such right hand side (for example, if the stress–energy tensor is chosen to be that of the perfect fluid, a spherically symmetric solution can serve as a stellar model)
Inserting this metric into Einstein's field equations relate the evolution of this scale factor to the pressure and energy of the matter in the universe. With the stress–energy tensor for a perfect fluid, results in the equations are described below. [4]: 73
is the Cauchy stress tensor. is the flow velocity. and is the heat flux vector. Because it expresses conservation of total energy, this is sometimes referred to as the energy balance equation of continuous media.
In particular, the averaged null energy condition is satisfied in the Casimir effect. Indeed, for energy–momentum tensors arising from effective field theories on Minkowski spacetime, the averaged null energy condition holds for everyday quantum fields. Extending these results is an open problem.
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