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Perfect fluids admit a Lagrangian formulation, which allows the techniques used in field theory, in particular, quantization, to be applied to fluids. Perfect fluids are used in general relativity to model idealized distributions of matter , such as the interior of a star or an isotropic universe.
A scalar field can be viewed as a sort of perfect fluid with equation of state = ˙ ˙ + (), where ˙ is the time-derivative of and () is the potential energy. A free ( V = 0 {\displaystyle V=0} ) scalar field has w = 1 {\displaystyle w=1} , and one with vanishing kinetic energy is equivalent to a cosmological constant: w = − 1 {\displaystyle ...
Let us assume a static, spherically symmetric perfect fluid. The metric components are similar to those for the Schwarzschild metric: [2] = = By the perfect fluid assumption, the stress-energy tensor is diagonal (in the central spherical coordinate system), with eigenvalues of energy density and pressure:
Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. From FLRW metric we compute Christoffel symbols, then the Ricci tensor. With the stress–energy tensor for a perfect fluid, we substitute them into Einstein's field equations and the resulting equations are described below.
Fluid solutions: must arise entirely from the stress–energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the energy, momentum, and stress (pressure and shear stress) of the matter comprising the fluid.
The last two are often used as cosmological models for (respectively) matter-dominated and radiation-dominated epochs. Notice that while in general it requires ten functions to specify a fluid, a perfect fluid requires only two, and dusts and radiation fluids each require only one function.
In general relativity, a dust solution is a fluid solution, a type of exact solution of the Einstein field equation, in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid that has positive mass density but vanishing pressure.
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.