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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.
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.
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.
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 ...
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.
Equations of state are also used for the modeling of the state of matter in the interior of stars, including neutron stars, dense matter (quark–gluon plasmas) and radiation fields. A related concept is the perfect fluid equation of state used in cosmology.
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.
These equations specify the evolution of the scale factor the universe in terms of the pressure and density of a perfect fluid. The evolving density is composed of different kinds of energy and matter, each with its own role in affecting the scale factor. [4]: 7 For example, a model might include baryons, photons, neutrinos, and dark matter.