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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.
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.
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 .
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 equations of motion are contained in the continuity equation of the stress–energy tensor: =, where is the covariant derivative. [5] For a perfect fluid, = (+) +. Here is the total mass-energy density (including both rest mass and internal energy density) of the fluid, is the fluid pressure, is the four-velocity of the fluid, and is the metric tensor. [2]
But if one requires an exact solution or a solution describing strong fields, the evolution of both the metric and the stress–energy tensor must be solved for at once. To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine the evolution of the stress–energy tensor):
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.
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