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Concept
Fluid solution
Summary
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. (It might help to think of a perfect gas as a special case of a perfect fluid.) In cosmology, fluid solutions are often used as cosmological models.
The stress–energy tensor of a relativistic fluid can be written in the form
Here
the world lines of the fluid elements are the integral curves of the velocity vector ,
the projection tensor projects other tensors onto hyperplane elements orthogonal to ,
the matter density is given by the scalar function ,
the pressure is given by the scalar function ,
the heat flux vector is given by ,
the viscous shear tensor is given by .
The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that
This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is symmetric and traceless, they have respectively three and five linearly independent components. Together with the density and pressure, this makes a total of 10 linearly independent components, which is the number of linearly independent components in a four-dimensional symmetric rank two tensor.
Several special cases of fluid solutions are noteworthy (here speed of light c = 1):
A perfect fluid has vanishing viscous shear and vanishing heat flux:
A dust is a pressureless perfect fluid:
A radiation fluid is a perfect fluid with :
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. It is much easier to find such solutions than it is to find a general fluid solution.
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