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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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Energy condition

In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of space cannot be negative" in a relativistically-phrased mathematical formulation. There are multiple possible alternative ways to express such a condition such that can be applied to the matter content of the theory. The hope is then that any reasonable matter theory will satisfy this condition or at least will preserve the condition if it is satisfied by the starting conditions.

Exact solutions in general relativity

In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field.

Kerr metric

The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find. The Kerr metric is a generalization to a rotating body of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically symmetric, and non-rotating body.

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