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In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
, 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 physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shea ...
. In
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the he ...
, 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 Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosophe ...
, fluid solutions are often used as cosmological models.


Mathematical definition

The
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress t ...
of a relativistic fluid can be written in the form :T^ = \mu \, u^a \, u^b + p \, h^ + \left( u^a \, q^b + q^a \, u^b \right) + \pi^ Here * the world lines of the fluid elements are the integral curves of the velocity vector u^a, * the projection tensor h_ = g_ + u_a \, u_b projects other tensors onto hyperplane elements orthogonal to u^a, * the matter density is given by the scalar function \mu, * the pressure is given by the scalar function p, * the heat flux vector is given by q^a, * the viscous shear tensor is given by \pi^. The heat flux vector and viscous shear tensor are ''transverse'' to the world lines, in the sense that :q_a \, u^a = 0, \; \; \pi_ \, u^b = 0 This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
and traceless, they have respectively three and five
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ...
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.


Special cases

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: ::T^ = (\mu + p) \, u^a \, u^b + p \, g^, * A
dust Dust is made of fine particles of solid matter. On Earth, it generally consists of particles in the atmosphere that come from various sources such as soil lifted by wind (an aeolian process), volcanic eruptions, and pollution. Dust in ...
is a pressureless perfect fluid: ::T^ = \mu \, u^a \, u^b, * A
radiation fluid In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes: * '' electromagnetic radiation'', such as radio waves, microwaves, infrared ...
is a perfect fluid with \mu = 3p: ::T^ = p \, \left( 4 \, u^a \, u^b + \, g^ \right). 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. Among the perfect fluids other than dusts or radiation fluids, by far the most important special case is that of the
static spherically symmetric perfect fluid In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as ssspf) is a spacetime equipped with suitable tensor fields which models a static round ba ...
solutions. These can always be matched to a
Schwarzschild vacuum In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...
across a spherical surface, so they can be used as interior solutions in a stellar model. In such models, the sphere r = r /math> where the fluid interior is matched to the vacuum exterior is the surface of the star, and the pressure must vanish in the limit as the radius approaches r_0. However, the density can be nonzero in the limit from below, while of course it is zero in the limit from above. In recent years, several surprisingly simple schemes have been given for obtaining ''all'' these solutions.


Einstein tensor

The components of a tensor computed with respect to a
frame field A frame field in general relativity (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime ...
rather than the coordinate basis are often called ''physical components'', because these are the components which can (in principle) be measured by an observer. In the special case of a ''perfect fluid'', an ''adapted frame'' :\vec_0, \; \vec_1, \; \vec_2, \; \vec_3 (the first is a timelike unit vector field, the last three are spacelike unit vector fields) can always be found in which the Einstein tensor takes the simple form :G^ = 8 \pi \, \left \begin \mu &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end \right where \mu is the ''energy density'' and p is the ''pressure'' of the fluid. Here, the timelike unit vector field \vec_0 is everywhere tangent to the world lines of observers who are comoving with the fluid elements, so the density and pressure just mentioned are those measured by comoving observers. These are the same quantities which appear in the general coordinate basis expression given in the preceding section; to see this, just put \vec = \vec_0. From the form of the physical components, it is easy to see that the isotropy group of any perfect fluid is isomorphic to the three dimensional Lie group SO(3), the ordinary rotation group. The fact that these results are exactly the same for curved spacetimes as for hydrodynamics in flat
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
is an expression of the equivalence principle.


Eigenvalues

The
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
of the Einstein tensor in a perfect fluid must have the form : \chi(\lambda) = \left( \lambda - 8 \pi \mu \right) \, \left( \lambda-8 \pi p \right)^3 where \mu, \, p are again the density and pressure of the fluid as measured by observers comoving with the fluid elements. (Notice that these quantities can ''vary'' within the fluid.) Writing this out and applying
Gröbner basis In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Grö ...
methods to simplify the resulting algebraic relations, we find that the coefficients of the characteristic must satisfy the following two
algebraically independent In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically i ...
(and invariant) conditions: : 12 a_4 + a_2^2 - 3 a_1 a_3 = 0 : a_1 a_2 a_3 - 9 a_3^2 - 9 a_1^2 a_4 + 32 a_2 a_4 = 0 But according to Newton's identities, the traces of the powers of the Einstein tensor are related to these coefficients as follows: : _a = t_1 = a_1 : _b \, _a = t_2 = a_1^2 - 2 a_2 : _b \, _c \, _a = t_3 = a_1^3 - 3 a_1 a_2 + 3 a_3 : _b \, _c \, _d \, _a = t_4 = a_1^4 - 4 a_1^2 a_2 + 4 a_1 a_3 + 2 a_2^2 - a_4 so we can rewrite the above two quantities entirely in terms of the traces of the powers. These are obviously scalar invariants, and they must vanish identically in the case of a perfect fluid solution: : t_2^3 + 4 t_3^2 + t_1^2 t_4 - 4 t_2 t_4 - 2 t_1 t_2 t_3 = 0 : t_1^4 + 7 t_2^2- 8 t_1^2 t_2 + 12 t_1 t_3 - 12 t_4 = 0 Notice that this assumes nothing about any possible
equation of state In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
relating the pressure and density of the fluid; we assume only that we have one simple and one triple eigenvalue. In the case of a dust solution (vanishing pressure), these conditions simplify considerably: : a_2 \, = a_3 = a_4 = 0 or : t_2 = t_1^2, \; \; t_3 = t_1^3, \; \; t_4 = t_1^4 In tensor gymnastics notation, this can be written using the Ricci scalar as: : _a = -R : _b \, _a = R^2 : _b \, _c \, _a = -R^3 : _b \, _c \, _d \, _a = -R^4 In the case of a radiation fluid, the criteria become :a_1 = 0, \; 27 \, a_3^2 + 8 a_2^3 = 0, \; 12 \, a_4 + a_2^2 = 0 or :t_1 = 0, 7 \, t_3^2 - t_2 \, t_4 = 0, \; 12 \, t_4 - 7 \, t_2^2 = 0 In using these criteria, one must be careful to ensure that the largest eigenvalue belongs to a ''timelike'' eigenvector, since there are
Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
s, satisfying this eigenvalue criterion, in which the large eigenvalue belongs to a ''spacelike'' eigenvector, and these cannot represent radiation fluids. The coefficients of the characteristic will often appear very complicated, and the traces are not much better; when looking for solutions it is almost always better to compute components of the Einstein tensor with respect to a suitably adapted frame and then to kill appropriate combinations of components directly. However, when no adapted frame is evident, these eigenvalue criteria can be sometimes be useful, especially when employed in conjunction with other considerations. These criteria can often be useful for spot checking alleged perfect fluid solutions, in which case the coefficients of the characteristic are often much simpler than they would be for a simpler imperfect fluid.


Examples

Noteworthy individual dust solutions are listed in the article on dust solutions. Noteworthy perfect fluid solutions which feature positive pressure include various radiation fluid models from cosmology, including * FRW radiation fluids, often referred to as the radiation-dominated FRW models. In addition to the family of static spherically symmetric perfect fluids, noteworthy rotating fluid solutions include *
Wahlquist fluid In general relativity, the Wahlquist fluid is an exact rotating perfect fluid solution to Einstein's equation with equation of state corresponding to constant gravitational mass density. Introduction The Wahlquist fluid was first discovered by ...
, which has similar symmetries to the
Kerr vacuum 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 ge ...
, leading to initial hopes (since dashed) that it might provide the interior solution for a simple model of a rotating star.


See also

* Dust solution, for the important special case of dust solutions, *
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 sh ...
, for exact solutions in general, *
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch phy ...
* Perfect fluid, for perfect fluids in physics in general, *
Relativistic disk In general relativity, the relativistic disk expression refers to a class of ''axi-symmetric'' self-consistent solutions to Einstein's field equations corresponding to the gravitational field generated by axi-symmetric isolated sources. To find s ...
s, for the interpretation of relativistic disks in terms of perfect fluids.


References

* Gives many examples of exact perfect fluid and dust solutions. *. See Chapter 8 for a discussion of relativistic fluids and thermodynamics. *. This review article surveys static spherically symmetric fluid solutions known up to about 1995. *. This article describes one of several schemes recently found for obtaining all the static spherically symmetric perfect fluid solutions in general relativity. {{DEFAULTSORT:Fluid Solution Exact solutions in general relativity