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| In [[general relativity]], a '''fluid solution''' is an [[exact solutions in general relativity|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]].
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| 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 [[physical cosmology|cosmology]], fluid solutions are often used as [[cosmological model]]s.
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| ==Mathematical definition==
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| The [[stress-energy tensor]] of a relativistic fluid can be written in the form
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| :<math>T^{ab} = \mu \, u^a \, u^b + p \, h^{ab} + \left( u^a \, q^b + q^a \, u^b \right) + \pi^{ab}</math>
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| Here
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| * the world lines of the fluid elements are the integral curves of the [[four-velocity|velocity vector]] <math>u^a</math>,
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| * the '''projection tensor''' <math>h_{ab} = g_{ab} + u_a \, u_b</math> projects other tensors onto hyperplane elements orthogonal to <math>u^a</math>,
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| * the '''matter density''' is given by the scalar function <math>\mu</math>,
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| * the '''pressure''' is given by the scalar function <math>p</math>,
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| * the '''heat flux vector''' is given by <math>q^a</math>,
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| * the '''viscous shear tensor''' is given by <math>\pi^{ab}</math>.
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| The heat flux vector and viscous shear tensor are ''transverse'' to the world lines, in the sense that
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| :<math>q_a \, u^a = 0, \; \; \pi_{ab} \, u^b = 0 </math>
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| This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is [[symmetric matrix|symmetric]] and [[traceless]], they have respectively 3 and 5 [[linear independence|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.
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| ==Special cases==
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| Several special cases of fluid solutions are noteworthy:
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| * A [[perfect fluid]] has vanishing viscous shear and vanishing heat flux:
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| :<math>T^{ab} = (\mu + p) \, u^a \, u^b + p \, g^{ab}</math>,
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| * A [[dust solution|dust]] is a pressureless perfect fluid:
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| :<math>T^{ab} = \mu \, u^a \, u^b </math>,
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| * A [[radiation fluid]] is a perfect fluid with <math>\mu = 3p</math>:
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| :<math>T^{ab} = p \, \left( 4 \, u^a \, u^b + \, g^{ab} \right)</math>
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| 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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| 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]] solutions. These can always be matched to a [[Schwarzschild metric|Schwarzschild vacuum]] across a spherical surface, so they can be used as '''interior solutions''' in a stellar model. In such models, the sphere <math>r = r[0]</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 <math>r_0</math>. 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.
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| ==Einstein tensor==
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| The components of a tensor computed with respect to a [[frame fields in general relativity|frame field]] rather than the coordinate basis are often called ''physical components'', because these are the components which can (in principle) be measured by an observer.
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| In the special case of a ''perfect fluid'', an ''adapted frame''
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| :<math>\vec{e}_0, \; \vec{e}_1, \; \vec{e}_2, \; \vec{e}_3</math>
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| (the first is a [[timelike]] unit [[vector field]], the last three are [[spacelike]] unit vector fields)
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| can always be found in which the Einstein tensor takes the simple form
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| :<math>G^{\hat{a}\hat{b}} = 8 \pi \, \left[ \begin{matrix} \mu &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix} \right] </math> | |
| where <math>\mu</math> is the ''density'' and <math>p</math> is the ''pressure'' of the fluid. Here, the timelike unit vector field <math>\vec{e}_0</math> 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 <math>\vec{u} = \vec{e}_0</math>. 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.
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| The fact that these results are exactly the same for curved spacetimes as for hydrodynamics in flat [[Minkowski spacetime]] is an expression of the [[equivalence principle]].
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| ==Eigenvalues==
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| The [[characteristic polynomial]] of the Einstein tensor in a perfect fluid must have the form
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| :<math> \chi(\lambda) = \left( \lambda - 8 \pi \mu \right) \, \left( \lambda-8 \pi p \right)^3 </math> | |
| where <math>\mu, \, p</math> 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]] methods to simplify the resulting algebraic relations, we find that the coefficients of the characteristic must satisfy the following two [[algebraically independent]] (and invariant) conditions:
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| :<math> 12 a_4 + a_2^2 - 3 a_1 a_3 = 0 </math>
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| :<math> a_1 a_2 a_3 - 9 a_3^2 - 9 a_1^2 a_4 + 32 a_2 a_4 = 0</math>
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| But according to [[Newton's identities]], the traces of the powers of the Einstein tensor are related to these coefficients as follows:
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| :<math> {G^a}_a = t_1 = a_1</math>
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| :<math> {G^a}_b \, {G^b}_a = t_2 = a_1^2 - 2 a_2</math>
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| :<math> {G^a}_b \, {G^b}_c \, {G^c}_a = t_3 = a_1^3 - 3 a_1 a_2 + 3 a_3</math>
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| :<math> {G^a}_b \, {G^b}_c \, {G^c}_d \, {G^d}_a = t_4 = a_1^4 - 4 a_1^2 a_2 + 4 a_1 a_3 + 2 a_2^2 - a_4</math>
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| 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:
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| :<math> 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 </math>
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| :<math> t_1^4 + 7 t_2^2- 8 t_1^2 t_2 + 12 t_1 t_3 - 12 t_4 = 0 </math>
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| Notice that this assumes nothing about any possible [[equation of state]] relating the pressure and density of the fluid; we assume only that we have one simple and one triple eigenvalue.
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| In the case of a dust solution (vanishing pressure), these conditions simplify considerably:
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| :<math> a_2 \, = a_3 = a_4 = 0 </math>
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| or
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| :<math> t_2 = t_1^2, \; \; t_3 = t_1^3, \; \; t_4 = t_1^4</math>
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| In tensor gymnastics notation, this can be written using the [[Ricci scalar]] as:
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| :<math> {G^a}_a = -R</math>
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| :<math> {G^a}_b \, {G^b}_a = R^2</math>
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| :<math> {G^a}_b \, {G^b}_c \, {G^c}_a = -R^3</math>
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| :<math> {G^a}_b \, {G^b}_c \, {G^c}_d \, {G^d}_a = -R^4</math>
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| In the case of a radiation fluid, the criteria become
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| :<math>a_1 = 0, \; 27 \, a_3^2 + 8 a_2^3 = 0, \; 12 \, a_4 + a_2^2 = 0</math>
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| or
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| :<math>t_1 = 0, 7 \, t_3^2 - t_2 \, t_4 = 0, \; 12 \, t_4 - 7 \, t_2^2 = 0</math>
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| In using these criteria, one must be careful to ensure that the largest eigenvalue belongs to a ''timelike'' eigenvector, since there are [[Lorentzian manifold]]s, satisfying this eigenvalue criterion, in which the large eigenvalue belongs to a ''spacelike'' eigenvector, and these cannot represent radiation fluids.
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| 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.
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| 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.
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| ==Examples==
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| Noteworthy individual dust solutions are listed in the article on [[dust solution]]s. Noteworthy perfect fluid solutions which feature positive pressure include various radiation fluid models from cosmology, including
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| *[[Friedmann-Lemaître-Robertson-Walker|FRW radiation fluids]], often referred to as the '''radiation-dominated''' FRW models.
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| In addition to the family of static spherically symmetric perfect fluids, noteworthy rotating fluid solutions include
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| *[[Wahlquist fluid]], which has similar symmetries to the [[Kerr vacuum]], leading to initial hopes (since dashed) that it might provide the interior solution for a simple model of a rotating star.
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| == See also ==
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| *[[Dust solution]], for the important special case of dust solutions,
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| *[[Exact solutions in general relativity]], for exact solutions in general,
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| *[[Lorentz group]]
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| *[[Perfect fluid]], for perfect fluids in physics in general,
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| *[[Relativistic disk]]s, for the interpretation of [[relativistic disk]]s in terms of perfect fluids.
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| == References ==
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| *{{cite book | author=Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E. | title=Exact Solutions of Einstein's Field Equations (2nd edn.) | location=Cambridge | publisher=Cambridge University Press | year=2003 | isbn=0-521-46136-7}} Gives many examples of exact perfect fluid and dust solutions.
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| *{{cite book | author=Stephani, Hans | title=General relativity (second edition) | location=Cambridge | publisher=Cambridge University Press | year=1996 | isbn=0-521-37941-5}}. See Chapter 8 for a discussion of relativistic fluids and thermodynamics.
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| *{{cite journal | author=Delgaty, M. S. R.; & Lake, Kayll | title=Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein's Equations | journal=Comput. Phys. Commun. | year=1998 | volume=115 | pages=395–415 | doi=10.1016/S0010-4655(98)00130-1 |arxiv=gr-qc/9809013|bibcode = 1998CoPhC.115..395D | issue=2–3 }}. This review article surveys static spherically symmetric fluid solutions known up to about 1995.
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| *{{cite journal | author=Lake, Kayll | title=All static spherically symmetric perfect fluid solutions of Einstein's Equations | journal= Phys. Rev. D | year=2003 | volume=67 | pages=104015 | doi=10.1103/PhysRevD.67.104015 |arxiv=gr-qc/0209104|bibcode = 2003PhRvD..67j4015L | issue=10 }}. This article describes one of several schemes recently found for obtaining all the static spherically symmetric perfect fluid solutions in general relativity.
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| {{DEFAULTSORT:Fluid Solution}}
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| [[Category:Exact solutions in general relativity]]
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