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${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$
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Ingeneral relativity, afluid solution is anexact solution of theEinstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of afluid.

Inastrophysics, fluid solutions are often employed asstellar models, since a perfect gas can be thought of as a special case of a perfect fluid. Incosmology, fluid solutions are often used ascosmological models.

Thestress–energy tensor of a relativistic fluid can be written in the form^[1]

${\displaystyle T^{\mu \nu }=\rho u^{\mu }{u}^{\nu }+p{h}^{\mu \nu }+\left(u^{\mu }{q}^{\nu }+q^{\mu }{u}^{\nu }\right)+{\pi }^{\mu \nu }}$

Here

• the world lines of the fluid elements are the integral curves of thevelocity vector${\displaystyle u^{\mu }}$,
• theprojection tensor${\displaystyle h_{\mu \nu }=g_{\mu \nu }+u_{\mu }{u}_{\nu }}$ projects other tensors onto hyperplane elements orthogonal to${\displaystyle u^{\mu }}$,
• thematter density is given by the scalar function${\displaystyle \rho }$,
• thepressure is given by the scalar function${\displaystyle p}$,
• theheat flux vector is given by${\displaystyle q^{\mu }}$,
• theviscous shear tensor is given by${\displaystyle {\pi }^{\mu \nu }}$.

The heat flux vector and viscous shear tensor aretransverse to the world lines, in the sense that

${\displaystyle q_{\mu }{u}^{\mu }=0,{\pi }_{\mu \nu }{u}^{\nu }=0}$

This means that they are effectively three-dimensional quantities, and since the viscous stress tensor issymmetric andtraceless, they have respectively three and fivelinearly 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 lightc = 1 and the metric sign convention used is${\displaystyle g_{\mu \nu }=\mathrm{diag}(-1,1,1,1)}$):

• Aperfect fluid has vanishing viscous shear and vanishing heat flux:

${\displaystyle T^{\mu \nu }=(\rho +p)u^{\mu }{u}^{\nu }+p{g}^{\mu \nu },}$

• Adust is a pressureless perfect fluid:

${\displaystyle T^{\mu \nu }=\rho u^{\mu }{u}^{\nu },}$

• Aradiation fluid is a perfect fluid with${\displaystyle \rho =3p}$:

${\displaystyle T^{\mu \nu }=p\left(4u^{\mu }{u}^{\nu }+g^{\mu \nu }\right).}$

The last two are often used as cosmological models for (respectively)matter-dominated andradiation-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 thestatic spherically symmetric perfect fluid solutions. These can always be matched to aSchwarzschild vacuum across a spherical surface, so they can be used asinterior solutions in a stellar model. In such models, the sphere${\displaystyle r=r[0]}$ 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${\displaystyle r_0}$. 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 obtainingall these solutions.

The components of a tensor computed with respect to aframe field rather than the coordinate basis are often calledphysical components, because these are the components which can (in principle) be measured by an observer.

In the special case of aperfect fluid, anadapted frame

${\displaystyle {\overrightarrow{e}}_0,{\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3}$

(the first is atimelike unitvector field, the last three arespacelike unit vector fields)can always be found in which the Einstein tensor takes the simple form

where${\displaystyle \rho }$ is theenergy density and${\displaystyle p}$ is thepressure of the fluid. Here, the timelike unit vector field${\displaystyle {\overrightarrow{e}}_0}$ 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${\displaystyle \overrightarrow{u}={\overrightarrow{e}}_0}$. From the form of the physical components, it is easy to see that theisotropy 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 flatMinkowski spacetime is an expression of theequivalence principle.

Thecharacteristic polynomial of the Einstein tensor in a perfect fluid must have the form

${\displaystyle \chi (\lambda )=\left(\lambda -8\pi \rho \right){\left(\lambda -8\pi p\right)}^3}$

where${\displaystyle \rho ,p}$ are again the density and pressure of the fluid as measured by observers comoving with the fluid elements. (Notice that these quantities canvary within the fluid.) Writing this out and applyingGröbner basis methods to simplify the resulting algebraic relations, we find that the coefficients of the characteristic must satisfy the following twoalgebraically independent (and invariant) conditions:

${\displaystyle 12a_4+a_2^2-3a_1{a}_3=0}$

${\displaystyle a_1{a}_2{a}_3-9a_3^2-9a_1^2{a}_4+32a_2{a}_4=0}$

But according toNewton's identities, the traces of the powers of the Einstein tensor are related to these coefficients as follows:

${\displaystyle {G^a}_a=t_1=a_1}$

${\displaystyle {G^a}_b{G^b}_a=t_2=a_1^2-2a_2}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_a=t_3=a_1^3-3a_1{a}_2+3a_3}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_d{G^d}_a=t_4=a_1^4-4a_1^2{a}_2+4a_1{a}_3+2a_2^2-a_4}$

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:

${\displaystyle t_2^3+4t_3^2+t_1^2{t}_4-4t_2{t}_4-2t_1{t}_2{t}_3=0}$

${\displaystyle t_1^4+7t_2^2-8t_1^2{t}_2+12t_1{t}_3-12t_4=0}$

Notice that this assumes nothing about any possibleequation of state 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:

${\displaystyle a_2=a_3=a_4=0}$

or

${\displaystyle t_2=t_1^2,t_3=t_1^3,t_4=t_1^4}$

In tensor gymnastics notation, this can be written using theRicci scalar as:

${\displaystyle {G^a}_a=-R}$

${\displaystyle {G^a}_b{G^b}_a=R^2}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_a=-R^3}$

${\displaystyle {G^a}_b{G^b}_c{G^c}_d{G^d}_a=-R^4}$

In the case of a radiation fluid, the criteria become

${\displaystyle a_1=0,27a_3^2+8a_2^3=0,12a_4+a_2^2=0}$

or

${\displaystyle t_1=0,7t_3^2-t_2{t}_4=0,12t_4-7t_2^2=0}$

In using these criteria, one must be careful to ensure that the largest eigenvalue belongs to atimelike eigenvector, since there areLorentzian manifolds, satisfying this eigenvalue criterion, in which the large eigenvalue belongs to aspacelike 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.

Noteworthy individual dust solutions are listed in the article ondust solutions. Noteworthy perfect fluid solutions which feature positive pressure include various radiation fluid models from cosmology, including

• FRW radiation fluids, often referred to as theradiation-dominated FRW models.

In addition to the family of static spherically symmetric perfect fluids, noteworthy rotating fluid solutions include

• Wahlquist fluid, which has similar symmetries to theKerr vacuum, leading to initial hopes (since dashed) that it might provide the interior solution for a simple model of a rotating star.

• Dust solution, for the important special case of dust solutions,
• Exact solutions in general relativity, for exact solutions in general,
• Lorentz group
• Perfect fluid, for perfect fluids in physics in general,
• Relativistic disks, for the interpretation of relativistic disks in terms of perfect fluids.

• Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003).Exact Solutions of Einstein's Field Equations (2nd edn.). Cambridge:Cambridge University Press.ISBN 0-521-46136-7. Gives many examples of exact perfect fluid and dust solutions.
• Stephani, Hans (1996).General relativity (second ed.). Cambridge: Cambridge University Press.ISBN 0-521-37941-5.. See Chapter 8 for a discussion of relativistic fluids and thermodynamics.
• Delgaty, M. S. R.; Lake, Kayll (1998). "Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein's Equations".Comput. Phys. Commun.115 (2–3):395–415.arXiv:gr-qc/9809013.Bibcode:1998CoPhC.115..395D.doi:10.1016/S0010-4655(98)00130-1.S2CID 17957408.. This review article surveys static spherically symmetric fluid solutions known up to about 1995.
• Lake, Kayll (2003). "All static spherically symmetric perfect fluid solutions of Einstein's Equations".Phys. Rev. D.67 (10) 104015.arXiv:gr-qc/0209104.Bibcode:2003PhRvD..67j4015L.doi:10.1103/PhysRevD.67.104015.S2CID 119447644.. This article describes one of several schemes recently found for obtaining all the static spherically symmetric perfect fluid solutions in general relativity.

• Exact solutions in general relativity

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