Dust solution

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In general relativity, a dust solution is a fluid solution, a type of exact solution of the Einstein field equation, in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid which has positive mass density but vanishing pressure. Dust solutions are the most important[according to whom?] special case of fluid solutions in general relativity.

Dust model

The pressureless perfect fluid in a dust solution can be interpreted as a model of a configuration of dust particles which interact with each other only gravitationally. For this reason, dust models are often employed in cosmology as models of a toy universe, in which the dust particles are considered as highly idealized models of galaxies, clusters, or superclusters. In astrophysics, dust solutions have been employed as models of gravitational collapse. Dust solutions can also be used to model finite rotating disks of dust grains; some examples are listed below. If superimposed somehow on a stellar model comprising a ball of fluid surrounded by vacuum, a dust solution could be used to model an accretion disk around a massive object; however, no such exact solutions modeling rotating accretion disks are yet known due to the extreme mathematical difficulty of constructing them.

Mathematical definition

The stress–energy tensor of a relativistic pressureless fluid can be written in the simple form

${\displaystyle T^{\mu \nu }=\rho U^{\mu }U^{\nu }}$

Here

• the world lines of the dust particles are the integral curves of the four-velocity ${\displaystyle U^{\mu }}$,
• the matter density is given by the scalar function ${\displaystyle \rho }$.

Eigenvalues

${\displaystyle \chi (\lambda )=\lambda ^{4}+a_{3}\,\lambda ^{3}+a_{2}\,\lambda ^{2}+a_{1}\,\lambda +a_{0}}$

of the Einstein tensor in a dust solution must[why?] have the form

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

Multiplying out this product, we find that the coefficients must satisfy the following three algebraically independent (and invariant) conditions:

${\displaystyle a_{0}\,=a_{1}=a_{2}=0}$

Using Newton's identities, in terms of the sums of the powers of the roots (eigenvalues), which are also the traces of the powers of the Einstein tensor itself, these conditions become:

${\displaystyle t_{2}=t_{1}^{2},\;\;t_{3}=t_{1}^{3},\;\;t_{4}=t_{1}^{4}}$

In index notation, this can be written using the Ricci scalar as:

${\displaystyle {G^{a}}_{a}=-R}$
${\displaystyle {G^{a}}_{b}\,{G^{b}}_{a}=R^{2}}$
${\displaystyle {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{a}=-R^{3}}$
${\displaystyle {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}=R^{4}}$

This eigenvalue criterion is sometimes useful in searching for dust solutions, since it shows that very few Lorentzian manifolds could possibly admit an interpretation, in general relativity, as a dust solution.

Examples

Null dust solution

A null dust solution is a dust solution where the Einstein tensor is null.[further explanation needed]

Bianchi dust

A Bianchi dust models exhibits various[which?] types of Lie algebras of Killing vector fields.

Special cases include FLRW and Kasner dust.[further explanation needed]

Kasner dust

A Kasner dusts is the simplest[according to whom?] cosmological model exhibiting anisotropic expansion.[further explanation needed]

FLRW dust

Friedmann–Lemaître–Robertson–Walker (FLRW) dusts arehomogeneous[clarification needed] and isotropic[clarification needed]. These solutions often referred to as the matter-dominated FLRW models.[why?]

Rotating dust

The van Stockum dust is a cylindrically symmetric rotating dust.[further explanation needed]

The Neugebauer–Meinel dust models a rotating disk of dust matched to an axisymmetric vacuum exterior.[further explanation needed] This solution has been called[according to whom?], the most remarkable exact solution discovered since the Kerr vacuum.

Other solutions

Noteworthy individual dust solutions include: