# Dust solution

General relativity :
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$ :
Introduction
HistoryTimelineTests
Mathematical formulation :
Fundamental concepts
Equivalence principle
Special relativity
World line
Pseudo-Riemannian manifold :
Phenomena
Kepler problem
Gravitational lensing
Gravitational redshift
Gravitational time dilation
Gravitational waves
Frame-dragging
Geodetic effect
Event horizon
Singularity
Black hole
Spacetime
Spacetime diagrams
Minkowski spacetime
Einstein–Rosen bridge :
EquationsFormalisms
Equations
Linearized gravity
Einstein field equations
Friedmann
Geodesics
Mathisson–Papapetrou–Dixon
Hamilton–Jacobi–Einstein
Formalisms
ADM
BSSN
Post-Newtonian
Advanced theory
Kaluza–Klein theory
Quantum gravity :
Solutions
Schwarzschild (interior)
Reissner–Nordström
Einstein–Rosen waves
Wormhole
Gödel
Kerr
Kerr–Newman
Kerr–Newman–de Sitter
Kasner
Lemaître–Tolman
Taub–NUT
Milne
Robertson–Walker
Oppenheimer–Snyder
pp-wave
van Stockum dust
Weyl−Lewis−Papapetrou
Hartle–Thorne :
Scientists
Einstein
Lorentz
Hilbert
Poincaré
Schwarzschild
de Sitter
Reissner
Nordström
Weyl
Eddington
Friedmann
Milne
Zwicky
Lemaître
Oppenheimer
Gödel
Wheeler
Robertson
Bardeen
Walker
Kerr
Chandrasekhar
Ehlers
Penrose
Hawking
Raychaudhuri
Taylor
Hulse
van Stockum
Taub
Newman
Yau
Thorne
others :
Physics portal
 Category :
vte :
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 that has positive mass density but vanishing pressure. Dust solutions are an important special case of fluid solutions in general relativity.

## Dust model

A perfect and pressureless fluid can be interpreted as a model of a configuration of dust particles that locally move in concert and interact with each other only gravitationally, from which the name is derived. 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 models 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 that model 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
    $T^{\mu \nu }=\rho _{0}U^{\mu }U^{\nu }.$
Here, the world lines of the dust particles are the integral curves of the four-velocity $U^{\mu }$ and the matter density in dust's rest frame is given by the scalar function $\rho _{0}$.

## Eigenvalues

Because the stress-energy tensor is a rank-one matrix, a short computation shows that the characteristic polynomial
    $\chi (\lambda )=\lambda ^{4}+a_{3}\,\lambda ^{3}+a_{2}\,\lambda ^{2}+a_{1}\,\lambda +a_{0}$
of the Einstein tensor in a dust solution will have the form
    $\chi (\lambda )=\left(\lambda -8\pi \mu \right)\,\lambda ^{3}$
Multiplying out this product, we find that the coefficients must satisfy the following three algebraically independent (and invariant) conditions:
    $a_{0}\,=a_{1}=a_{2}=0$
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:
    $t_{2}=t_{1}^{2},\;\;t_{3}=t_{1}^{3},\;\;t_{4}=t_{1}^{4}$
In tensor index notation, this can be written using the Ricci scalar as:
    ${G^{a}}_{a}=-R$
    ${G^{a}}_{b}\,{G^{b}}_{a}=R^{2}$
    ${G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{a}=-R^{3}$
    ${G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}=R^{4}$
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.

### Bianchi dust

A Bianchi dust models exhibits various types of Lie algebras of Killing vector fields.
Special cases include FLRW and Kasner dust.

#### Kasner dust

A Kasner dust is the simplest cosmological model exhibiting anisotropic expansion.

#### FLRW dust

Friedmann–Lemaître–Robertson–Walker (FLRW) dusts are homogeneous and isotropic. These solutions often referred to as the matter-dominated FLRW models.

### Rotating dust

The van Stockum dust is a cylindrically symmetric rotating dust.
The Neugebauer–Meinel dust models a rotating disk of dust matched to an axisymmetric vacuum exterior. This solution has been called, the most remarkable exact solution discovered since the Kerr vacuum.

### Other solutions

Noteworthy individual dust solutions include:
* Lemaître–Tolman–Bondi (LTB) dusts (some of the simplest inhomogeneous cosmological models, often employed as models of gravitational collapse)
* Kantowski–Sachs dusts (cosmological models which exhibit perturbations from FLRW models)
* Gödel metric