Dust solution

This article includes a improve this article by introducing more precise citations. (May 2016) (Learn how and when to remove this template message )

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational red shift Gravitational time dilation Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms 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 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
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedman 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
v t e

In

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

${\displaystyle T^{\mu \nu }=\rho _{0}U^{\mu }U^{\nu }.}$

Here, the world lines of the dust particles are the integral curves of the four-velocity ${\displaystyle U^{\mu }}$ and the matter density in dust's rest frame is given by the scalar function ${\displaystyle \rho _{0}}$.

Eigenvalues

Because the stress-energy tensor is a rank-one matrix, a short computation shows that the characteristic polynomial

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

of the Einstein tensor in a dust solution will 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

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

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

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

${\displaystyle {G^{a}}_{b}\,{G^{b}}_{a}=R^{2}}$

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

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

This eigenvalue criterion is sometimes useful in searching for dust solutions, since it shows that very few

Examples

Null dust solution

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

This section needs expansion. You can help by adding to it. (May 2017)

Bianchi dust

A

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

Kasner dust

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

This section needs expansion. You can help by adding to it. (May 2017)

FLRW dust

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

This section needs expansion. You can help by adding to it. (May 2017)

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^{[according to whom?]}, 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

See also

• Lorentz group

References

• Schutz, Bernard F. (2009), "4. Perfect fluids in special relativity", A first course in general relativity (2 ed.), Cambridge University Press,
  ISBN 978-0-521-88705-2
• 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 dust solutions.