dust solution

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.

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

:$T^\{\backslash mu\backslash nu\}\; =\; \backslash rho\_0\; U^\backslash mu\; U^\backslash nu.$

Here, the world lines of the dust particles are the integral curves of the four-velocity $U^\backslash mu$ and the matter density in dust's rest frame is given by the scalar function $\backslash rho\_0$.

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

:$\backslash chi(\backslash lambda)\; =\; \backslash lambda^4\; +\; a\_3\; \backslash ,\; \backslash lambda^3\; +\; a\_2\; \backslash ,\; \backslash lambda^2\; +\; a\_1\; \backslash ,\; \backslash lambda\; +\; a\_0$

of the Einstein tensor in a dust solution will have the form

:$\backslash chi(\backslash lambda)\; =\; \backslash left(\; \backslash lambda\; -\; 8\; \backslash pi\; \backslash mu\; \backslash right)\; \backslash ,\; \backslash lambda^3$

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

:$a\_0\; \backslash ,\; =\; a\_1\; =\; a\_2\; =\; 0$

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:

:$t\_2\; =\; t\_1^2,\; \backslash ;\; \backslash ;\; t\_3\; =\; t\_1^3,\; \backslash ;\; \backslash ;\; t\_4\; =\; t\_1^4$

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

:$\{G^a\}\_a\; =\; -R$

:$\{G^a\}\_b\; \backslash ,\; \{G^b\}\_a\; =\; R^2$

:$\{G^a\}\_b\; \backslash ,\; \{G^b\}\_c\; \backslash ,\; \{G^c\}\_a\; =\; -R^3$

:$\{G^a\}\_b\; \backslash ,\; \{G^b\}\_c\; \backslash ,\; \{G^c\}\_d\; \backslash ,\; \{G^d\}\_a\; =\; R^4$

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.

{{main|Null dust solution}}

A null dust solution is a dust solution where the Einstein tensor is null.{{elucidate|date=May 2017}}

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A Bianchi dust models exhibits various{{which|date=May 2017}} types of Lie algebras of Killing vector fields.

Special cases include FLRW and Kasner dust.{{elucidate|date=May 2017}}

A Kasner dust is the simplest{{whom|date=May 2017}} cosmological model exhibiting anisotropic expansion.{{elucidate|date=May 2017}}

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

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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{{whom|date=May 2017}}, the most remarkable exact solution discovered since the Kerr vacuum.

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

• Lorentz group

• {{Citation

| last=Schutz

| first=Bernard F.

| chapter=4. Perfect fluids in special relativity

| title=A first course in general relativity

| publisher=Cambridge University Press

| edition=2

| year=2009

| isbn=978-0-521-88705-2

| url-access=registration

| url=https://archive.org/details/firstcourseingen00bern_0

}}

• {{cite book |author1=Stephani, H. |author2=Kramer, D. |author3=MacCallum, M. |author4=Hoenselaers, C. |author5=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 dust solutions.

Category:Exact solutions in general relativity