Dust solution

In general relativity, a dust solution is an 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 by far the most important special case of fluid solutions in general relativity.

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

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

Here

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

Eigenvalues

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 must 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 gymnastics 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

Noteworthy individual dust solutions include:

• Friedmann–Lemaître–Robertson–Walker (FLRW) dusts (homogeneous and isotropic solutions often referred to as the matter-dominated FLRW models)
• Kasner dusts (the simplest cosmological model exhibiting anisotropic expansion)
• Bianchi dust models (generalizations of FLRW and Kasner models, exhibiting various types of Lie algebras of Killing vector fields)
• 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)
• van Stockum dust (a cylindrically symmetric rotating dust)
• the Neugebauer-Meinel dust (which models a rotating disk of dust matched to an axisymmetric vacuum exterior; this solution has been called, with some justice, the most remarkable exact solution discovered since the Kerr vacuum)
• Gödel metric

See also

• Exact solutions in general relativity, for exact solutions in general,
• Fluid solution, for perfect fluid solutions in general relativity (a generalization of dust solutions),
• 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 0-521-88705-4
• 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.