Dust solution

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 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 tensor 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.

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