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.
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.
The stress–energy tensor of a relativistic pressureless fluid can be written in the simple form Here, the world lines of the dust particles are the integral curves of the four-velocity
and the matter density in dust's rest frame is given by the scalar function
Because the stress-energy tensor is a rank-one matrix, a short computation shows that the characteristic polynomial of the Einstein tensor in a dust solution will have the form Multiplying out this product, we find that the coefficients must satisfy the following three algebraically independent (and invariant) conditions: 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: In tensor index notation, this can be written using the Ricci scalar as: 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.