Dust Solution - Eigenvalues

Eigenvalues

The characteristic polynomial

of the Einstein tensor in a dust solution must 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 gymnastics 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.