Fluid Solution - Eigenvalues

Eigenvalues

The characteristic polynomial of the Einstein tensor in a perfect fluid must have the form

where are again the density and pressure of the fluid as measured by observers comoving with the fluid elements. (Notice that these quantities can vary within the fluid.) Writing this out and applying Gröbner basis methods to simplify the resulting algebraic relations, we find that the coefficients of the characteristic must satisfy the following two algebraically independent (and invariant) conditions:

But according to Newton's identities, the traces of the powers of the Einstein tensor are related to these coefficients as follows:

so we can rewrite the above two quantities entirely in terms of the traces of the powers. These are obviously scalar invariants, and they must vanish identically in the case of a perfect fluid solution:

Notice that this assumes nothing about any possible equation of state relating the pressure and density of the fluid; we assume only that we have one simple and one triple eigenvalue.

In the case of a dust solution (vanishing pressure), these conditions simplify considerably:

or

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

In the case of a radiation fluid, the criteria become

or

In using these criteria, one must be careful to ensure that the largest eigenvalue belongs to a timelike eigenvector, since there are Lorentzian manifolds, satisfying this eigenvalue criterion, in which the large eigenvalue belongs to a spacelike eigenvector, and these cannot represent radiation fluids.

The coefficients of the characteristic will often appear very complicated, and the traces are not much better; when looking for solutions it is almost always better to compute components of the Einstein tensor with respect to a suitably adapted frame and then to kill appropriate combinations of components directly. However, when no adapted frame is evident, these eigenvalue criteria can be sometimes be useful, especially when employed in conjunction with other considerations.

These criteria can often be useful for spot checking alleged perfect fluid solutions, in which case the coefficients of the characteristic are often much simpler than they would be for a simpler imperfect fluid.