Eigenvalues
The characteristic polynomial of the Einstein tensor of an non-null electrovacuum must have the form
Using Newton's identities, this condition can be re-expressed in terms of the traces of the powers of the Einstein tensor as
where
This necessary criterion can be useful for checking that a putative non-null electrovacuum solution is plausible, and is sometimes useful for finding non-null electrovacuum solutions.
The characteristic polynomial of a null electrovacuum vanishes identically, even if the energy density is nonzero. This possibility is a tensor analogue of the well known that a null vector always has vanishing length, even if it is not the zero vector. Thus, every null electrovacuum has one quadruple eigenvalue, namely zero.
Read more about this topic: Electrovacuum Solution