components which can (in principle) be measured by an observer. In the case of an electrovacuum solution, an adapted frame can always be found in which the Einstein tensor has a particularly simple appearance. Here, the first vector is understood to be a timelike unit vector field; this is everywhere tangent to the world lines of the corresponding family of adapted observers, whose motion is "aligned" with the electromagnetic field. The last three are spacelike unit vector fields. For a non-null electrovacuum, an adapted frame can be found in which the Einstein tensor takes the form where is the energy density of the electromagnetic field, as measured by any adapted observer. From this expression, it is easy to see that the isotropy group of our non-null electrovacuum is generated by boosts in the direction and rotations about the axis. In other words, the isotropy group of any non-null electrovacuum is a two-dimensional abelian Lie group isomorphic to SO(1,1) x SO(2). For a null electrovacuum, an adapted frame can be found in which the Einstein tensor takes the form From this it is easy to see that the isotropy group of our null electrovacuum includes rotations about the axis; two further generators are the two parabolic Lorentz transformations aligned with the direction given in the article on the Lorentz group. In other words, the isotropy group of any null electrovacuum is a three-dimensional Lie group isomorphic to E(2), the isometry group of the euclidean plane. The fact that these results are exactly the same in curved spacetimes as for electrodynamics in flat Minkowski spacetime is one expression of the equivalence principle. Eigenvalues The characteristic polynomial of the Einstein tensor of a 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. Rainich conditions In 1925, George Yuri Rainich presented purely mathematical conditions which are both necessary