and sufficient for a Lorentzian manifold to admit an interpretation in general relativity as a non-null electrovacuum. These comprise three algebraic conditions and one differential condition. The conditions are sometimes useful for checking that a putative non-null electrovacuum really is what it claims, or even for finding such solutions. Analogous necessary and sufficient conditions for a null electrovacuum have been found by Charles Torre. Test fields Sometimes one can assume that the field energy of any electromagnetic field is so small that its gravitational effects can be neglected. Then, to obtain an approximate electrovacuum solution, we need only solve the Maxwell equations on a given vacuum solution. In this case, the electromagnetic field is often called a test field, in analogy with the term test particle (denoting a small object whose mass is too small to contribute appreciably to the ambient gravitational field). Here, it is useful to know that any Killing vectors which may be present will (in the case of a vacuum solution) automatically satisfy the curved spacetime Maxwell equations. Note that this procedure amounts to assuming that the electromagnetic field, but not the gravitational field, is "weak". Sometimes we can go even further; if the gravitational field is also considered "weak", we can independently solve the linearised Einstein field equations and the (flat spacetime) Maxwell equations on a Minkowksi vacuum background. Then the (weak) metric tensor gives the approximate geometry; the Minkowski background is unobservable by physical means, but mathematically much simpler to work with, whenever we can get away with such a sleight-of-hand. Examples Noteworthy individual non-null electrovacuum solutions include: Reissner–Nordström electrovacuum (which describes the geometry around a charged spherical mass), Kerr–Newman electrovacuum (which describes the geometry around a charged, rotating object), Melvin electrovacuum (a model of a cylindrically symmetric magnetostatic field), Garfinkle–Melvin electrovacuum (like the preceding, but including a gravitational wave traveling along the axis of symmetry), Bertotti–Robinson electrovacuum: this is a simple spacetime having a remarkable product structure; it arises from a kind of "blow up" of the horizon of the Reissner–Nordström electrovacuum, Witten electrovacuums (discovered by Louis Witten, father of Edward Witten). Noteworthy individual null electrovacuum solutions include: the monochromatic electromagnetic plane wave, an exact solution which is the general relativitistic analogue of the plane waves in classical electromagnetism, Bell–Szekeres electrovacuum (a colliding plane wave model). Some well known families of electrovacuums are: Weyl–Maxwell electrovacuums: this is the family of all static axisymmetric