In general relativity, the geometric setting for physical phenomena is a Lorentzian manifold, which is interpreted as a curved spacetime, and which is specified by defining a metric tensor
In general relativity, they can be interpreted as geometric manifestations (curvature and forces) of the gravitational field.
To be classified as an electrovacuum solution, these two tensors are required to satisfy two following conditions The first Maxwell equation is satisfied automatically if we define the field tensor in terms of an electromagnetic potential vector
In terms of the dual covector (or potential one-form) and the electromagnetic two-form, we can do this by setting
Then we need only ensure that the divergences vanish (i.e. that the second Maxwell equation is satisfied for a source-free field) and that the electromagnetic stress–energy matches the Einstein tensor.
In the case of an electrovacuum solution, an adapted frame can always be found in which the Einstein tensor has a particularly simple appearance.
For a non-null electrovacuum, an adapted frame can be found in which the Einstein tensor takes the form where
From this expression, it is easy to see that the isotropy group of our non-null electrovacuum is generated by boosts in the
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.
In 1925, George Yuri Rainich presented purely mathematical conditions which are both necessary and sufficient for a Lorentzian manifold to admit an interpretation in general relativity as a non-null electrovacuum.
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.
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.
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.